TM 663 Operations Planning September 12, 2011 Paula Jensen 2nd Session: Chapter 2: Inventory Control From EOQ to ROP

Agenda Chapter 2: Inventory Control From EOQ to ROP

Tentative Schedule Chapters Assigned 8/29/2011 0,1 9/5/2011 Holiday 8/29/2011 0,1 9/5/2011 Holiday 9/12/2011 2 9/19/2011 2, 3 9/26/2011 4, 5 10/3/2011 6, 7 10/10/2011 Holiday 10/17/2011 Exam 1 10/24/2011 8,9 10/31/2011 9,10 11/7/2011 11, 12 11/14/2011 13, 14 Chapters Assigned 11/21/2011 Exam 2 11/28/2011 15 p: 1-3 12/5/2011 16 p:1-4, 17 p1 12/13/2011 Final 18, 19 Not covered We may rearrange a bit. We could skip chapter 11 &12 and do 18 &19

Inventory Control There is a trade-off between set-ups and inventory. There is a trade-off between customer service and inventory There is a trade-off between variability and inventory

The EOQ Model To a pessimist, the glass is half empty. to an optimist, it is half full. – Anonymous

EOQ History Introduced in 1913 by Ford W. Harris, “How Many Parts to Make at Once” Interest on capital tied up in wages, material and overhead sets a maximum limit to the quantity of parts which can be profitably manufactured at one time; “set-up” costs on the job fix the minimum. Experience has shown one manager a way to determine the economical size of lots. Early application of mathematical modeling to Scientific Management

MedEquip Example Small manufacturer of medical diagnostic equipment. Purchases standard steel “racks” into which components are mounted. Metal working shop can produce (and sell) racks more cheaply if they are produced in batches due to wasted time setting up shop. MedEquip doesn’t want to tie up too much precious capital in inventory. Question: how many racks should MedEquip order at once?

EOQ Modeling Assumptions 1. Production is instantaneous – there is no capacity constraint and the entire lot is produced simultaneously. 2. Delivery is immediate – there is no time lag between production and availability to satisfy demand. 3. Demand is deterministic – there is no uncertainty about the quantity or timing of demand. 4. Demand is constant over time – in fact, it can be represented as a straight line, so that if annual demand is 365 units this translates into a daily demand of one unit. 5. A production run incurs a fixed setup cost – regardless of the size of the lot or the status of the factory, the setup cost is constant. 6. Products can be analyzed singly – either there is only a single product or conditions exist that ensure separability of products.

Notation D demand rate (units per year) c unit production cost, not counting setup or inventory costs (dollars per unit) A fixed or setup cost to place an order (dollars) h holding cost (dollars per year); if the holding cost is consists entirely of interest on money tied up in inventory, then h = ic where i is an annual interest rate. Q the unknown size of the order or lot size decision variable

Inventory vs Time in EOQ Model Q/D 2Q/D 3Q/D 4Q/D Time

Costs Holding Cost: Setup Costs: A per lot, so Production Cost: c per unit Cost Function:

MedEquip Example Costs D = 1000 racks per year c = $250 A = $500 (estimated from supplier’s pricing) h = (0.1)($250) + $10 = $35 per unit per year

Costs in EOQ Model

Economic Order Quantity EOQ Square Root Formula MedEquip Solution

EOQ Modeling Assumptions 1. Production is instantaneous – there is no capacity constraint and the entire lot is produced simultaneously. 2. Delivery is immediate – there is no time lag between production and availability to satisfy demand. 3. Demand is deterministic – there is no uncertainty about the quantity or timing of demand. 4. Demand is constant over time – in fact, it can be represented as a straight line, so that if annual demand is 365 units this translates into a daily demand of one unit. 5. A production run incurs a fixed setup cost – regardless of the size of the lot or the status of the factory, the setup cost is constant. 6. Products can be analyzed singly – either there is only a single product or conditions exist that ensure separability of products. relax via EPL model

Notation – EPL Model D demand rate (units per year) P production rate (units per year), where P>D c unit production cost, not counting setup or inventory costs (dollars per unit) A fixed or setup cost to place an order (dollars) h holding cost (dollars per year); if the holding cost is consists entirely of interest on money tied up in inventory, then h = ic where i is an annual interest rate. Q the unknown size of the production lot size decision variable

Inventory vs Time in EPL Model Production run of Q takes Q/P time units (P-D)(Q/P) -D P-D (P-D)(Q/P)/2 Inventory Time

Solution to EPL Model Annual Cost Function: Solution (by taking derivative and setting equal to zero): setup holding production tends to EOQ as P otherwise larger than EOQ because replenishment takes longer

The Key Insight of EOQ Order Frequency: Inventory Investment: There is a tradeoff between lot size and inventory Order Frequency: Inventory Investment:

EOQ Tradeoff Curve

Sensitivity of EOQ Model to Quantity Optimal Unit Cost: Optimal Annual Cost: Multiply Y* by D and simplify, We neglect unit cost, c, since it does not affect Q*

Sensitivity of EOQ Model to Quantity (cont.) Annual Cost from Using Q': Ratio: Example: If Q' = 2Q*, then the ratio of the actual to optimal cost is (1/2)[2 + (1/2)] = 1.25

Sensitivity of EOQ Model to Order Interval Order Interval: Let T represent time (in years) between orders (production runs) Optimal Order Interval:

Sensitivity of EOQ Model to Order Interval (cont.) Ratio of Actual to Optimal Costs: If we use T' instead of T* Powers-of-Two Order Intervals: The optimal order interval, T* must lie within a multiplicative factor of 2 of a “power-of-two.” Hence, the maximum error from using the best power-of-two is

The “Root-Two” Interval divide by less than 2 to get to 2m multiply by less than 2 to get to 2m+1

Medequip Example Optimum: Q*=169, so T*=Q*/D =169/1000 years = 62 days Round to Nearest Power-of-Two: 62 is between 32 and 64, but since 322=45.25, it is “closest” to 64. So, round to T’=64 days or Q’= T’D=(64/365)1000=175. Only 0.07% error because we were lucky and happened to be close to a power-of-two. But we can’t do worse than 6%.

Powers-of-Two Order Intervals Order Interval Week 0 1 2 3 4 5 6 7 8

EOQ Takeaways Batching causes inventory (i.e., larger lot sizes translate into more stock). Under specific modeling assumptions the lot size that optimally balances holding and setup costs is given by the square root formula: Total cost is relatively insensitive to lot size (so rounding for other reasons, like coordinating shipping, may be attractive).

The Wagner-Whitin Model Change is not made without inconvenience, even from worse to better. – Robert Hooker

EOQ Assumptions 1. Instantaneous production. 2. Immediate delivery. 3. Deterministic demand. 4. Constant demand. 5. Known fixed setup costs. 6. Single product or separable products. WW model relaxes this one

Dynamic Lot Sizing Notation t a period (e.g., day, week, month); we will consider t = 1, … ,T, where T represents the planning horizon. Dt demand in period t (in units) ct unit production cost (in dollars per unit), not counting setup or inventory costs in period t At fixed or setup cost (in dollars) to place an order in period t ht holding cost (in dollars) to carry a unit of inventory from period t to period t +1 Qt the unknown size of the order or lot size in period t decision variables

Wagner-Whitin Example Data Lot-for-Lot Solution

Wagner-Whitin Example (cont.) Fixed Order Quantity Solution

Wagner-Whitin Property Under an optimal lot-sizing policy either the inventory carried to period t+1 from a previous period will be zero or the production quantity in period t+1 will be zero.

Basic Idea of Wagner-Whitin Algorithm By WW Property I, either Qt=0 or Qt=D1+…+Dk for some k. If jk* = last period of production in a k period problem then we will produce exactly Dk+…DT in period jk*. We can then consider periods 1, … , jk*-1 as if they are an independent jk*-1 period problem.

Wagner-Whitin Example Step 1: Obviously, just satisfy D1 (note we are neglecting production cost, since it is fixed). Step 2: Two choices, either j2* = 1 or j2* = 2.

Wagner-Whitin Example (cont.) Step3: Three choices, j3* = 1, 2, 3.

Wagner-Whitin Example (cont.) Step 4: Four choices, j4* = 1, 2, 3, 4.

Planning Horizon Property If jt*=t, then the last period in which production occurs in an optimal t+1 period policy must be in the set t, t+1,…t+1. In the Example: We produce in period 4 for period 4 of a 4 period problem. We would never produce in period 3 for period 5 in a 5 period problem.

Wagner-Whitin Example (cont.) Step 5: Only two choices, j5* = 4, 5. Step 6: Three choices, j6* = 4, 5, 6. And so on.

Wagner-Whitin Example Solution Produce in period 1 for 1, 2, 3 (20 + 50 + 10 = 80 units) Produce in period 4 for 4, 5, 6, 7 (50 + 50 + 10 + 20 = 130 units) Produce in period 8 for 8, 9, 10 (40 + 20 + 30 = 90 units

Wagner-Whitin Example Solution (cont.) Optimal Policy: Produce in period 8 for 8, 9, 10 (40 + 20 + 30 = 90 units) Produce in period 4 for 4, 5, 6, 7 (50 + 50 + 10 + 20 = 130 units) Produce in period 1 for 1, 2, 3 (20 + 50 + 10 = 80 units) Note: we produce in 7 for an 8 period problem, but this never comes into play in optimal solution.

Problems with Wagner-Whitin 1. Fixed setup costs. 2. Deterministic demand and production (no uncertainty) 3. Never produce when there is inventory (WW Property I). safety stock (don't let inventory fall to zero) random yields (can't produce for exact no. periods)

Statistical Reorder Point Models When your pills get down to four, Order more. – Anonymous, from Hadley &Whitin

EOQ Assumptions EPL model relaxes this one 1. Instantaneous production. 2. Immediate delivery. 3. Deterministic demand. 4. Constant demand. 5. Known fixed setup costs. 6. Single product or separable products. lags can be added to EOQ or other models newsvendor and (Q,r) relax this one WW model relaxes this one can use constraint approach Chapter 17 extends (Q,r) to multiple product cases

Modeling Philosophies for Handling Uncertainty 1. Use deterministic model – adjust solution - EOQ to compute order quantity, then add safety stock - deterministic scheduling algorithm, then add safety lead time 2. Use stochastic model - news vendor model - base stock and (Q,r) models - variance constrained investment models

The Newsvendor Approach Assumptions: 1. single period 2. random demand with known distribution 3. linear overage/shortage costs 4. minimum expected cost criterion Examples: newspapers or other items with rapid obsolescence Christmas trees or other seasonal items capacity for short-life products

Newsvendor Model Notation

Newsvendor Model Cost Function: Note: for any given day, we will be either over or short, not both. But in expectation, overage and shortage can both be positive.

Newsvendor Model (cont.) Optimal Solution: taking derivative of Y(Q) with respect to Q, setting equal to zero, and solving yields: Notes: Critical Ratio is probability stock covers demand 1 G(x) Q*

Newsvendor Example – T Shirts Scenario: Demand for T-shirts is exponential with mean 1000 (i.e., G(x) = P(X  x) = 1- e-x/1000). (Note - this is an odd demand distribution; Poisson or Normal would probably be better modeling choices.) Cost of shirts is $10. Selling price is $15. Unsold shirts can be sold off at $8. Model Parameters: cs = 15 – 10 = $5 co = 10 – 8 = $2

Newsvendor Example – T Shirts (cont.) Solution: Sensitivity: If co = $10 (i.e., shirts must be discarded) then

Newsvendor Model with Normal Demand Suppose demand is normally distributed with mean  and standard deviation . Then the critical ratio formula reduces to: (z) z Note: Q* increases in both  and  if z is positive (i.e., if ratio is greater than 0.5).

Multiple Period Problems Difficulty: Technically, Newsvendor model is for a single period. Extensions: But Newsvendor model can be applied to multiple period situations, provided: demand during each period is iid, distributed according to G(x) there is no setup cost associated with placing an order stockouts are either lost or backordered Key: make sure co and cs appropriately represent overage and shortage cost.

Example Scenario: Problem: how should they set order amounts? GAP orders a particular clothing item every Friday mean weekly demand is 100, std dev is 25 wholesale cost is $10, retail is $25 holding cost has been set at $0.5 per week (to reflect obsolescence, damage, etc.) Problem: how should they set order amounts?

Example (cont.) Newsvendor Parameters: Solution: c0 = $0.5 cs = $15 Every Friday, they should order-up-to 146, that is, if there are x on hand, then order 146-x.

Newsvendor Takeaways Inventory is a hedge against demand uncertainty. Amount of protection depends on “overage” and “shortage” costs, as well as distribution of demand. If shortage cost exceeds overage cost, optimal order quantity generally increases in both the mean and standard deviation of demand.

The (Q,r) Approach Assumptions: Decision Variables: 1. Continuous review of inventory. 2. Demands occur one at a time. 3. Unfilled demand is backordered. 4. Replenishment lead times are fixed and known. Decision Variables: Reorder Point: r – affects likelihood of stockout (safety stock). Order Quantity: Q – affects order frequency (cycle inventory).

Inventory vs Time in (Q,r) Model

Base Stock Model Assumptions 1. There is no fixed cost associated with placing an order. 2. There is no constraint on the number of orders that can be placed per year. That is, we can replenish one at a time (Q=1).

Base Stock Notation l = delivery lead time Q = 1, order quantity (fixed at one) r = reorder point R = r +1, base stock level l = delivery lead time q = mean demand during l  = std dev of demand during l p(x) = Prob{demand during lead time l equals x} G(x) = Prob{demand during lead time l is less than x} h = unit holding cost b = unit backorder cost S(R) = average fill rate (service level) B(R) = average backorder level I(R) = average on-hand inventory level

Inventory Balance Equations inventory position = on-hand inventory - backorders + orders Under Base Stock Policy inventory position = R

Inventory Profile for Base Stock System (R=5)

Service Level (Fill Rate) Let: X = (random) demand during lead time l so E[X] = . Consider a specific replenishment order. Since inventory position is always R, the only way this item can stock out is if X  R. Expected Service Level:

Backorder Level Note: At any point in time, number of orders equals number demands during last l time units (X) so from our previous balance equation: R = on-hand inventory - backorders + orders on-hand inventory - backorders = R - X Note: on-hand inventory and backorders are never positive at the same time, so if X=x, then Expected Backorder Level: simpler version for spreadsheet computing

Inventory Level Observe: Result: on-hand inventory - backorders = R-X E[X] =  from data E[backorders] = B(R) from previous slide Result: I(R) = R -  + B(R)

Base Stock Example q = 10 units (per month) Assume Poisson demand, so l = one month q = 10 units (per month) Assume Poisson demand, so Note: Poisson demand is a good choice when no variability data is available.

Base Stock Example Calculations

Base Stock Example Results Service Level: For fill rate of 90%, we must set R-1= r =14, so R=15 and safety stock s = r- = 4. Resulting service is 91.7%. Backorder Level: B(R) = B(15) = 0.103 Inventory Level: I(R) = R -  + B(R) = 15 - 10 + 0.103 = 5.103

“Optimal” Base Stock Levels Objective Function: Y(R) = hI(R) + bB(R) = h(R-+B(R)) + bB(R) = h(R- ) + (h+b)B(R) Solution: if we assume G is continuous, we can use calculus to get holding plus backorder cost Implication: set base stock level so fill rate is b/(h+b). Note: R* increases in b and decreases in h.

Base Stock Normal Approximation If G is normal(,), then where (z)=b/(h+b). So R* =  + z  Note: R* increases in  and also increases in  provided z>0.

“Optimal” Base Stock Example Data: Approximate Poisson with mean 10 by normal with mean 10 units/month and standard deviation 10 = 3.16 units/month. Set h=$15, b=$25. Calculations: since (0.32) = 0.625, z=0.32 and hence R* =  + z = 10 + 0.32(3.16) = 11.01  11 Observation: from previous table fill rate is G(10) = 0.583, so maybe backorder cost is too low.

Inventory Pooling Situation: Specialized Inventory: n different parts with lead time demand normal(, ) z=2 for all parts (i.e., fill rate is around 97.5%) Specialized Inventory: base stock level for each item =  + 2  total safety stock = 2n  Pooled Inventory: suppose parts are substitutes for one another lead time demand is normal (n ,n ) base stock level (for same service) = n +2 n  ratio of safety stock to specialized safety stock = 1/ n cycle stock safety stock

Effect of Pooling on Safety Stock Conclusion: cycle stock is not affected by pooling, but safety stock falls dramatically. So, for systems with high safety stock, pooling (through product design, late customization, etc.) can be an attractive strategy.

Pooling Example PC’s consist of 6 components (CPU, HD, CD ROM, RAM, removable storage device, keyboard) 3 choices of each component: 36 = 729 different PC’s Each component costs $150 ($900 material cost per PC) Demand for all models is Poisson distributed with mean 100 per year Replenishment lead time is 3 months (0.25 years) Use base stock policy with fill rate of 99%

Pooling Example - Stock PC’s Base Stock Level for Each PC:  = 100  0.25 = 25, so using Poisson formulas, G(R-1)  0.99 R = 38 units On-Hand Inventory for Each PC: I(R) = R -  + B(R) = 38 - 25 + 0.0138 = 13.0138 units Total On-Hand Inventory : 13.0138 729  $900 = $8,538,358

Pooling Example - Stock Components 729 models of PC 3 types of each comp. Necessary Service for Each Component: S = (0.99)1/6 = 0.9983 Base Stock Level for Components: = (100  729/3)0.25 = 6075, so G(R-1)  0.9983 R = 6306 On-Hand Inventory Level for Each Component: I(R) = R -  + B(R) = 6306-6075+0.0363 = 231.0363 units Total On-Hand Inventory : 231.0363  18  $150 = $623,798 93% reduction!

Base Stock Insights Reorder points control prob of stockouts by establishing safety stock. To achieve a given fill rate, the required base stock level (and hence safety stock) is an increasing function of mean and (provided backorder cost exceeds shortage cost) std dev of demand during replenishment lead time. The “optimal” fill rate is an increasing in the backorder cost and a decreasing in the holding cost. We can use either a service constraint or a backorder cost to determine the appropriate base stock level. Base stock levels in multi-stage production systems are very similar to kanban systems and therefore the above insights apply. Base stock model allows us to quantify benefits of inventory pooling.

The Single Product (Q,r) Model Motivation: Either 1. Fixed cost associated with replenishment orders and cost per backorder. 2. Constraint on number of replenishment orders per year and service constraint. Objective: Under (1) As in EOQ, this makes batch production attractive.

Summary of (Q,r) Model Assumptions One-at-a-time demands. Demand is uncertain, but stationary over time and distribution is known. Continuous review of inventory level. Fixed replenishment lead time. Constant replenishment batch sizes. Stockouts are backordered.

(Q,r) Notation

(Q,r) Notation (cont.) Decision Variables: Performance Measures:

Inventory and Inventory Position for Q=4, r=4 uniformly distributed between r+1=5 and r+Q=8

Costs in (Q,r) Model Fixed Setup Cost: AF(Q) Stockout Cost: kD(1-S(Q,r)), where k is cost per stockout Backorder Cost: bB(Q,r) Inventory Carrying Costs: cI(Q,r)

Fixed Setup Cost in (Q,r) Model Observation: since the number of orders per year is D/Q,

Stockout Cost in (Q,r) Model Key Observation: inventory position is uniformly distributed between r+1 and r+Q. So, service in (Q,r) model is weighted sum of service in base stock model. Result: Note: this form is easier to use in spreadsheets because it does not involve a sum.

Service Level Approximations Type I (base stock): Type II: Note: computes number of stockouts per cycle, underestimates S(Q,r) Note: neglects B(r,Q) term, underestimates S(Q,r)

Backorder Costs in (Q,r) Model Key Observation: B(Q,r) can also be computed by averaging base stock backorder level function over the range [r+1,r+Q]. Result: Notes: 1. B(Q,r) B(r) is a base stock approximation for backorder level. 2. If we can compute B(x) (base stock backorder level function), then we can compute stockout and backorder costs in (Q,r) model.

Inventory Costs in (Q,r) Model Approximate Analysis: on average inventory declines from Q+s to s+1 so Exact Analysis: this neglects backorders, which add to average inventory since on-hand inventory can never go below zero. The corrected version turns out to be

Inventory vs Time in (Q,r) Model Expected Inventory Actual Inventory Exact I(Q,r) = Approx I(Q,r) + B(Q,r) s+Q Inventory r Approx I(Q,r) s+1=r-+1 Time

Expected Inventory Level for Q=4, r=4, q=2

(Q,r) Model with Backorder Cost Objective Function: Approximation: B(Q,r) makes optimization complicated because it depends on both Q and r. To simplify, approximate with base stock backorder formula, B(r):

Results of Approximate Optimization Assumptions: Q,r can be treated as continuous variables G(x) is a continuous cdf Results: Note: this is just the EOQ formula Note: this is just the base stock formula if G is normal(,), where (z)=b/(h+b)

(Q,r) Example D = 14 units per year c = $150 per unit Stocking Repair Parts: D = 14 units per year c = $150 per unit h = 0.1 × 150 + 10 = $25 per unit l = 45 days q = (14 × 45)/365 = 1.726 units during replenishment lead time A = $10 b = $40 Demand during lead time is Poisson

Values for Poisson(q) Distribution 95

Calculations for Example

Performance Measures for Example

Observations on Example Orders placed at rate of 3.5 per year Fill rate fairly high (90.4%) Very few outstanding backorders (0.049 on average) Average on-hand inventory just below 3 (2.823)

Varying the Example Change: suppose we order twice as often so F=7 per year, then Q=2 and: which may be too low, so increase r from 2 to 3: This is better. For this policy (Q=2, r=4) we can compute B(2,3)=0.026, I(Q,r)=2.80. Conclusion: this has higher service and lower inventory than the original policy (Q=4, r=2). But the cost of achieving this is an extra 3.5 replenishment orders per year.

(Q,r) Model with Stockout Cost Objective Function: Approximation: Assume we can still use EOQ to compute Q* but replace S(Q,r) by Type II approximation and B(Q,r) by base stock approximation:

Results of Approximate Optimization Assumptions: Q,r can be treated as continuous variables G(x) is a continuous cdf Results: Note: this is just the EOQ formula Note: another version of base stock formula (only z is different) if G is normal(,), where (z)=kD/(kD+hQ)

Backorder vs. Stockout Model Backorder Model when real concern is about stockout time because B(Q,r) is proportional to time orders wait for backorders useful in multi-level systems Stockout Model when concern is about fill rate better approximation of lost sales situations (e.g., retail) Note: We can use either model to generate frontier of solutions Keep track of all performance measures regardless of model B-model will work best for backorders, S-model for stockouts

Lead Time Variability Problem: replenishment lead times may be variable, which increases variability of lead time demand. Notation: L = replenishment lead time (days), a random variable l = E[L] = expected replenishment lead time (days) L = std dev of replenishment lead time (days) Dt = demand on day t, a random variable, assumed independent and identically distributed d = E[Dt] = expected daily demand D = std dev of daily demand (units)

Including Lead Time Variability in Formulas Standard Deviation of Lead Time Demand: Modified Base Stock Formula (Poisson demand case): if demand is Poisson Inflation term due to lead time variability Note:  can be used in any base stock or (Q,r) formula as before. In general, it will inflate safety stock.

Single Product (Q,r) Insights Basic Insights: Safety stock provides a buffer against stockouts. Cycle stock is an alternative to setups/orders. Other Insights: 1. Increasing D tends to increase optimal order quantity Q. 2. Increasing q tends to increase the optimal reorder point. (Note: either increasing D or l increases q.) 3. Increasing the variability of the demand process tends to increase the optimal reorder point (provided z > 0). 4. Increasing the holding cost tends to decrease the optimal order quantity and reorder point.

TM 663 Operations Planning September 19, 2011 Paula Jensen 3rd Session: Chapter 2: Inventory Control From EOQ to ROP continued Chapter 3:The MRP Crusade

Material Requirements Planning (MRP) Unlike many other approaches and techniques, material requirements planning “works” which is its best recommendation. – Joseph Orlicky, 1974

History Begun around 1960 as computerized approach to purchasing and production scheduling. Joseph Orlicky, Oliver Wight, and others. APICS launched “MRP Crusade” in 1972 to promote MRP.

Key Insight Independent Demand – finished products Dependent Demand – components It makes no sense to independently forecast dependent demands.

Assumptions 1. Known deterministic demands. 2. Fixed, known production leadtimes. 3. Infinite capacity. Idea is to “back out” demand for components by using leadtimes and bills of material.

MRP Procedure 1. Netting: net requirements against projected inventory 2. Lot Sizing: planned order quantities 3. Time Phasing: planned orders backed out by leadtime 4. BOM Explosion: gross requirements for components

Inputs Master Production Schedule (MPS): due dates and quantities for all top level items Bills of Material (BOM): for all parent items Inventory Status: (on hand plus scheduled receipts) for all items Planned Leadtimes: for all items

Example - Stool Indented BOM Graphical BOM Stool Base (1) Legs (4) Bolts (2) Seat (1) Stool Level 0 Base (1) Seat (1) Level 1 Legs (4) Bolts (4) Bolts (2) Level 2 Note: bolts are treated at lowest level in which they occur for MRP calcs. Actually, they might be left off BOM altogether in practice.

Example

Example (cont.) BOM explosion

Terminology Level Code: lowest level on any BOM on which part is found Planning Horizon: should be longer than longest cumulative leadtime for any product Time Bucket: units planning horizon is divided into Lot-for-Lot: batch sizes equal demands (other lot sizing techniques, e.g., EOQ or Wagner-Whitin can be used) Pegging: identify gross requirements with next level in BOM (single pegging) or customer order (full pegging) that generated it. Single usually used because full is difficult due to lot-sizing, yield loss, safety stocks, etc.

More Terminology Firm Planned Orders (FPO’s): planned order that the MRP system does not automatically change when conditions change – can stabilize system Service Parts: parts used in service and maintenance – must be included in gross requirements Order Launching: process of releasing orders to shop or vendors – may include inflation factor to compensate for shrinkage Exception Codes: codes to identify possible data inaccuracy (e.g., dates beyond planning horizon, exceptionally large or small order quantities, invalid part numbers, etc.) or system diagnostics (e.g., orders open past due, component delays, etc.)

Lot Sizing in MRP Lot-for-lot – “chase” demand Fixed order quantity method – constant lot sizes EOQ – using average demand Fixed order period method – use constant lot intervals Part period balancing – try to make setup/ordering cost equal to holding cost Wagner-Whitin – “optimal” method

Lot Sizing Example Wagner-Whitin: $560 Lot-for-Lot: $1000 Note: WW is “optimal” given this objective.

Lot Sizing Example (cont.) Fixed Order Quantity (using EOQ):

Lot Sizing Example (cont.) Fixed Order Period (FOP): 3 periods

Nervousness Note: we are using FOP lot-sizing rule.

Nervousness Example (cont.) * Past Due Note: Small reduction in requirements caused large change in orders and made schedule infeasible.

Reducing Nervousness Reduce Causes of Plan Changes: Stabilize MPS (e.g., frozen zones and time fences) Reduce unplanned demands by incorporating spare parts forecasts into gross requirements Use discipline in following MRP plan for releases Control changes in safety stocks or leadtimes Alter Lot-Sizing Procedures: Fixed order quantities at top level Lot for lot at intermediate levels Fixed order intervals at bottom level Use Firm Planned Orders: Planned orders that do not automatically change when conditions change Managerial action required to change a FPO

Handling Change New order in MPS Order completed late Scrap loss Causes of Change: New order in MPS Order completed late Scrap loss Engineering changes in BOM Responses to Change: Regenerative MRP: completely re-do MRP calculations starting with MPS and exploding through BOMs. Net Change MRP: store material requirements plan and alter only those parts affected by change (continuously on-line or batched daily). Comparison: Regenerative fixes errors. Net change responds faster but must be regenerated periodically.

Rescheduling Top Down Planning: use MRP system with changes (e.g., altered MPS or scheduled receipts) to recompute plan can lead to infeasibilities (exception codes) Orlicky suggested using minimum leadtimes bottom line is that MPS may be infeasible Bottom Up Replanning: use pegging and firm planned orders to guide rescheduling process pegging allows tracing of release to sources in MPS FPO’s allow fixing of releases necessary for firm customer orders compressed leadtimes (expediting) are often used to justify using FPO’s to override system leadtimes

Safety Stocks and Safety Leadtimes generate net requirements to ensure min level of inventory at all times used as hedge against quantity uncertainties (e.g., yield loss) Safety Leadtimes: inflate production leadtimes in part record used as hedge against time uncertainty (e.g., delivery delays)

Safety Stock Example Note: safety stock level is 20.

Safety Stock vs. Safety Leadtime

Safety Stock vs. Safety Leadtime (cont.)

Manufacturing Resource Planning (MRP II) Sometime called MRP, in contrast with mrp (“little” mrp); more recent implementations are called ERP (Enterprise Resource Planning). Extended MRP into: Master Production Scheduling (MPS) Rough Cut Capacity Planning (RCCP) Capacity Requirements Planning (CRP) Production Activity Control (PAC)

MRP II Planning Hierarchy Demand Forecast Resource Planning Aggregate Production Planning Rough-cut Capacity Planning Master Production Scheduling Bills of Material Material Requirements Planning Inventory Status Job Pool Capacity Requirements Planning Job Release Routing Data Job Dispatching

Master Production Scheduling (MPS) MPS drives MRP Should be accurate in near term (firm orders) May be inaccurate in long term (forecasts) Software supports forecasting order entry netting against inventory Frequently establishes a “frozen zone” in MPS

Rough Cut Capacity Planning (RCCP) Quick check on capacity of key resources Use Bill of Resource (BOR) for each item in MPS Generates usage of resources by exploding MPS against BOR (offset by leadtimes) Infeasibilities addressed by altering MPS or adding capacity (e.g., overtime)

Capacity Requirements Planning (CRP) Uses routing data (work centers and times) for all items Explodes orders against routing information Generates usage profile of all work centers Identifies overload conditions More detailed than RCCP No provision for fixing problems Leadtimes remain fixed despite queueing

Production Activity Control (PAC) Sometimes called “shop floor control” Provides routing/standard time information Sets planned start times Can be used for prioritizing/expediting Can perform input-output control (compare planned with actual throughput) Modern term is MES (Manufacturing Execution System), which represents functions between Planning and Control.

Enterprise Resources Planning SCM BPR MRP MRP II ERP Goal: link information across entire enterprise: manufacturing distribution accounting financial personnel IT

“Integrated” ERP Approach Advantages: integrated functionality consistent user interfaces integrated database single vendor and contract unified architecture unified product support Disadvantages: incompatibility with existing systems long and expensive implementation incompatibility with existing management practices loss of flexibility to use tactical point systems long product development and implementation cycles long payback period lack of technological innovation

Other Planning Tools Manufacturing Execution Systems (MES): automated implementation of shop floor control data tracking (WIP, yield, quality, etc.) merging with ERP? Advanced Planning Systems (APS): algorithms for performing specific functions finite capacity scheduling, forecasting, available to promise, demand management, warehouse management, distribution, etc. partnerships between developers and ERP vendors

Conclusions Insight: distinction between independent and dependent demands Advantages: General approach Supports planning hierarchy (MRP II, ERP) Problems: Assumptions – especially infinite capacity Cultural factors – e.g., data accuracy, training, etc. Focus – authority delegated to computer

TM 663 Operations Planning September 26, 2011 Paula Jensen Chapter 4: From the JIT Revolution to Lean Manufacturing Chapter 5: What Went Wrong?

Agenda Airplanes Factory Physics Chapter 4: From the JIT Revolution to Lean Chapter 5: What Went Wrong (New Assignment Chapter 4 Study Questions: 4,5,6,7 Chapter 5 Study Questions: 2,3,4,6 )

Just In Time (JIT) I tip my hat to the new constitution Take a bow for the new revolution Smile and grin at the change all around Pick up my guitar and play Just like yesterday Then I get on my knees and pray WE DON'T GET FOOLED AGAIN! –The Who

Origins of JIT Japanese firms, particularly Toyota, in 1970's and 1980's Taiichi Ohno and Shigeo Shingo Geographical and cultural roots Japanese objectives “catch up with America” (within 3 years of 1945) small lots of many models Japanese motivation Japanese domestic production in 1949 – 25,622 trucks, 1,008 cars American to Japanese productivity ratio – 9:1 Era of “slow growth” in 1970's

Toyota Production System Pillars: 1. just-in-time, and 2. autonomation, or automation with a human touch Practices: setup reduction (SMED) worker training vendor relations quality control foolproofing (baka-yoke) many others

Supermarket Stimulus Customers get only what they need Stock replenished quickly But, who holds inventory?

Auto-Activated Loom Stimulus Automatically detect problems and shut down Foolproofing Automation with a human touch

Zero Inventories Metaphorical Writing: – Shingo 1990 – Ohno 1988 The Toyota production wrings water out of towels that are already dry. There is nothing more important than planting “trees of will”. – Shingo 1990 5W = 1H – Ohno 1988 Platonic Ideal: Zero Inventories connotes a level of perfection not ever attainable in a production process. However, the concept of a high level of excellence is important because it stimulates a quest for constant improvement through imaginative attention to both the overall task and to the minute details. – Hall 1983

The Seven Zeros Zero Defects: To avoid delays due to defects. (Quality at the source) Zero (Excess) Lot Size: To avoid “waiting inventory” delays. (Usually stated as a lot size of one.) Zero Setups: To minimize setup delay and facilitate small lot sizes. Zero Breakdowns: To avoid stopping tightly coupled line. Zero (Excess) Handling: To promote flow of parts. Zero Lead Time: To ensure rapid replenishment of parts (very close to the core of the zero inventories objective). Zero Surging: Necessary in system without WIP buffers.

The Environment as a Control Constraints or Controls? machine setup times vendor deliveries quality levels (scrap, rework) production schedule (e.g. customer due dates) product designs Impact: the manufacturing system can be made much easier to manage by improving the environment.

Implementing JIT Production Smoothing: relatively constant volumes relatively constant product mix Mixed Model Production (heijunka): 10,000 per month (20 working days) 500 per day (2 shifts) 250 per shift (480 minutes) 1 unit every 1.92 minutes

Implementing JIT (cont.) Production Sequence: Mix of 50% A, 25% B, 25% C in daily production of 500 units 0.5  500 = 250 units of A 0.25  500 = 125 units of B 0.25  500 = 125 units of C A – B – A – C – A – B – A – C – A – B – A – C – A – B – A – C …

Inherent Inflexibility of JIT Sources of Inflexibility: Stable volume Stable mix Precise sequence Rapid (instant?) replenishment Measures to Promote Flexibility: Capacity buffers Setup reduction Cross training Plant layout

Capacity Buffers Problems: Buffer Capacity: JIT is intrinsically rigid (volume, mix, sequence) No explicit link between production and customers How to deal with quota shortfalls Buffer Capacity: Protection against quota shortfalls Regular flow allows matching against customer demands Two shifting: 4 – 8 – 4 – 8 Contrast with WIP buffers found in MRP systems

Setup Reduction Motivation: Small lot sequences not feasible with large setups. Internal vs. External Setups: External – performed while machine is still running Internal – performed while machine is down Approach: 1. Separate the internal setup from the external setup 2. Convert as much as possible of the internal setup to the external setup 3. Eliminate the adjustment process 4. Abolish the setup itself (e.g., uniform product design, combined production, parallel machines)

Cross Training Adds flexibility to inherently inflexible system Allows capacity to float to smooth flow Reduces boredom Fosters appreciation for overall picture Increase potential for idea generation

Workforce Agility Cross-Trained Workers: Shared Tasks: float where needed appreciate line-wide perspective provide more heads per problem area Shared Tasks: can be done by adjacent stations reduces variability in tasks, and hence line stoppages/quality problems

Plant Layout Promote flow with little WIP Facilitate workers staffing multiple machines U-shaped cells Maximum visibility Minimum walking Flexible in number of workers Facilitates monitoring of work entering and leaving cell Workers can conveniently cooperate to smooth flow and address problems

Layout for JIT Cellular Layout: Advanced Material Handling: Proximity for flow control, material handling, floating labor, etc. May require duplication of machinery (decreased utilization?) logical cells? Advanced Material Handling: Avoid large transfer batches Close coordination of physically separate operations Inbound Stock Outbound Stock

Focused Factories Pareto Analysis: Dedicated Lines: Small percentage of sku’s represent large percentage of volume Large percentage of sku’s represent little volume but much complexity Dedicated Lines: for families of high runners few setups little complexity Job Shop Environment: for low runners many setups poorer performance, but only on smaller portion of business Saw Lathe Mill Drill Saw Mill Drill Paint Stores Assembly Warehouse Grind Mill Drill Paint Weld Grind Lathe Drill Saw Grind Paint Stores Lathe Assembly Warehouse Mill Drill

Total Quality Management Origins: Americans (Shewhart, Deming, Juran, Feigenbaum) Fertility of Japan: Japanese abhorrence for wasting scarce resources The Japanese innate resistance to specialists (including QA) Integrality to JIT: JIT requires high quality to work JIT promotes high quality identification of problems facilitates rapid detection of problems pressure to improve quality

Total Quality Management (cont.) Techniques: Process Control (SPC) Easy-to-See Quality Insistence on Compliance (quality first, output second) Line Stop Correcting One's Own Errors (no rework loops) 100 Percent Check (not statistical sampling) Continual Improvement Housekeeping Small Lots Vendor Certification Total Preventive Maintenance

Kanban Definition: A “kanban” is a sign-board or card in Japanese and is the name of the flow control system developed by Toyota. Role: Kanban is a tool for realizing just-in-time. For this tool to work fairly well, the production process must be managed to flow as much as possible. This is really the basic condition. Other important conditions are leveling production as much as possible and always working in accordance with standard work methods. – Ohno 1988 Push vs. Pull: Kanban is a “pull system” Push systems schedule releases Pull systems authorize releases

One-Card Kanban Outbound stockpoint Outbound stockpoint Production Completed parts with cards enter outbound stockpoint. Production cards When stock is removed, place production card in hold box. Production card authorizes start of work.

Two-Card Kanban Inbound stockpoint Outbound stockpoint Move stock to inbound stock point. Move card authorizes pickup of parts. When stock is removed, place production card in hold box. Remove move card and place in hold box. Production cards Move cards Production card authorizes start of work.

MRP versus Kanban MRP … Kanban … … Kanban Signals Full Containers Lover Level Inven-tory … Assem-bly Kanban Lover Level Inven-tory … Assem-bly … Kanban Signals Full Containers

Motorola won the 1988 Malcolm Baldridge Quality Award Goodbye JIT, Hello Lean In 1990 lean appeared in The Machine That Changed the World (Womack, Jones, Roos) Lean is a neater package than JIT; it focuses on flow, the value stream, and elimination of waste. Lean became the rage. Lean appears to be more successful than JIT in achieving results. SixSigma appeared in 1985-1987 at Motorola as a method of creating radically better products 4.5 PPM Motorola won the 1988 Malcolm Baldridge Quality Award ABB, GE, Allied Signal, and Kodak have big successes

The Lessons of JIT/Lean The production environment itself is a control Operational details matter strategically Controlling WIP is important Speed and flexibility are important assets Quality can come first Continual improvement is a condition for survival

Key Insights of TQM/SixSigma Quality and Logistics must be improved together “If you don’t have time to do it right, when will you find time to do it over.” Variability must be identified and reduced Determine the root cause Eliminate the root cause

What Went Wrong? Look ma, the emperor has no clothes! – Hans Christian Andersen Our task is not to fix the blame for the past, but to fix the course for the future. – John F. Kennedy

American Manufacturing Trouble in 1980s Slowdown in productivity growth Severe decline in market share in various markets Widespread perception of inferior quality Persistently large trade deficit

Causes Cultural factors Governmental policies Poor product design Marketing mistakes Counterproductive financial strategies Poor operations management

Management Tools Quantitative Methods Material Requirements Planning inventory scheduling plant layout facility location Material Requirements Planning Just-in-Time

Trouble with Quantitative Methods Cultural Factors: The frontier ethic – best and brightest shun OM Faith in the scientific method – emphasis on mathematical precision Combined Effect: Top management out of OM loop Sophisticated techniques for narrower and narrower problems

EOQ Unrealistic Assumptions: Ill Effects: fixed, known setup cost constant, deterministic demand instantaneous delivery single product or no product interactions Ill Effects: Inefficiency in lot-sizing Wasted effort in trying to fit model Myopic perspective about lot-sizing Missed importance of setup reduction Missed value of splitting move lots

Scheduling 2 & 3 machine min makespan problem (Johnson 1954) Virtually no applications Mathematically challenging Hundreds of follow-on papers At this time, it appears that one research paper (that by Johnson) set a wave of research in motion that devoured scores of person-years of research time on an intractable problem of little practical consequence. – Dudek Panwalkar, Smith, 1992

OM Trends Engineering Courses: became virtually math courses Management Courses: anecdotal case studies Calls for Changes: Strategic importance of operational details OM is technical We need a science of manufacturing

Trouble with MRP MRP Successes: But … Number of MRP systems in America grew from a handful in the early 1960's, to 150 in 1971 APICS MRP Crusade in 1972 spurred number of MRP systems in the U.S. as high as 8000 In 1984, 16 companies sold $400 million in MRP software In 1989, $1.2 billion worth of MRP software was sold to American industry, constituting just under one-third of the entire American market for computer services By late 1990’s, ERP was a $10 billion industry (ERP consulting even bigger); SAP was world’s fourth largest software company But …

Surveys of MRP Users 1980 Survey of Over 1,100 Firms: much less than 10% of U.S. and European companies recoup MRP investment within two years 1982 Survey of 679 APICS Members: 9.5% regarded their companies as being Class A users 60% reported their firms as being Class C or Class D users This from an APICS survey of materials managers 1986 Survey of 33 S. Carolina MRP Users: Similar responses to 1982 survey Average eventual investment in hardware, software, personnel, and training for an MRP system was $795,000 with a standard deviation of $1,191,000

APICS Explanations 1. Lack of top management commitment, 2. Lack of education of those who use the system, 3. An unrealistic master production schedule, 4. Inaccurate data, including bills of material and inventory records.

The Fundamental Flaw of MRP … an MRP system is capacity-insensitive, and properly so, as its function is to determine what materials and components will be needed and when, in order to execute a given master production schedule. There can be only one correct answer to that, and it cannot therefore vary depending on what capacity does or does not exist. – Orlicky 1975 But, lead times do depend on loading when capacity is finite Incentive to inflate leadtimes Result is increased congestion, increased WIP, decreased customer service

Historical Interpretation of MRP MRP is the quintessential American production control system When Scientific Management (developed here) met the computer (developed here), MRP was the result Unfortunately, the computer that Scientific Management met was a computer of the 1960's Insufficient RAM to process parts simultaneously Fixed leadtimes allow transaction based system

MRP Patches MRP II provides planning hierarchy and data management features CRP is the sin of MRP repeated over and over Approaches like closed-loop MRP either: wait for WIP explosion to modify releases, or fail to consider PAC in plan ERP extended MRP to supply chains but did not by itself change underlying paradigm Can MES save MRP? wide variety of commercial approaches to MES interface between planning and execution still critical

Trouble With JIT $64,000 Question: Is JIT a system, and, if so, is it transportable? Answers: “Unquestionably” and “Yes” –Schonberger, Hall, Monden “Maybe not” and “To a limited extent” –Hayes Conjecture: JIT is a system of beliefs, but a collection of methods

Romantic versus Pragmatic JIT Romantic JIT: An aesthetic ideal Simplicity in the extreme Almost trivial to implement Phrased in stirring rhetoric Pragmatic JIT: setup time reduction (SMED) plant layout (e.g., U-shaped cells) quality-control preventive maintenance design for manufacturability many others

Mortals Emulating Genius Persistence: Toyota took 25 years to reduce setups from 2-3 hours to 3 minutes. Environmental Factors: Harder to address than direct procedures. Some people imagine that Toyota has put on a smart new set of clothes, the kanban system, so they go out and purchase the same outfit and try it on. They quickly discover that they are much too fat to wear it. – Shingo Prioritization: Systems view is first thing to get lost. Deliberate obfuscation? If the U.S. had understood what Toyota was doing, it would have been no good for us. – Ohno

A Matter of Perspective Policies conflict Romantic JIT bans the t-word (Schonberger) Japanese originators creatively balanced objectives subtly, implicitly pursued policies across functions context-specific procedures Dangers of lack of perspective management by slogan inventory is the root of all evil water and rocks analogy effort wasted on chubchiks (e.g., unnecessary setup reduction) failure to coordinate efforts (e.g., cells running large batches of parts)

3rd Edition Updates were not in the slides in August Trends in Manufacturing 6s an improvement methodology w/ training program Lean philosophy promotes the right incentives IT (SCM and ERP) provide data to make decisions MISSING A SCIENTIFIC FRAMEWORK relationships of between cycle time, production rate, utilization, inventory, WIP, capacity, variability in demand, variability in manufacturing process

Business Process Re-engineering Systems Analysis applied to Management “the fundamental rethinking and radical redesign of business practices to achieve dramatic improvements in critical, contemporary measures of performance, such as cost, quality, service, and speed. Most resign schemes included eliminating jobs, so it was associated with downsizing Revolutionary aspect paved the way for ERP, which required restructuring manufacturing processes to fit software.

Lean Manufacturing Uses value stream mapping (VSM): a variation of process flow mapping. It has problems: No exact definition of “value-added” Value-added time is so short is does not offer a reasonable target for a cycle time VSM does not provide a means of diagnosing causes of long cycle times VSM collects capacity and demand data, but does not compute utilization No feasibility check for a “future state”.

Six Sigma (6s) Emphasizes the experimental aspect of the scientific method Define Measure Analyze Improve Control Story about 6s students ignoring Factory Physics training

Where from Here? Problems with Traditional Approaches: Reality: OM (quantitative methods) has stressed math over realism MRP is fundamentally flawed, in the basics, not the details JIT is a collection of methods and slogans, not a system Reality: manufacturing is large scale, complex, and varied continual improvement is essential no “technological silver bullet” can save us

Where from Here? A Science of Manufacturing... What Can We Hope For? Better Education basics intuition synthesis Better Tools descriptive models prescriptive models integrated framework A Science of Manufacturing...

ENGM 663 Paula Jensen Chapter 6: A Science of Manufacturing Chapter 7: Basic Factory Dynamics

Agenda Factory Physics Chapter 6: A Science of Manufacturing (From 2nd Ed) Chapter 7: Basic Factory Dynamics (New Assignment Chapter 6: Problem 1 Chapter 7: Problems 5, 8, 10) Test 1 Study Guide

Objectives, Measures, and Controls I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind; it may be the beginning of knowledge, but have scarcely, in your thoughts, advanced to the stage of Science, whatever the matter may be. – Lord Kelvin

Why a Science of Manufacturing? Confusion in Industry: too many “revolutions” management by buzzword sales glitz over substance Confusion in Academia: high-powered methodology applied to non-problems huge variation in what is taught Example of Other Fields: Civil Engineering–statics, dynamics Electrical Engineering – electricity and magnetism Many others

Automobile Design Requirements: Can we do it? Answer: Mass of car of 1000 kg Acceleration of 2.7 meters per second squared (zero to 60 in 10 seconds) Engine with no more than 200 Newtons of force Can we do it? Answer: No way! F = ma 200 Nt  (1000 kg) (2.7 m/s2) = 2,700 Nt.

? Factory Design Requirements: Can we do it? Answer: 3000 units per day, with a lead time of not greater than 10 days, and with a service level (percent of jobs that finish on time) of at least 90%. Can we do it? Answer: ? Who knows?

Factory Tradeoff Curves

Goals of a Science of Manufacturing Tools: descriptive models prescriptive models solution techniques Terminology: rationalize buzzwords recognize commonalities across environments Perspective: basics intuition synthesis

The Nature of Science Purpose: Steps: The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of hypothesis or axioms. --- Albert Einstein Steps: 1. Observation. 2. Classification. 3. Theoretical Conjecture. 4. Experimental verification/refutation. 5. Repeat.

Systems Analysis Definition: Systems analysis is a structured approach to problem-solving that involves 1. Identification of objectives (what you want to accomplish), measures (for comparing alternatives), and controls (what you can change). 2. Generation of specific alternatives. 3. Modeling (some form of abstraction from reality to facilitate comparison of alternatives). 4. Optimization (at least to the extent of ranking alternatives and choosing “best” one). 5. Iteration (going back through the process as new facets arise).

System Analysis Paradigm REAL WORLD OPERATIONS ANALYSIS ANALOG WORLD Conjecture Objectives Verify constraints Identify Alternatives Choose Measures of Effectiveness Specify Parameters and Controls Model Interactions Verify & Validate Model SYSTEMS DESIGN Compare Alternatives Choose Policies Ask “What If” Questions Compare Controls Optimize Control Levels Sensitivity Analysis Implement Policies Train Users Fine Tune System IMPLEMENTATION EVALUATION Evaluate System Performance Look For Oversights Identify Future Opportunities Validate Model Predictions Question Assumptions Identify Other Controls

General Measures and Objectives Fundamental Objective: elementary starting point source of agreement example - make money over the long-term Hierarchy of Objectives: more basic objectives that support fundamental objective closer to improvement policies Tradeoffs:objectives conflict we need models

Hierarchical Objectives High Profitability Low Costs High Sales Low Unit Costs Quality Product High Customer Service High Throughput High Utilization Low Inventory Fast Response Many products Less Variability Short Cycle Times Low Utilization High Inventory More Variability

Corporate Measures and Objectives Fundamental Objective: Maximize the wealth and well-being of the stakeholders over the long term. Financial Performance Measures: 1. Net-profit. 2. Return on investment. Components: 1. Revenue. 2. Expenses. 3. Assets.

Plant Measures and Objectives Throughput: product that is high quality and is sold. Costs: Operating budget of plant. Assets: Capital equipment and WIP. Objectives: Maximize profit. Minimize unit costs. Tradeoffs: we would like (but can’t always have) Throughput Cost Assets

Systems Analysis Tools Process Mapping: identify main sequence of activities highlight bottlenecks clarify critical connections across business systems Workshops: structured interaction between various parties many methods: Nominal Group Technique, Delphi, etc. roles of moderator and provocateur are critical

Systems Analysis Tools (cont.) Conjecture and Refutation: promotes group ownership of ideas places critical thinking in a constructive mode everyday use of the scientific method Modeling: always done with specific purpose value of model is its usefulness modeling is an iterative process

The Need for Process Mapping Example: North American Switch Manufacturer -- 10-12 week leadtimes in spite of dramatically reduced factory cycle times: 10% Sales 15% Order Entry 15% Order Coding 20% Engineering 10% Order Coding 15% Scheduling 5% Premanufacturing and Manufacturing 10% Delivery and Prep Conclusion: Lead time reduction must address entire value delivery system.

Process Mapping Activities Purpose: understand current system by identifying main sequence of activities highlighting bottlenecks clarifying critical connections across business system Types of Maps: Assembly Flowchart: diagram of activities to assembly product. Process Flowchart: diagram of how pieces of system interrelate in an organization. Relationship Map: diagram of specific steps to accomplish a task, without indication of functions or subsystems. Cross-Functional Process Map: diagram of specific steps to accomplish a task organized by function or subsystem responsible for the step.

Sample Assembly Flowchart CELL 1 START PANASERT 1050 ROBOT 1100 ROBOT 1150 ROBOT 1200 CIM FLEX 1250 ROBOT 1300 ROBOT 1350 ROBOT 1375 ROBOT 1380 SOLDER STATION 1000 DECODER SINGULATION ROBOT 1500 RECEIVER SINGULATION ROBOT 1750 LASER TRIM 1775 CELL 2 EOL TEST 1550 LEGEND UNIX CELL CONTROLLER TEST BAY

Process Flowchart for Order Entry Receive Customer Order Form Generate Standard Layout Plan Customer Approval? No Yes Review Plan/Lists Generate Parts Lists Approval? No Yes Enter Parts Lists into System End of Bucket? No Yes Generate Cutting Orders

Sample Relationship Map Operating departments make independent decision Production Control - controls work flow Customers Warehouse Production control Salesmen Order Processing Production Scheduling Design Fabricating Finishing Shipping Salesman controls the order processing and design flow

Sample Cross-Functional Process Map Field Offices Customer needs observed Field support needs reviewed Field support planned Market opportunity defined New product evaluated Price and distribution options reviewed Price point set Roll-out planned Marketing New product concept floated New product prototype developed Engineering Final product engineered Process feasibility review and cost estimating Tooling and capacity planned Production readiness planned Manufacturing Production TIME

Conclusions Science of Manufacturing: Systems Approach: Modeling: important for practice provides a structure for OM education Systems Approach: one of the most powerful engineering tools a key management skill as well (e.g., re-engineering) Modeling: part, but not all, of systems analysis key to a science of manufacturing more descriptive models are needed

Basic Factory Dynamics Physics should be explained as simply as possible, but no simpler. – Albert Einstein

HAL Case Large Panel Line: produces unpopulated printed circuit boards Line runs 24 hr/day (but 19.5 hrs of productive time) Recent Performance: throughput = 1,400 panels per day (71.8 panels/hr) WIP = 47,600 panels CT = 34 days (663 hr at 19.5 hr/day) customer service = 75% on-time delivery Is HAL lean? What data do we need to decide?

HAL - Large Panel Line Processes Lamination (Cores): press copper and prepreg into core blanks Machining: trim cores to size Internal Circuitize: etch circuitry into copper of cores Optical Test and Repair (Internal): scan panels optically for defects Lamination (Composites): press cores into multiple layer boards External Circuitize: etch circuitry into copper on outside of composites Optical Test and Repair (External): scan composites optically for defects Drilling: holes to provide connections between layers Copper Plate: deposits copper in holes to establish connections Procoat: apply plastic coating to protect boards Sizing: cut panels into boards End of Line Test: final electrical test

HAL Case - Science? External Benchmarking Internal Benchmarking but other plants may not be comparable Internal Benchmarking capacity data: what is utilization? but this ignores WIP effects Need relationships between WIP, TH, CT, service!

Definitions Workstations: a collection of one or more identical machines. Parts: a component, sub-assembly, or an assembly that moves through the workstations. End Items: parts sold directly to customers; relationship to constituent parts defined in bill of material. Consumables: bits, chemicals, gasses, etc., used in process but do not become part of the product that is sold. Routing: sequence of workstations needed to make a part. Order: request from customer. Job: transfer quantity on the line.

Definitions (cont.) Throughput (TH): for a line, throughput is the average quantity of good (non-defective) parts produced per unit time. Work in Process (WIP): inventory between the start and endpoints of a product routing. Raw Material Inventory (RMI): material stocked at beginning of routing. Crib and Finished Goods Inventory (FGI): crib inventory is material held in a stockpoint at the end of a routing; FGI is material held in inventory prior to shipping to the customer. Cycle Time (CT): time between release of the job at the beginning of the routing until it reaches an inventory point at the end of the routing.

Factory Physics® Definition: A manufacturing system is a goal-oriented network of processes through which parts flow. Structure: Plant is made up of routings (lines), which in turn are made up of processes. Focus: Factory Physics® is concerned with the network and flows at the routing (line) level.

Parameters Descriptors of a Line: 1) Bottleneck Rate (rb): Rate (parts/unit time or jobs/unit time) of the process center having the highest long-term utilization. 2) Raw Process Time (T0): Sum of the long-term average process times of each station in the line. 3) Congestion Coefficient (): A unitless measure of congestion. Zero variability case, a = 0. “Practical worst case,” a = 1. “Worst possible case,” a = W0. Note: we won’t use  quantitatively, but point it out to recognize that lines with same rb and T0 can behave very differently.

Parameters (cont.) Relationship: Critical WIP (W0): WIP level in which a line having no congestion would achieve maximum throughput (i.e., rb) with minimum cycle time (i.e., T0). W0 = rb T0

The Penny Fab Characteristics: Parameters: Four identical tools in series. Each takes 2 hours per piece (penny). No variability. CONWIP job releases. Parameters: rb = T0 = W0 = a = 0.5 pennies/hour 8 hours 0.5  8 = 4 pennies 0 (no variability, best case conditions)

The Penny Fab

The Penny Fab (WIP=1) Time = 0 hours

The Penny Fab (WIP=1) Time = 2 hours

The Penny Fab (WIP=1) Time = 4 hours

The Penny Fab (WIP=1) Time = 6 hours

The Penny Fab (WIP=1) Time = 8 hours

The Penny Fab (WIP=1) Time = 10 hours

The Penny Fab (WIP=1) Time = 12 hours

The Penny Fab (WIP=1) Time = 14 hours

The Penny Fab (WIP=1) Time = 16 hours

Penny Fab Performance

The Penny Fab (WIP=2) Time = 0 hours

The Penny Fab (WIP=2) Time = 2 hours

The Penny Fab (WIP=2) Time = 4 hours

The Penny Fab (WIP=2) Time = 6 hours

The Penny Fab (WIP=2) Time = 8 hours

The Penny Fab (WIP=2) Time = 10 hours

The Penny Fab (WIP=2) Time = 12 hours

The Penny Fab (WIP=2) Time = 14 hours

The Penny Fab (WIP=2) Time = 16 hours

The Penny Fab (WIP=2) Time = 18 hours

Penny Fab Performance

The Penny Fab (WIP=4) Time = 0 hours

The Penny Fab (WIP=4) Time = 2 hours

The Penny Fab (WIP=4) Time = 4 hours

The Penny Fab (WIP=4) Time = 6 hours

The Penny Fab (WIP=4) Time = 8 hours

The Penny Fab (WIP=4) Time = 10 hours

The Penny Fab (WIP=4) Time = 12 hours

The Penny Fab (WIP=4) Time = 14 hours

Penny Fab Performance

The Penny Fab (WIP=5) Time = 0 hours

The Penny Fab (WIP=5) Time = 2 hours

The Penny Fab (WIP=5) Time = 4 hours

The Penny Fab (WIP=5) Time = 6 hours

The Penny Fab (WIP=5) Time = 8 hours

The Penny Fab (WIP=5) Time = 10 hours

The Penny Fab (WIP=5) Time = 12 hours

Penny Fab Performance

TH vs. WIP: Best Case rb 1/T0 W0

CT vs. WIP: Best Case 1/rb T0 W0

Best Case Performance Best Case Law: The minimum cycle time (CTbest) for a given WIP level, w, is given by The maximum throughput (THbest) for a given WIP level, w is given by,

Best Case Performance (cont.) Example: For Penny Fab, rb = 0.5 and T0 = 8, so W0 = 0.5  8 = 4, which are exactly the curves we plotted.

A Manufacturing Law Little's Law: The fundamental relation between WIP, CT, and TH over the long-term is: Insights: Fundamental relationship Simple units transformation Definition of cycle time (CT = WIP/TH)

Penny Fab Two 2 hr 5 hr 3 hr 10 hr

Penny Fab Two 0.5 0.4 0.6 0.67 0.4 p/hr 20 hr 8 pennies rb = ____________ T0 = ____________ W0 = ____________

Penny Fab Two Simulation (Time=0) 2 2 hr 5 hr 3 hr 10 hr

Penny Fab Two Simulation (Time=2) 7 4 2 hr 5 hr 3 hr 10 hr

Penny Fab Two Simulation (Time=4) 7 6 9 2 hr 5 hr 3 hr 10 hr

Penny Fab Two Simulation (Time=6) 7 8 9 2 hr 5 hr 3 hr 10 hr

Penny Fab Two Simulation (Time=7) 17 12 8 9 2 hr 5 hr 3 hr 10 hr

Penny Fab Two Simulation (Time=8) 17 12 10 9 2 hr 5 hr 3 hr 10 hr

Penny Fab Two Simulation (Time=9) 17 19 12 10 14 2 hr 5 hr 3 hr 10 hr

Penny Fab Two Simulation (Time=10) 17 19 12 12 14 2 hr 5 hr 3 hr 10 hr

Penny Fab Two Simulation (Time=12) 17 19 17 22 14 14 2 hr 5 hr 3 hr 10 hr

Penny Fab Two Simulation (Time=14) 17 19 17 22 16 19 24 2 hr 5 hr 3 hr 10 hr

Penny Fab Two Simulation (Time=16) 17 19 17 22 19 24 2 hr 5 hr 3 hr 10 hr

Penny Fab Two Simulation (Time=17) 27 19 22 22 20 19 24 2 hr 5 hr 3 hr 10 hr

Penny Fab Two Simulation (Time=19) 27 29 22 22 20 24 24 22 2 hr 5 hr 3 hr 10 hr

Penny Fab Two Simulation (Time=20) 27 Note: job will arrive at bottleneck just in time to prevent starvation. 29 22 22 22 24 24 22 2 hr 5 hr 3 hr 10 hr

Penny Fab Two Simulation (Time=22) 27 29 27 32 25 24 24 24 2 hr 5 hr 3 hr Note: job will arrive at bottleneck just in time to prevent starvation. 10 hr

Penny Fab Two Simulation (Time=24) 27 29 27 32 25 29 34 27 2 hr 5 hr 3 hr And so on…. Bottleneck will just stay busy; all others will starve periodically 10 hr

Worst Case Observation: The Best Case yields the minimum cycle time and maximum throughput for each WIP level. Question: What conditions would cause the maximum cycle time and minimum throughput? Experiment: set average process times same as Best Case (so rb and T0 unchanged) follow a marked job through system imagine marked job experiences maximum queueing

Worst Case Penny Fab Time = 0 hours

Worst Case Penny Fab Time = 8 hours

Worst Case Penny Fab Time = 16 hours

Worst Case Penny Fab Time = 24 hours

Worst Case Penny Fab Time = 32 hours Note: CT = 32 hours = 4 8 = wT0 TH = 4/32 = 1/8 = 1/T0

TH vs. WIP: Worst Case Best Case rb Worst Case 1/T0 W0

CT vs. WIP: Worst Case Worst Case Best Case T0 W0

Worst Case Performance Worst Case Law: The worst case cycle time for a given WIP level, w, is given by, CTworst = w T0 The worst case throughput for a given WIP level, w, is given by, THworst = 1 / T0 Randomness? None - perfectly predictable, but bad!

Practical Worst Case Observation: There is a BIG GAP between the Best Case and Worst Case performance. Question: Can we find an intermediate case that: divides “good” and “bad” lines, and is computable? Experiment: consider a line with a given rb and T0 and: single machine stations balanced lines variability such that all WIP configurations (states) are equally likely

PWC Example – 3 jobs, 4 stations clumped up states spread out states

Practical Worst Case Let w = jobs in system, N = no. stations in line, and t = process time at all stations: CT(single) = (1 + (w-1)/N) t CT(line) = N [1 + (w-1)/N] t = Nt + (w-1)t = T0 + (w-1)/rb TH = WIP/CT = [w/(w+W0-1)]rb From Little’s Law

Practical Worst Case Performance Practical Worst Case Definition: The practical worst case (PWC) cycle time for a given WIP level, w, is given by, The PWC throughput for a given WIP level, w, is given by, where W0 is the critical WIP.

TH vs. WIP: Practical Worst Case Best Case rb PWC Good (lean) Bad (fat) Worst Case 1/T0 W0

CT vs. WIP: Practical Worst Case PWC Bad (fat) Best Case Good (lean) T0 W0

Penny Fab Two Performance Note: process times in PF2 have var equal to PWC. But… unlike PWC, it has unbalanced line and multi machine stations. Best Case rb Penny Fab 2 Practical Worst Case 1/T0 Worst Case W0

Penny Fab Two Performance (cont.) Worst Case Practical Worst Case Penny Fab 2 1/rb T0 Best Case W0

Back to the HAL Case - Capacity Data

HAL Case - Situation Critical WIP: rbT0 = 114  33.9 = 3,869 Actual Values: CT = 34 days = 663 hours (at 19.5 hr/day) WIP = 47,600 panels TH = 71.8 panels/hour Conclusions: Throughput is 63% of capacity WIP is 12.3 times critical WIP CT is 24.1 times raw process time

HAL Case - Analysis Conclusion: actual system is much worse than PWC! TH Resulting from PWC with WIP = 47,600? Much higher than actual TH! WIP Required for PWC to Achieve TH = 0.63rb? Much lower than actual WIP! Conclusion: actual system is much worse than PWC!

HAL Internal Benchmarking Outcome “Lean" Region “Fat" Region

Labor Constrained Systems Motivation: performance of some systems are limited by labor or a combination of labor and equipment. Full Flexibility with Workers Tied to Jobs: WIP limited by number of workers (n) capacity of line is n/T0 Best case achieves capacity and has workers in “zones” ample capacity case also achieves full capacity with “pick and run” policy

Labor Constrained Systems (cont.) Full Flexibility with Workers Not Tied to Jobs: TH depends on WIP levels THCW(n)  TH(w)  THCW(w) need policy to direct workers to jobs (focus on downstream is effective) Agile Workforce Systems bucket brigades kanban with shared tasks worksharing with overlapping zones many others

Factory Dynamics Takeaways Performance Measures: throughput WIP cycle time service Range of Cases: best case practical worst case worst case Diagnostics: simple assessment based on rb, T0, actual WIP,actual TH evaluate relative to practical worst case

TM 663 Operations Planning October 24, 2011 Paula Jensen Chapter 8: Variability Basics Chapter 9: The Corrupting Influence of Variability

Agenda Paula’s Idea Factory Physics Chapter 8: Variability Basics Chapter 9: The Corrupting Influence of Variability (New Assignment To be assigned later Chapter 8: Problem Chapter 9: Problems)

Variability Basics God does not play dice with the universe. – Albert Einstein Stop telling God what to do. – Niels Bohr

Variability Makes a Difference! Little’s Law: TH = WIP/CT, so same throughput can be obtained with large WIP, long CT or small WIP, short CT. The difference? Penny Fab One: achieves full TH (0.5 j/hr) at WIP=W0=4 jobs if it behaves like Best Case, but requires WIP=27 jobs to achieve 95% of capacity if it behaves like the Practical Worst Case. Why? Variability! Variability!

Tortise and Hare Example Two machines: subject to same workload: 69 jobs/day (2.875 jobs/hr) subject to unpredictable outages (availability = 75%) Hare X19: long, but infrequent outages Tortoise 2000: short, but more frequent outages Performance: Hare X19 is substantially worse on all measures than Tortoise 2000. Why? Variability!

Variability Views Variability: Randomness: Any departure from uniformity Random versus controllable variation Randomness: Essential reality? Artifact of incomplete knowledge? Management implications: robustness is key

Probabilistic Intuition Uses of Intuition: driving a car throwing a ball mastering the stock market First Moment Effects: Throughput increases with machine speed Throughput increases with availability Inventory increases with lot size Our intuition is good for first moments g

Probabilistic Intuition (cont.) Second Moment Effects: Which is more variable – processing times of parts or batches? Which are more disruptive – long, infrequent failures or short frequent ones? Our intuition is less secure for second moments Misinterpretation – e.g., regression to the mean

Variability Definition: Variability is anything that causes the system to depart from regular, predictable behavior. Sources of Variability: setups • workpace variation machine failures • differential skill levels materials shortages • engineering change orders yield loss • customer orders rework • product differentiation operator unavailability • material handling

Measuring Process Variability Note: we often use the “squared coefficient of variation” (SCV), ce2

Variability Classes in Factory Physics® Low variability (LV) Moderate variability (MV) High variability (HV) Effective Process Times: actual process times are generally LV effective process times include setups, failure outages, etc. HV, LV, and MV are all possible in effective process times Relation to Performance Cases: For balanced systems MV – Practical Worst Case LV – between Best Case and Practical Worst Case HV – between Practical Worst Case and Worst Case ce 0.75 1.33

Measuring Process Variability – Example Question: can we measure ce this way? Answer: No! Won’t consider “rare” events properly.

Natural Variability Definition: variability without explicitly analyzed cause Sources: operator pace material fluctuations product type (if not explicitly considered) product quality Observation: natural process variability is usually in the LV category.

Down Time – Mean Effects Definitions:

Down Time – Mean Effects (cont.) Availability: Fraction of time machine is up Effective Processing Time and Rate:

Totoise and Hare - Availability Hare X19: t0 = 15 min 0 = 3.35 min c0 = 0 /t0 = 3.35/15 = 0.05 mf = 12.4 hrs (744 min) mr = 4.133 hrs (248 min) cr = 1.0 Availability: Tortoise: t0 = 15 min 0 = 3.35 min c0 = 0 /t0 = 3.35/15 = 0.05 mf = 1.9 hrs (114 min) mr = 0.633 hrs (38 min) cr = 1.0 A = A = No difference between machines in terms of availability.

Down Time – Variability Effects Effective Variability: Conclusions: Failures inflate mean, variance, and CV of effective process time Mean (te) increases proportionally with 1/A SCV (ce2) increases proportionally with mr SCV (ce2) increases proportionally in cr2 For constant availability (A), long infrequent outages increase SCV more than short frequent ones Variability depends on repair times in addition to availability

Tortoise and Hare - Variability Hare X19: te = ce2 = Tortoise 2000 te = ce2 = Hare X19 is much more variable than Tortoise 2000!

Setups – Mean and Variability Effects Analysis:

Setups – Mean and Variability Effects (cont.) Observations: Setups increase mean and variance of processing times. Variability reduction is one benefit of flexible machines. However, the interaction is complex.

Setup – Example Fast, inflexible machine – 2 hr setup every 10 jobs Data: Fast, inflexible machine – 2 hr setup every 10 jobs Slower, flexible machine – no setups Traditional Analysis? No difference!

Setup – Example (cont.) Factory Physics® Approach: Compare mean and variance Fast, inflexible machine – 2 hr setup every 10 jobs

Setup – Example (cont.) Conclusion: Slower, flexible machine – no setups Conclusion: Flexibility can reduce variability.

Setup – Example (cont.) New Machine: Consider a third machine same as previous machine with setups, but with shorter, more frequent setups Analysis: Conclusion: Shorter, more frequent setups induce less variability.

Other Process Variability Inflators Sources: operator unavailability recycle batching material unavailability et cetera, et cetera, et cetera Effects: inflate te inflate ce Consequences: Effective process variability can be LV, MV,or HV.

Illustrating Flow Variability Low variability arrivals t smooth! High variability arrivals t bursty!

Measuring Flow Variability

Propagation of Variability ce2(i) Single Machine Station: where u is the station utilization given by u = rate Multi-Machine Station: where m is the number of (identical) machines and ca2(i) cd2(i) = ca2(i+1) i i+1 departure var depends on arrival var and process var

Propagation of Variability – High Utilization Station HV LV HV LV LV HV Conclusion: flow variability out of a high utilization station is determined primarily by process variability at that station.

Propagation of Variability – Low Utilization Station HV LV HV LV LV HV Conclusion: flow variability out of a low utilization station is determined primarily by flow variability into that station.

Variability Interactions Importance of Queueing: manufacturing plants are queueing networks queueing and waiting time comprise majority of cycle time System Characteristics: Arrival process Service process Number of servers Maximum queue size (blocking) Service discipline (FCFS, LCFS, EDD, SPT, etc.) Balking Routing Many more

Kendall's Classification A/B/C A: arrival process B: service process C: number of machines M: exponential (Markovian) distribution G: completely general distribution D: constant (deterministic) distribution. B A C Queue Server

Queueing Parameters ra = the rate of arrivals in customers (jobs) per unit time (ta = 1/ra = the average time between arrivals). ca = the CV of inter-arrival times. m = the number of machines. re = the rate of the station in jobs per unit time = m/te. ce = the CV of effective process times. u = utilization of station = ra/re. Note: a station can be described with 5 parameters.

Queueing Measures Measures: Relationships: CTq = the expected waiting time spent in queue. CT = the expected time spent at the process center, i.e., queue time plus process time. WIP = the average WIP level (in jobs) at the station. WIPq = the expected WIP (in jobs) in queue. Relationships: CT = CTq + te WIP = ra  CT WIPq = ra  CTq Result: If we know CTq, we can compute WIP, WIPq, CT.

The G/G/1 Queue Formula: Observations: Useful model of single machine workstations Separate terms for variability, utilization, process time. CTq (and other measures) increase with ca2 and ce2 Flow variability, process variability, or both can combine to inflate queue time. Variability causes congestion!

The G/G/m Queue Formula: Observations: Useful model of multi-machine workstations Extremely general. Fast and accurate. Easily implemented in a spreadsheet (or packages like MPX).

VUT Spreadsheet basic data failures setups yield measures

Effects of Blocking VUT Equation: characterizes stations with infinite space for queueing useful for seeing what will happen to WIP, CT without restrictions But real world systems often constrain WIP: physical constraints (e.g., space or spoilage) logical constraints (e.g., kanbans) Blocking Models: estimate WIP and TH for given set of rates, buffer sizes much more complex than non-blocking (open) models, often require simulation to evaluate realistic systems

The M/M/1/b Queue Model of Station 2 Note: there is room for b=B+2 jobs in system, B in the buffer and one at each station. Infinite raw materials B buffer spaces Model of Station 2 Goes to u/(1-u) as b Always less than WIP(M/M/1) Goes to ra as b Always less than TH(M/M/1) Little’s law Note: u>1 is possible; formulas valid for u1

Blocking Example M/M/1/b system has less WIP and less TH than M/M/1 system 18% less TH 90% less WIP

Seeking Out Variability General Strategies: look for long queues (Little's law) look for blocking focus on high utilization resources consider both flow and process variability ask “why” five times Specific Targets: equipment failures setups rework operator pacing anything that prevents regular arrivals and process times

Variability Pooling Basic Idea: the CV of a sum of independent random variables decreases with the number of random variables. Example (Time to process a batch of parts):

Safety Stock Pooling Example PC’s consist of 6 components (CPU, HD, CD ROM, RAM, removable storage device, keyboard) 3 choices of each component: 36 = 729 different PC’s Each component costs $150 ($900 material cost per PC) Demand for all models is normally distributed with mean 100 per year, standard deviation 10 per year Replenishment lead time is 3 months, so average demand during LT is  = 25 for computers and  = 25(729/3) = 6075 for components Use base stock policy with fill rate of 99%

Pooling Example - Stock PC’s cycle stock Base Stock Level for Each PC: R =  + zs = 25 + 2.33( 25) = 37 On-Hand Inventory for Each PC: I(R) = R -  + B(R)  R -  = zs = 37 - 25 = 12 units Total (Approximate) On-Hand Inventory : 12 729  $900 = $7,873,200 safety stock

Pooling Example - Stock Components Necessary Service for Each Component: S = (0.99)1/6 = 0.9983 zs = 2.93 Base Stock Level for Each Component: R =  + zs = 6075 + 2.93( 6075) = 6303 On-Hand Inventory Level for Each Component: I(R) = R -  + B(R)  R -  = zs = 6303-6075 = 228 units Total Safety Stock: 228  18  $150 = $615,600 cycle stock safety stock 92% reduction!

Basic Variability Takeaways Variability Measures: CV of effective process times CV of interarrival times Components of Process Variability failures setups many others - deflate capacity and inflate variability long infrequent disruptions worse than short frequent ones Consequences of Variability: variability causes congestion (i.e., WIP/CT inflation) variability propagates variability and utilization interact pooled variability less destructive than individual variability

The Corrupting Influence of Variability When luck is on your side, you can do without brains. – Giordano Bruno,burned at the stake in 1600 The more you know the luckier you get. – “J.R. Ewing” of Dallas

Performance of a Serial Line Measures: Throughput Inventory (RMI, WIP, FGI) Cycle Time Lead Time Customer Service Quality Evaluation: Comparison to “perfect” values (e.g., rb, T0) Relative weights consistent with business strategy? Links to Business Strategy: Would inventory reduction result in significant cost savings? Would CT (or LT) reduction result in significant competitive advantage? Would TH increase help generate significantly more revenue? Would improved customer service generate business over the long run? Remember – standards change over time!

Capacity Laws Capacity Law: In steady state, all plants will release work at an average rate that is strictly less than average capacity. Utilization Law: If a station increases utilization without making any other change, average WIP and cycle time will increase in a highly nonlinear fashion. Notes: Cannot run at full capacity (including overtime, etc.) Failure to recognize this leads to “fire fighting”

Cycle Time vs. Utilization

What Really Happens: System with Insufficient Capacity

What Really Happens: Two Cases with Releases at 100% of Capacity

What Really Happens: Two Cases with Releases at 82% of Capacity

Overtime Vicious Cycle Release work at plant capacity. Variability causes WIP to increase. Jobs are late, customers complain,… Authorize one-time use of overtime. WIP falls, cycle times go down, backlog is reduced. Breathe sigh of relief. Go to Step 1!

Mechanics of Overtime Vicious Cycle

Influence of Variability Variability Law: Increasing variability always degrades the performance of a production system. Examples: process time variability pushes best case toward worst case higher demand variability requires more safety stock for same level of customer service higher cycle time variability requires longer lead time quotes to attain same level of on-time delivery

Variability Buffering Buffering Law: Systems with variability must be buffered by some combination of: 1. inventory 2. capacity 3. time. Interpretation: If you cannot pay to reduce variability, you will pay in terms of high WIP, under-utilized capacity, or reduced customer service (i.e., lost sales, long lead times, and/or late deliveries).

Variability Buffering Examples Ballpoint Pens: can’t buffer with time (who will backorder a cheap pen?) can’t buffer with capacity (too expensive, and slow) must buffer with inventory Ambulance Service: can’t buffer with inventory (stock of emergency services?) can’t buffer with time (violates strategic objectives) must buffer with capacity Organ Transplants: can’t buffer with WIP (perishable) can’t buffer with capacity (ethically anyway) must buffer with time

Simulation Studies TH Constrained System (push) 1 2 3 4 ra, ca B(1)= te(1), ce(1) B(2)= te(2), ce(2) B(3)= te(3), ce(3) B(4)= te(4), ce(4) WIP Constrained System (pull) Infinite raw materials 1 2 3 4 te(1), ce(1) B(2) te(2), ce(2) B(3) te(3), ce(3) B(4) te(4), ce(4)

Variability in Push Systems Notes: ra = 0.8, ca = ce(i) in all cases. B(i) = , i = 1-4 in all cases. Observations: TH is set by release rate in a push system. Increasing capacity (rb) reduces need for WIP buffering. Reducing process variability reduces WIP, CT, and CT variability for a given throughput level.

Variability in Pull Systems Notes: Station 1 pulls in job whenever it becomes empty. B(i) = 0, i = 1, 2, 4 in all cases, except case 6, which has B(2) = 1.

Variability in Pull Systems (cont.) Observations: Capping WIP without reducing variability reduces TH. WIP cap limits effect of process variability on WIP/CT. Reducing process variability increases TH, given same buffers. Adding buffer space at bottleneck increases TH. Magnitude of impact of adding buffers depends on variability. Buffering less helpful at non-bottlenecks. Reducing process variability reduces CT variability. Conclusion: consequences of variability are different in push and pull systems, but in either case the buffering law implies that you will pay for variability somehow.

Example – Discrete Parts Flowline process buffer process buffer process Inventory Buffers: raw materials, WIP between processes, FGI Capacity Buffers: overtime, equipment capacity, staffing Time Buffers: frozen zone, time fences, lead time quotes Variability Reduction: smaller WIP & FGI , shorter cycle times

Example – Batch Chemical Process reactor column reactor column reactor column tank tank Inventory Buffers: raw materials, WIP in tanks, finished goods Capacity Buffers: idle time at reactors Time Buffers: lead times in supply chain Variability Reduction: WIP is tightly constrained, so target is primarily throughput improvement, and maybe FGI reduction.

Example – Moving Assembly Line in-line buffer fabrication lines final assembly line Inventory Buffers: components, in-line buffers Capacity Buffers: overtime, rework loops, warranty repairs Time Buffers: lead time quotes Variability Reduction: initially directed at WIP reduction, but later to achieve better use of capacity (e.g., more throughput)

Buffer Flexibility Buffer Flexibility Corollary: Flexibility reduces the amount of variability buffering required in a production system. Examples: Flexible Capacity: cross-trained workers Flexible Inventory: generic stock (e.g., assemble to order) Flexible Time: variable lead time quotes

Variability from Batching VUT Equation: CT depends on process variability and flow variability Batching: affects flow variability affects waiting inventory Conclusion: batching is an important determinant of performance

Process Batch Versus Move Batch Dedicated Assembly Line: What should the batch size be? Process Batch: Related to length of setup. The longer the setup the larger the lot size required for the same capacity. Move (transfer) Batch: Why should it equal process batch? The smaller the move batch, the shorter the cycle time. The smaller the move batch, the more material handling. Lot Splitting: Move batch can be different from process batch. 1. Establish smallest economical move batch. 2. Group batches of like families together at bottleneck to avoid setups. 3. Implement using a “backlog”.

Process Batching Effects Types of Process Batching: 1. Serial Batching: processes with sequence-dependent setups “batch size” is number of jobs between setups batching used to reduce loss of capacity from setups 2. Parallel Batching: true “batch” operations (e.g., heat treat) “batch size” is number of jobs run together batching used to increase effective rate of process

Process Batching Process Batching Law: In stations with batch operations or significant changeover times: The minimum process batch size that yields a stable system may be greater than one. As process batch size becomes large, cycle time grows proportionally with batch size. Cycle time at the station will be minimized for some process batch size, which may be greater than one. Basic Batching Tradeoff: WIP versus capacity

Serial Batching Parameters: Time to process batch: te = kt + s ts k t0 setup t0 ra,ca forming batch queue of batches te = 10(1) + 5 = 15

Process Batching Effects (cont.) Arrival rate of batches: ra/k Utilization: u = (ra/k)(kt + s) For stability: u < 1 requires ra = 0.4/10 = 0.04 u = 0.04(10·1+5) = 0.6 minimum batch size required for stability of system...

Process Batching Effects (cont.) Average queue time at station: Average cycle time depends on move batch size: Move batch = process batch Move batch = 1 Note: we assume arrival CV of batches is ca regardless of batch size – an approximation... Note: splitting move batches reduces wait for batch time.

Cycle Time vs. Batch Size – 5 hr setup Optimum Batch Sizes

Cycle Time vs. Batch Size – 2.5 hr setup Optimum Batch Sizes

Setup Time Reduction Where? Steps: Stations where capacity is expensive Excess capacity may sometimes be cheaper Steps: 1. Externalize portions of setup 2. Reduce adjustment time (guides, clamps, etc.) 3. Technological advancements (hoists, quick-release, etc.) Caveat: Don’t count on capacity increase; more flexibility will require more setups.

Parallel Batching Parameters: Time to form batch: Time to process batch: te = t t ra,ca k W = ((10 – 1)/2)(1/0.005) = 90 forming batch queue of batches te = 90

Parallel Batching (cont.) Arrival of batches: ra/k Utilization: u = (ra/k)(t) For stability: u < 1 requires ra/k = 0.05/10 = 0.005 u = (0.005)(90) = 0.45 minimum batch size required for stability of system... k > 0.05(90) = 4.5

Parallel Batching (cont.) Average wait-for-batch time: Average queue plus process time at station: Total cycle time: batch size affects both wait-for-batch time and queue time

Cycle Time vs. Batch Size in a Parallel Operation queue time due to utilization wait for batch time Optimum Batch Size B

Variable Batch Sizes Observation: Waiting for full batch in parallel batch operation may not make sense. Could just process whatever is there when operation becomes available. Example: Furnace has space for 120 wrenches Heat treat requires 1 hour Demand averages 100 wrenches/hr Induction coil can heat treat 1 wrench in 30 seconds What is difference between performance of furnace and coil?

Variable Batch Sizes (cont.) Furnace: Ignoring queueing due to variability Process starts every hour 100 wrenches in furnace 50 wrenches waiting on average 150 total wrenches in WIP CT = WIP/TH = 150/100 = 3/2 hr = 90 min Induction Coil: Capacity same as furnace (120 wrenches/hr), but CT = 0.5 min = 0.0083 hr WIP = TH × CT = 100 × 0.0083 = 0.83 wrenches Conclusion: Dramatic reduction in WIP and CT due to small batches—independent of variability or other factors. 100 50

Move Batching Move Batching Law: Cycle times over a segment of a routing are roughly proportional to the transfer batch sizes used over that segment, provided there is no waiting for the conveyance device. Insights: Basic Batching Tradeoff: WIP vs. move frequency Queueing for conveyance device can offset CT reduction from reduced move batch size Move batching intimately related to material handling and layout decisions

Move Batching Problem: Two machines in series First machine receives individual parts at rate ra with CV of ca(1) and puts out batches of size k. First machine has mean process time of te(1) for one part with CV of ce(1). Second machine receives batches of k and put out individual parts. How does cycle time depend on the batch size k? te(1),ce(1) k ra,ca(1) te(2),ce(2) single job batch Station 1 Station 2

Move Batching Calculations Time at First Station: Average time before batching is: Average time forming the batch is: Average time spent at the first station is: regular VUT equation... first part waits (k-1)(1/ra), last part doesn’t wait, so average is (k-1)(1/ra)/2

Move Batching Calculations (cont.) Output of First Station: Time between output of individual parts into the batch is ta. Time between output of batches of size k is kta. Variance of interoutput times of parts is cd2(1)ta2, where Variance of batches of size k is kcd2(1)ta2. SCV of batch arrivals to station 2 is: because cd2(1)=d2/ta2 by def of CV because departures are independent, so variances add variance divided by mean squared...

Move Batching Calculations (cont.) Time at Second Station: Time to process a batch of size k is kte(2). Variance of time to process a batch of size k is kce2(2)te2(2). SCV for a batch of size k is: Mean time spent in partial batch of size k is: So, average time spent at the second station is: independent process times... first part doesn’t wait, last part waits (k-1)te(2), so average is (k-1)te(2)/2 VUT equation to compute queue time of batches...

Move Batching Calculations (cont.) Total Cycle Time: Insight: Cycle time increases with k. Inflation term does not involve CV’s Congestion from batching is more bad control than randomness. inflation factor due to move batching

Assembly Operations Assembly Operations Law: The performance of an assembly station is degraded by increasing any of the following: Number of components being assembled. Variability of component arrivals. Lack of coordination between component arrivals. Observations: This law can be viewed as special instance of variability law. Number of components affected by product/process design. Arrival variability affected by process variability and production control. Coordination affected by scheduling and shop floor control.

Attacking Variability Objectives reduce cycle time increase throughput improve customer service Levers reduce variability directly buffer using inventory buffer using capacity buffer using time increase buffer flexibility

Cycle Time Definition (Station Cycle Time): The average cycle time at a station is made up of the following components: cycle time = move time + queue time + setup time + process time + wait-to-batch time + wait-in-batch time + wait-to-match time Definition (Line Cycle Time): The average cycle time in a line is equal to the sum of the cycle times at the individual stations less any time that overlaps two or more stations. delay times typically make up 90% of CT

Reducing Queue Delay CTq = V U t Reduce Variability failures setups uneven arrivals, etc. Reduce Utilization arrival rate (yield, rework, etc.) process rate (speed, time, availability, etc)

Reducing Batching Delay CTbatch = delay at stations + delay between stations Reduce Process Batching Optimize batch sizes Reduce setups Stations where capacity is expensive Capacity vs. WIP/CT tradeoff Reduce Move Batching Move more frequently Layout to support material handling (e.g., cells)

Reducing Matching Delay CTbatch = delay due to lack of synchronization Reduce Variability on high utilization fabrication lines usual variability reduction methods Improve Coordination scheduling pull mechanisms modular designs Reduce Number of Components product redesign kitting

Increasing Throughput TH = P(bottleneck is busy)  bottleneck rate Reduce Blocking/Starving buffer with inventory (near bottleneck) reduce system “desire to queue” Increase Capacity add equipment increase operating time (e.g. spell breaks) increase reliability reduce yield loss/rework CTq = V U t Reduce Variability Reduce Utilization Note: if WIP is limited, then system degrades via TH loss rather than WIP/CT inflation

Customer Service Elements of Customer Service: lead time fill rate (% of orders delivered on-time) quality Law (Lead Time): The manufacturing lead time for a routing that yields a given service level is an increasing function of both the mean and standard deviation of the cycle time of the routing.

Improving Customer Service LT = CT + z CT Reduce CT Visible to Customer delayed differentiation assemble to order stock components Reduce Average CT queue time batch time match time Reduce CT Variability generally same as methods for reducing average CT: improve reliability improve maintainability reduce labor variability improve quality improve scheduling, etc.

Cycle Time and Lead Time CT = 10 CT = 3 CT = 10 CT = 6

Diagnostics Using Factory Physics® Situation: Two machines in series; machine 2 is bottleneck ca2 = 1 Machine 1: Machine 2: Space at machine 2 for 20 jobs of WIP Desired throughput 2.4 jobs/hr, not being met

Diagnostic Example (cont.) Proposal: Install second machine at station 2 Expensive Very little space Analysis Tools: Analysis: Step 1: At 2.4 job/hr CTq at first station is 645 minutes, average WIP is 25.8 jobs. CTq at second station is 892 minutes, average WIP is 35.7 jobs. Space requirements at machine 2 are violated! VUT equation propogation equation Ask why five times...

Diagnostic Example (cont.) Step 2: Why is CTq at machine 2 so big? Break CTq into The 23.11 min term is small. The 12.22 correction term is moderate (u  0.9244) The 3.16 correction is large. Step 3: Why is the correction term so large? Look at components of correction term. ce2 = 1.04, ca2 = 5.27. Arrivals to machine are highly variable.

Diagnostic Example (cont.) Step 4: Why is ca2 to machine 2 so large? Recall that ca2 to machine 2 equals cd2 from machine 1, and ce2 at machine 1 is large. Step 5: Why is ce2 at machine 1 large? Effective CV at machine 1 is affected by failures, The inflation due to failures is large. Reducing MTTR at machine 1 would substantially improve performance.

Procoat Case – Situation Problem: Current WIP around 1500 panels Desired capacity of 3000 panels/day (19.5 hr day with breaks/lunches) Typical output of 1150 panels/day Outside vendor being used to make up slack Proposal: Expose is bottleneck, but in clean room Expansion would be expensive Suggested alternative is to add bake oven for touchups

Procoat Case – Layout IN OUT Loader Clean Coat 1 Coat 2 Unloader Touchup D&I Inspect Unloader Bake Develop Loader Manufacturing Inspect Expose Clean Room OUT

Procoat Case – Capacity Calculations rb = 2,879 p/day T0 = 546 min = 0.47 days W0 = rbT0 = 1,343 panels

Procoat Case – Benchmarking TH Resulting from PWC with WIP = 1,500: Conclusion: actual system is significantly worse than PWC. Higher than actual TH Question: what to do?

Procoat Case – Factory Physics® Analysis Bottleneck Capacity - rate: - time: Bottleneck Starving - process variability: - flow variability: (Expose) operator training, setup reduction break spelling, shift changes reduces “desire to queue” so that clean room buffer is adequate operator training coater line – field ready replacements

Procoat Case – Outcome 3300 3000 Best Case 2700 "Good" Region After 2400 Practical Worst Case 2100 1800 TH (panels/day) "Bad" Region 1500 1200 Before 900 600 300 Worst Case -300 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 WIP (panels)

Corrupting Influence Takeaways Variance Degrades Performance: many sources of variability planned and unplanned Variability Must be Buffered: inventory capacity time Flexibility Reduces Need for Buffering: still need buffers, but smaller ones

Corrupting Influence Takeaways (cont.) Variability and Utilization Interact: congestion effects multiply utilization effects are highly nonlinear importance of bottleneck management Batching is an Important Source of Variability: process and move batching serial and parallel batching wait-to-batch time in addition to variability effects

Corrupting Influence Takeaways (cont.) Assembly Operations Magnify Impact of Variability: wait-to-match time caused by lack of synchronization Variability Propagates: flow variability is as disruptive as process variability non-bottlenecks can be major problems

TM 663 Operations Planning October 31, 2011 Paula Jensen Chapter 9: The Corrupting Influence of Variability(continued) Chapter 10:Push & Pull Production Systems

Agenda Factory Physics (New Assignment Chapter 8: Problem 6, 8 Chapter 9: Problems 1-4 Chapter 10: Problems 1, 2, 3, 5) Video

Setup Time Reduction Where? Steps: Stations where capacity is expensive Excess capacity may sometimes be cheaper Steps: 1. Externalize portions of setup 2. Reduce adjustment time (guides, clamps, etc.) 3. Technological advancements (hoists, quick-release, etc.) Caveat: Don’t count on capacity increase; more flexibility will require more setups.

Parallel Batching Parameters: Time to form batch: Time to process batch: te = t t ra,ca k W = ((10 – 1)/2)(1/0.005) = 90 forming batch queue of batches te = 90

Parallel Batching (cont.) Arrival of batches: ra/k Utilization: u = (ra/k)(t) For stability: u < 1 requires ra/k = 0.05/10 = 0.005 u = (0.005)(90) = 0.45 minimum batch size required for stability of system... k > 0.05(90) = 4.5

Parallel Batching (cont.) Average wait-for-batch time: Average queue plus process time at station: Total cycle time: batch size affects both wait-for-batch time and queue time

Cycle Time vs. Batch Size in a Parallel Operation queue time due to utilization wait for batch time Optimum Batch Size B

Variable Batch Sizes Observation: Waiting for full batch in parallel batch operation may not make sense. Could just process whatever is there when operation becomes available. Example: Furnace has space for 120 wrenches Heat treat requires 1 hour Demand averages 100 wrenches/hr Induction coil can heat treat 1 wrench in 30 seconds What is difference between performance of furnace and coil?

Variable Batch Sizes (cont.) Furnace: Ignoring queueing due to variability Process starts every hour 100 wrenches in furnace 50 wrenches waiting on average 150 total wrenches in WIP CT = WIP/TH = 150/100 = 3/2 hr = 90 min Induction Coil: Capacity same as furnace (120 wrenches/hr), but CT = 0.5 min = 0.0083 hr WIP = TH × CT = 100 × 0.0083 = 0.83 wrenches Conclusion: Dramatic reduction in WIP and CT due to small batches—independent of variability or other factors. 100 50

Move Batching Move Batching Law: Cycle times over a segment of a routing are roughly proportional to the transfer batch sizes used over that segment, provided there is no waiting for the conveyance device. Insights: Basic Batching Tradeoff: WIP vs. move frequency Queueing for conveyance device can offset CT reduction from reduced move batch size Move batching intimately related to material handling and layout decisions

Move Batching Problem: Two machines in series First machine receives individual parts at rate ra with CV of ca(1) and puts out batches of size k. First machine has mean process time of te(1) for one part with CV of ce(1). Second machine receives batches of k and put out individual parts. How does cycle time depend on the batch size k? te(1),ce(1) k ra,ca(1) te(2),ce(2) single job batch Station 1 Station 2

Move Batching Calculations Time at First Station: Average time before batching is: Average time forming the batch is: Average time spent at the first station is: regular VUT equation... first part waits (k-1)(1/ra), last part doesn’t wait, so average is (k-1)(1/ra)/2

Move Batching Calculations (cont.) Output of First Station: Time between output of individual parts into the batch is ta. Time between output of batches of size k is kta. Variance of interoutput times of parts is cd2(1)ta2, where Variance of batches of size k is kcd2(1)ta2. SCV of batch arrivals to station 2 is: because cd2(1)=d2/ta2 by def of CV because departures are independent, so variances add variance divided by mean squared...

Move Batching Calculations (cont.) Time at Second Station: Time to process a batch of size k is kte(2). Variance of time to process a batch of size k is kce2(2)te2(2). SCV for a batch of size k is: Mean time spent in partial batch of size k is: So, average time spent at the second station is: independent process times... first part doesn’t wait, last part waits (k-1)te(2), so average is (k-1)te(2)/2 VUT equation to compute queue time of batches...

Move Batching Calculations (cont.) Total Cycle Time: Insight: Cycle time increases with k. Inflation term does not involve CV’s Congestion from batching is more bad control than randomness. inflation factor due to move batching

Assembly Operations Assembly Operations Law: The performance of an assembly station is degraded by increasing any of the following: Number of components being assembled. Variability of component arrivals. Lack of coordination between component arrivals. Observations: This law can be viewed as special instance of variability law. Number of components affected by product/process design. Arrival variability affected by process variability and production control. Coordination affected by scheduling and shop floor control.

Attacking Variability Objectives reduce cycle time increase throughput improve customer service Levers reduce variability directly buffer using inventory buffer using capacity buffer using time increase buffer flexibility

Cycle Time Definition (Station Cycle Time): The average cycle time at a station is made up of the following components: cycle time = move time + queue time + setup time + process time + wait-to-batch time + wait-in-batch time + wait-to-match time Definition (Line Cycle Time): The average cycle time in a line is equal to the sum of the cycle times at the individual stations less any time that overlaps two or more stations. delay times typically make up 90% of CT

Reducing Queue Delay CTq = V U t Reduce Variability failures setups uneven arrivals, etc. Reduce Utilization arrival rate (yield, rework, etc.) process rate (speed, time, availability, etc)

Reducing Batching Delay CTbatch = delay at stations + delay between stations Reduce Process Batching Optimize batch sizes Reduce setups Stations where capacity is expensive Capacity vs. WIP/CT tradeoff Reduce Move Batching Move more frequently Layout to support material handling (e.g., cells)

Reducing Matching Delay CTbatch = delay due to lack of synchronization Reduce Variability on high utilization fabrication lines usual variability reduction methods Improve Coordination scheduling pull mechanisms modular designs Reduce Number of Components product redesign kitting

Increasing Throughput TH = P(bottleneck is busy)  bottleneck rate Reduce Blocking/Starving buffer with inventory (near bottleneck) reduce system “desire to queue” Increase Capacity add equipment increase operating time (e.g. spell breaks) increase reliability reduce yield loss/rework CTq = V U t Reduce Variability Reduce Utilization Note: if WIP is limited, then system degrades via TH loss rather than WIP/CT inflation

Customer Service Elements of Customer Service: lead time fill rate (% of orders delivered on-time) quality Law (Lead Time): The manufacturing lead time for a routing that yields a given service level is an increasing function of both the mean and standard deviation of the cycle time of the routing.

Improving Customer Service LT = CT + z CT Reduce CT Visible to Customer delayed differentiation assemble to order stock components Reduce Average CT queue time batch time match time Reduce CT Variability generally same as methods for reducing average CT: improve reliability improve maintainability reduce labor variability improve quality improve scheduling, etc.

Cycle Time and Lead Time CT = 10 CT = 3 CT = 10 CT = 6

Diagnostics Using Factory Physics® Situation: Two machines in series; machine 2 is bottleneck ca2 = 1 Machine 1: Machine 2: Space at machine 2 for 20 jobs of WIP Desired throughput 2.4 jobs/hr, not being met

Diagnostic Example (cont.) Proposal: Install second machine at station 2 Expensive Very little space Analysis Tools: Analysis: Step 1: At 2.4 job/hr CTq at first station is 645 minutes, average WIP is 25.8 jobs. CTq at second station is 892 minutes, average WIP is 35.7 jobs. Space requirements at machine 2 are violated! VUT equation propogation equation Ask why five times...

Diagnostic Example (cont.) Step 2: Why is CTq at machine 2 so big? Break CTq into The 23.11 min term is small. The 12.22 correction term is moderate (u  0.9244) The 3.16 correction is large. Step 3: Why is the correction term so large? Look at components of correction term. ce2 = 1.04, ca2 = 5.27. Arrivals to machine are highly variable.

Diagnostic Example (cont.) Step 4: Why is ca2 to machine 2 so large? Recall that ca2 to machine 2 equals cd2 from machine 1, and ce2 at machine 1 is large. Step 5: Why is ce2 at machine 1 large? Effective CV at machine 1 is affected by failures, The inflation due to failures is large. Reducing MTTR at machine 1 would substantially improve performance.

Procoat Case – Situation Problem: Current WIP around 1500 panels Desired capacity of 3000 panels/day (19.5 hr day with breaks/lunches) Typical output of 1150 panels/day Outside vendor being used to make up slack Proposal: Expose is bottleneck, but in clean room Expansion would be expensive Suggested alternative is to add bake oven for touchups

Procoat Case – Layout IN OUT Loader Clean Coat 1 Coat 2 Unloader Touchup D&I Inspect Unloader Bake Develop Loader Manufacturing Inspect Expose Clean Room OUT

Procoat Case – Capacity Calculations rb = 2,879 p/day T0 = 546 min = 0.47 days W0 = rbT0 = 1,343 panels

Procoat Case – Benchmarking TH Resulting from PWC with WIP = 1,500: Conclusion: actual system is significantly worse than PWC. Higher than actual TH Question: what to do?

Procoat Case – Factory Physics® Analysis Bottleneck Capacity - rate: - time: Bottleneck Starving - process variability: - flow variability: (Expose) operator training, setup reduction break spelling, shift changes reduces “desire to queue” so that clean room buffer is adequate operator training coater line – field ready replacements

Procoat Case – Outcome 3300 3000 Best Case 2700 "Good" Region After 2400 Practical Worst Case 2100 1800 TH (panels/day) "Bad" Region 1500 1200 Before 900 600 300 Worst Case -300 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 WIP (panels)

Corrupting Influence Takeaways Variance Degrades Performance: many sources of variability planned and unplanned Variability Must be Buffered: inventory capacity time Flexibility Reduces Need for Buffering: still need buffers, but smaller ones

Corrupting Influence Takeaways (cont.) Variability and Utilization Interact: congestion effects multiply utilization effects are highly nonlinear importance of bottleneck management Batching is an Important Source of Variability: process and move batching serial and parallel batching wait-to-batch time in addition to variability effects

Corrupting Influence Takeaways (cont.) Assembly Operations Magnify Impact of Variability: wait-to-match time caused by lack of synchronization Variability Propagates: flow variability is as disruptive as process variability non-bottlenecks can be major problems

Push and Pull Production Systems You say yes. I say no. You say stop. and I say go, go, go! – The Beatles

The Key Difference Between Push and Pull Push Systems: schedule work releases based on demand. inherently due-date driven control release rate, observe WIP level Pull Systems: authorize work releases based on system status. inherently rate driven control WIP level, observe throughput

Push vs. Pull Mechanics PUSH PULL Push systems do not limit (Exogenous) Schedule (Endogenous) Stock Void Production Process Production Process Job Job Push systems do not limit WIP in the system. Pull systems deliberately establish a limit on WIP.

What Pull is Not! Make-to-Order: Make-to-Stock: Forecast Free: MRP with firm orders on MPS is make-to-order. But it does not limit WIP and is therefore a push system. Make-to-Stock: Pull systems do replenish inventory voids. But jobs can be associated with customer orders. Forecast Free: Toyota’s classic system made cars to forecasts. Use of takt times or production smoothing often involves production without firm orders (and hence forecasts).

Push and Pull Examples Are the following systems essentially push or essentially pull? Kinko’s copy shop: Soda vending machine: “Pure” MRP system: Doctor’s office: Supermarket (goods on shelves): Tandem line with finite interstation buffers: Runway at O’Hare during peak periods: Order entry server at Amazon.com: PUSH PULL PUSH PUSH – into office, PULL into exam room PULL PULL PULL PUSH

Push and Pull Line Schematics Pure Push (MRP) Stock Point . . . Stock Point Pure Pull (Kanban) Stock Point . . . Stock Point … CONWIP Stock Point . . . Stock Point Authorization Signals Full Containers

Pulling with Kanban Outbound stockpoint Outbound stockpoint Production Completed parts with cards enter outbound stockpoint. Production cards When stock is removed, place production card in hold box. Production card authorizes start of work.

Inventory/Order Interface Concept: Make-to-stock and make-to-order can be used in same system. Dividing point is called the inventory/order interface. This is sometimes called the push/pull interface, but since WIP could be limited or unlimited in both segments, this is not a strictly accurate term. Benefit: eliminate entire portion of cycle time seen by customers by building to stock. Implementation: kanban late customization (postponement)

Example – Custom Taco Production Line I/O Interface Make-to-Stock Make-to-Order Refrigerator Cooking Assembly Packaging Sales Customer

Example – Quick Taco Production Line I/O Interface Make-to-Order Make-to-Stock Refrigerator Cooking Assembly Packaging Sales Warming Table Customer Notes: I/O interface can differ by time of day (or season). I/O interface can differ by product.

Example – IBM Panel Plant Original Line Treater Prepreg, Copper Lamination Machining Circuitize Drilling Copper Plate Procoat Sizing, Test I/O Interface process that gives boards “personality” Revised Line Treater Prepreg, Copper Lamination Machining Core Blanks Circuitize Drilling Copper Plate Procoat I/O Interface Sizing, Test Notes: Moving I/O interface closer to customer shortens leadtime seen by customer. Small number of core blanks presents opportunity to make them to stock.

Example – HP Deskjet Supply Chain U.S. DC Customer Integrated Circuit Manufacturing Printed Circuit Assembly & Test Final Assembly and Test European DC Customer Far East DC Customer I/O Interface Notes: I/O interface located in markets to achieve quick response to customers Delayed differentiation of products (power supplies for different countries) enables pooling of safety stocks

I/O Interface Conclusions Basic Tradeoff: responsiveness vs. inventory (time vs. money) moving PPI closer to customer increases responsiveness and (usually) inventory Optimal Position of I/O Interface: need for responsiveness cost of carrying inventory  product diversification Levers: product design (postponement) process design (quick response manufacturing)

Advantages of Pull Systems Low Unit Cost: low inventory reduced space little rework High External Quality: high internal quality pressure for good quality promotion of good quality (e.g., defect detection) Good Customer Service: short cycle times steady, predictable output stream Flexibility: avoids committing jobs too early encourages floating capacity

The Magic of Pull Pulling Everywhere? – Hall 1983 You don’t never make nothin’ and send it no place. Somebody has to come get it. – Hall 1983 No! It’s the WIP Cap: Kanban – WIP cannot exceed number of cards “WIP explosions” are impossible WIP

Pull Benefits Achieved by WIP Cap Reduces Costs: prevents WIP explosions reduces average WIP reduces engineering changes Improves Quality: pressure for higher quality improved defect detection improved communication Improves Customer Service: reduces cycle time variability pressure to reduce sources of process variability promotes shorter lead times and better on-time performance Maintains Flexibility: avoids early release (like air traffic control) less direct congestion less reliance on forecasts promotes floating capacity

CONWIP Assumptions: 1. Single routing 2. WIP measured in units Mechanics: allow next job to enter line each time a job leaves (i.e., maintain a WIP level of m jobs in the line at all times). Modeling: MRP looks like an open queueing network CONWIP looks like a closed queueing network Kanban looks like a closed queueing network with blocking . . .

CONWIP Controller . . . LAN PC PC Work Backlog Indicator Lights PN Quant –— ––––– LAN Indicator Lights R G PC PC . . . Workstations

CONWIP vs. Pure Push Push/Pull Laws: A CONWIP system has the following advantages over an equivalent pure push system: 1) Observability: WIP is observable; capacity is not. 2) Efficiency: A CONWIP system requires less WIP on average to attain a given level of throughput. 3) Robustness: A profit function of the form Profit = pTh - hWIP is more sensitive to errors in TH than WIP.

CONWIP Efficiency Example Equipment Data: 5 machines in tandem, all with capacity of one part/hr (u=TH·te=TH) exponential (moderate variability) process times CONWIP System: Pure Push System: PWC formula 5 M/M/1 queues

CONWIP Efficiency Example (cont.) How much WIP is required for push to match TH attained by CONWIP system with WIP=w? In this example, WIP is always 25% higher for same TH in push than in CONWIP In general, the increase won’t always be 25%, but it will always take more WIP to get same TH under push than under pull.

CONWIP Robustness Example Profit Function: CONWIP: Push: Key Question: what happens when we don’t choose optimum values (as we never will)? need to find “optimal” WIP level need to find “optimal” TH level (i.e., release rate)

CONWIP vs. Pure Push Comparisons Optimum CONWIP Efficiency Robustness Push

Modeling CONWIP with Mean-Value Analysis Notation: Basic Approach: Compute performance measures for increasing w assuming job arriving to line “sees” other jobs distributed according to average behavior with w-1 jobs.

Mean-Value Analysis Formulas Starting with WIPj(0)=0 and TH(0)=0, compute for w=1,2,…

Computing Inputs for MVA

Output of MVA

Using MVA to Evaluate Line Performance

Implementing Pull Pull is Rigid: replenishing stocks quickly (just in time) requires level mix, volume, sequence. JIT Practices: Support Rigidity: production smoothing/mix stabilization Mitigate Rigidity in Production System capacity buffers setup reduction flexible labor facility layout product design (postponement, etc.) Mitigate Rigidity in Organization TQM vendor management, etc. this is the “genius” of pull!

Capacity Buffers Motivation: facilitate rapid replenishments with minimal WIP Benefits: Protection against quota shortfalls Regular flow allows matching against customer demands Can be more economical in long run than WIP buffers in push systems Techniques: Planned underutilization (e.g., use u = 75% in aggregate planning) Two shifting: 4 – 8 – 4 – 8 Schedule dummy jobs to allow quick response to hot jobs

Setup Reduction Motivation: Small lot sequences not feasible with large setups. Internal vs. External Setups: External – performed while machine is still running Internal – performed while machine is down Approach: 1. Separate the internal setup from the external setup. 2. Convert as much internal setup as possible to external setup. 3. Eliminate the adjustment process. 4. Abolish the setup itself (e.g., uniform product design, combined production, parallel machines).

Flexible Labor Cross-Trained Workers: Shared Tasks: float where needed appreciate line-wide perspective provide more heads per problem area Shared Tasks: can be done by adjacent stations reduces variability in tasks, and hence line stoppages/quality problems work can float to workers, or workers can float to work…

Cellular Layout Advantages: Challenges: Better flow control Improved material handling (smaller transfer batches) Ease of communication (e.g., for floating labor) Challenges: May require duplicate equipment Product to cell assignment Inbound Stock Outbound Stock

Focused Factories Pareto Analysis: Dedicated Lines: Small percentage of sku’s represent large percentage of volume Large percentage of sku’s represent little volume but much complexity Dedicated Lines: for families of high runners few setups can use pull effectively Job Shop Environment: for low runners many setups poorer performance, but only on smaller portion of business may need to use push Saw Lathe Mill Drill Saw Mill Drill Paint Stores Assembly Warehouse Grind Mill Drill Paint Weld Grind Lathe Drill Saw Grind Paint Stores Lathe Assembly Warehouse Mill Drill

Push/Pull Takeaways Magic of Pull: the WIP cap MTS/MTO Hybrids: locating the I/O interface Logistical Benefits of Pull: observability efficiency robustness (this is the key one) Overcoming Rigidity of Pull: capacity buffers setup reduction flexible labor facility layout, etc.

TM 663 Operations Planning November 7, 2011 Paula Jensen Chapter 11: The Human Element in Operations Management Chapter 12: Total Quality Manufacturing

IIE Meeting 11/8 In the classroom on 2nd floor! 11:00 Free Pizza for lunch! We will be discussing conference, socials, and much more! See you there!!

Agenda Schedule Factory Physics (New Assignment Chapter 11: Study Q’s 1-4 Chapter 12: Problems 1, 2, 4, 5)

The Human Element in Operations Management For as laws are necessary that good manners may be preserved, so there is a need of good manners that laws may be maintained. – Machiavelli We hold these truths to be self-evident. – Thomas Jefferson

Operations Management Frameworks Traditional Optimization Framework: perfect information perfect control leverage in quality of solution (policy) Factory Physics Framework: information captured in key measures (e.g., SCV's) intuition more important than control leverage from working with system's natural tendencies ICB Portfolio Framework: information system part of management problem control not always optimal buffers explicitly acknowledged

ICB Portfolio Contrasts

Human Connections Information: Buffers: Control: Implementing Change: complexity “off line” information Buffers: conceptual understanding flexibility Incentives: piecework “real” measures Control: skill levels learning curves Implementing Change: burnout champions

Self-Interest Implications: Self-Interest Law: People, not organizations are self-optimizing. Implications: “Optimal” strategies may not produce optimal results. Constraints can be good!

Relaxing Constraints in Optimization Problem

Diversity Theory X vs. Theory Y: Incentive Systems: Individuality Law: People are different. Theory X vs. Theory Y: empowerment officer vs. enlisted mentality Incentive Systems: team-oriented incentives social component of work

Toyota Sewn Products System Note: performance best when workers arranged slowest to fastest (i.e., because blocking is minimized).

Advocacy Upside of Champions: Downside of Champions: Advocacy Law: For any program, there exists a champion who can make it work—at least for a while. Upside of Champions: selling the “vision” motivating the troops Downside of Champions: risk of oversell overreliance can prevent institutionalization of change

Burnout Burnout Law: People get burned out. Why? Can you blame them? JIT ABC gurus ERP OPT TQM FMS MBO BPR FMS MBO SCM FMS benchmarking

Planning vs. Motivating Question: how high to set the bar?

Responsibility and Authority Responsibility Law: Responsibility without commensurate authority is demoralizing and counterproductive. Example: Deming's Red Bead Experiment

Deming’s Red Bead Experiment

Human Element Takeaways People act according to self-interest. Individuals are different. Champions can have powerful positive and negative consequences. People can burn out. There is a difference between planning and motivating. Responsibility should be commensurate with authority.

Total Quality Manufacturing Saw it on the tube Bought it on the phone Now you're home alone It's a piece of crap. I tried to plug it in I tried to turn it on When I got it home It was a piece of crap. – Neil Young

The Opportunity Rhetoric: Reality: customer-driven quality quality circles SQC courses “quality speak” Reality: many poor products unbelievably rude service uncoordinated use of SQC complacency?

The Opportunity (cont.) Quality Implications: quality promotes cycle time reduction and vice versa quality promotes variability reduction and vice versa quality promotes better management and vice versa Cycle Time Variability QUALITY Management

Attributes of Quality Quality Definitions: Transcendent: innate excellence or “I know it when I see it” view. Product-based: function of product attributes or “more is better” view. User-based: customer satisfaction or “beauty is in the eye of the beholder” view. Manufacturing-based: conformance to specifications, related to “do it right the first time” view. Value-based: price/performance or “affordable excellence” view.

Attributes of Quality (cont.) Customer Orientation: customer satisfaction depends on external quality external quality depends on internal quality quality must address product, process, system Promoting Internal Quality: error prevention inspection improvement environment enhancement

Dimensions of Quality Performance Features Serviceability Aesthetics Perceived Quality Reliability Conformance Durability Quality of design ........................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................ Quality of process conformance to design = process capability

Statistical Quality Control Acceptance Sampling: 100% inspection statistical sampling Process Control: continuous monitoring indication of “out of control” Design of Experiments: trace causes of problems many tools (factorial, block, nested designs, Taguchi, etc.)

Statistical Process Control Natural Variation relatively small due to uncontrollable sources Assignable Cause Variation larger can be traced to causes cause process to be out of control Challenge of SPC: separate assignable cause from natural variation.

Basic SPC Mechanics Null Hypothesis: samples are coming from a process with mean  and standard deviation . Procedure: Observe samples of size n. Under null hypothesis, these will have mean  and standard deviation . Compare sample mean, to control limits: If sample mean is outside of range between LCL and UCL, then observation is designated as assignable cause variation, indicating out-of-control situation.

SPC Example Problem: control diameter of hole in steel castings desired nominal diameter of  = 10 mm observations have shown  = 0.025 mm Process: every 2 hours a casting is randomly selected, so Note: variability would be reduced by taking n>1, due to pooling.

SPC Example Chart

Control Chart Patterns Pattern Description Possible Causes Normal Random Variation Lack of Stability Assignable (or special) causes (e.g. tool, material, operator, overcontrol ........................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................ Cumulative trend Tool Wear Cyclical Different work shifts, voltage fluctuations, seasonal effects

Continuous Improvement hypothesized process mean m m + 3s Upper Control Limit F(z) 99.74% Signal that a special cause has occurred SQC monitoring t m - 3s Lower Control Limit ........................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................ ........................................................................................................................................................................................................................................ Control Improvement driving improvement

Uses of SPC Product Quality Times Other Non-Quality Applications dimensions and other physical attributes fraction nonconforming range of attributes (for monitoring variability) Times process times repair times Other Non-Quality Applications tracking throughput due date quoting

Six Sigma Foundations

Six Sigma Foundations

Six Sigma Foundations

DMAIC: Define, Measure, Analyze, Improve, Control Six Sigma Terms DMAIC: Define, Measure, Analyze, Improve, Control Five Roles: Executive Leadership Champions: from ranks of upper management, mentor black belts Master Black Belts: Support Black Belts in Statistics & 6s Black Belts: Lead 6s projects Green Belts: Common training level, may lead projects Yellow Belts: Common training level, but not lead projects White Belts: minimal introductory training level

Quality and Logistics Quality and Cost: Quality Promotes Logistics: cost increases with quality? (e.g., better materials) cost decreases with quality? (e.g., less correction cost) reality is a balance Quality Promotes Logistics: Law: Variability degrades performance. Law: Congestion effects increase nonlinearly with utilization. yield loss and rework are major sources of variability and lost capacity. Logistics Promotes Quality: excess WIP obscures problems and delays/prevents diagnosis excess WIP magnifies losses excess cycle time degrades quality of service

Rework Rework Law: For a given throughput level, rework increases both the mean and standard deviation of the cycle time of a process. Implications: degraded performance through lost capacity increased variability Possible Cures: eliminate rework use non-bottleneck for reworking shorten rework loop

Rework on a Single Station 1- p r = 1/3 p

Rework in a Line 2/3 1 2/3 2/3 1- p p

Defect Detection 2/3 1 2/3 2/3 Prob q machine goes out of control Defects detected

Quality and the Supply Chain Importance: all manufacturing systems involve purchased parts trend toward outsourcing and “virtual manufacturing” a chain is only as good as its weakest link Vendor Quality: product quality service quality Assembly Systems: magnify impacts of vendor quality problems require effective vendor selection/management

Safety Lead Times in Assembly Systems Required Service: Single Component: 95% service level 10 Component Assembly: If each has 95% service then Prob{All components arrive on time} = (0.95)10 = 0.5987 so to get 95% service on the assembly we need each component to have p% service, where p10 = 0.95 p = 0.951/10 = 0.9949

Safety Lead Times in Assembly Systems (cont.) Consequences: Single Component: Supplier 1: 14 day lead time Supplier 2: 23 day lead time 10 Component Assembly: Supplier 1: 16.3 day lead time Supplier 2: 33.6 day lead time A B

Effect of Variability on Purchasing Lead Times

Effect of Variability on Purchasing Lead Times (cont.) 0.994 0.95 14 16.5 23 33.6

Circuitize: Current Situation Basic Problems: failure to make 3000 boards per day long CT (substantial part of 34 day CT) Symptoms: high WIP 6% defect level scrap at IP send aheads, test panels, rework at EP highly variable expose times (20 min for some operators, 40 min for others) clean room not very clean

Circuitize: Layout

Circuitize: Capacity Analysis Detractors: must account for setups, failures, rework, operator unavailability. IP Line: IP has tighter capacity than EP. Trouble spots are preclean/lamination/punch and expose. EP Line: EP has capacity for 3000 panels/day at 6% recycle (but not 10%). WIP is comparable to IP; EP is a variability bottleneck! Can’t make close to 3000 if first job is held for send aheads. Holding second job for send aheads has minor impact on capacity.

Circuitize: Recommendations Keep IP DES loaded as fully as possible Never starve for lack of operator. This controls IP throughput. Ensure capacity of IP Preclean/Lamination/Punch Cover preclean though breaks when room for WIP in clean room. Buy extra punch and maintain parallel dies to eliminate setup.

Circuitize: Recommendations (cont.) Improve IP Expose Capacity Certify operators (6 no recycle jobs 3 days in a row) Involve operators in hiring process. Tighten shift changes and use floaters to cover lunches. Use lead technicians to oversee flow (diazos, problems, etc.). Pursue extended life diazo program Add extra machine if necessary. Quality Improve cleanliness to increase yield Preserve old diazos to trace cause of defects Document effectiveness of policies (e.g., send aheads).

Circuitize: Outcome Steps: Results: Better housekeeping/training reduced recycle below 2%, making send aheads unnecessary. Extended life diazo and better personnel management made extra IP expose machine unnecessary. Line replicated in improved format to accommodate growing demand. Results: Capacity increased to near 3000 panels/day Dramatic decrease in CT to approximately one day. Improved line replicated to accommodate increased demand.

Conclusions Good quality supports good logistics Good logistics supports quality improvement Good quality at the supplier level promotes good logistics and quality at the plant level

TM 663 Operations Planning November 14, 2011 Paula Jensen Chapter 13: A Pull Planning Framework Chapter 14: Shop Floor Control

Agenda Schedule (New Assignment Chapter 13: Problem 1 Chapter 14: Problems 1, 2)

A Pull Planning Framework We think in generalities, we live in detail. –Alfred North Whitehead

Purpose of Production Control Objective: Meet customer expectations with on-time delivery of correct quantities of desired specification without excessive lead times or large inventory levels. Two Basic Approaches: Push Systems: Material Requirements Planning General. Provides a planning hierarchy. Underlying model often inappropriate. Pull Systems: Kanban, CONWIP Reduces congestion. Improves production environment. Suitable only for repetitive manufacturing.

Advantages of Pull Advantages: Magic of Pull: WIP Cap Observability: we can see WIP but not capacity. Efficiency: pull systems require less average WIP to attain same throughput as equivalent push system. Robustness: pull systems are less sensitive to errors in WIP level than push systems are to errors in release rate. Quality: pull systems require and promote improved quality. Magic of Pull: WIP Cap WIP

A Dilemma Question: If pull is so great, why do people still buy ERP systems? Answer: Manufacturing involves planning as well as execution. Execution

MRP II Planning Hierarchy Demand Forecast Resource Planning Aggregate Production Planning Rough-cut Capacity Planning Master Production Scheduling Bills of Material Material Requirements Planning Inventory Status Job Pool Capacity Requirements Planning Job Release Routing Data Job Dispatching

Hierarchical Pull Planning Framework Goals: To attain the benefits of a pull environment. To gain the generality of hierarchical production planning systems. The Environment: CONWIP production lines. Daily/Weekly production quota. The Hierarchy: Based on CONWIP for predictability and generality. Consistency between levels. Accommodate different implementations of modules for different environments. Use feedback.

Hierarchical Planning in a Pull System Personnel Plan FORECASTING CAPACITY/FACILITY PLANNING WORKFORCE Marketing Parameters Product/Process Labor Policies Capacity AGGREGATE Aggregate Strategy Work Schedule WIP/QUOTA SETTING DEMAND MANAGEMENT SEQUENCING & SCHEDULING Customer Demands Master Production SHOP FLOOR CONTROL WIP Position Tactics REAL-TIME SIMULATION PRODUCTION TRACKING Forecast Control

CONWIP as the Foundation Pull: jobs into the line whenever parts are used. jobs with the same routing. jobs for different part numbers. Push: jobs between stations on line. jobs into buffer storage between lines. A CONWIP Line: represents a level in a bill of material. is between stock points. maintains a constant amount of work in process. CONWIP

Benefits of CONWIP CONWIP vs. Push: CONWIP vs. Kanban: Easier and more robust control. Less congestion. Greater predictability. CONWIP vs. Kanban: Can accommodate a changing product mix. Can be used with setups. Suitable for short runs of small lots. More predictable. . . . . . . . . . …

Conveyor Model of CONWIP Predicting Completion Times: Practical production rate: rP parts per hour Minimum practical lead time: TP hours Xi is number of parts in job i on the backlog. Then the expected completion time of the nth job, cn, will be: Quoting Due Dates: need to add a “fudge factor” (which should consider cycle time variability) to ensure a reasonable service level. TP n rP

Aggregating Planning by Time Horizon

Other Levels of Aggregation Processes: Treat several workstations as one. Leave out unimportant (never bottleneck) workstations. Products: Group different part numbers into product families, which have have roughly the same routing have roughly the same price share setups Personnel: Categorize people according to management vs. labor shift workstation craft permanent vs. temporary

Forecasting Basic Problem: predict demand for planning purposes. Laws of Forecasting: 1. Forecasts are always wrong! 2. Forecasts always change! 3. The further into the future, the less reliable the forecast will be! Forecasting Tools: Qualitative: Delphi Analogies Many others Quantitative: Causal models (e.g., regression models) Time series models

Capacity/Facility Planning Basic Problem: how much and what kind of physical equipment is needed to support production goals? Issues: Basic Capacity Calculations: stand-alone capacities and congestion effects (e.g., blocking) Capacity Strategy: lead or follow demand Make-or-Buy: vendoring, long-term identity Flexibility: with regard to product, volume, mix Speed: scalability, learning curves

Workforce Planning Basic Problem: how much and what kind of labor is needed to support production goals? Issues: Basic Staffing Calculations: standard labor hours adjusted for worker availability. Working Environment: stability, morale, learning. Flexibility/Agility: ability of workforce to support plant's ability to respond to short and long term shifts. Quality: procedures are only as good as the people who carry them out.

Aggregate Planning Basic Problem: generate a long-term production plan that establishes a rough product mix, anticipates bottlenecks, and is consistent with capacity and workforce plans. Issues: Aggregation: product families and time periods must be set appropriately for the environment. Coordination: AP is the link between the high level functions of forecasting/capacity planning and intermediate level functions of quota setting and scheduling. Anticipating Execution: AP is virtually always done deterministically, while production is carried out in a stochastic environment. Linear Programming: is a powerful tool well-suited to AP and other optimization problems.

Quota Setting Basic Problem: set target production quota for pull system Issues: Larger quotas yield Benefits: Increased throughput. Increased utilization. Lower unit labor hour. Lower allocation of overhead. Costs: More overtime. Higher WIP levels. More expediting. Increased difficulties in quality control.

Planned Catch-Up Times Regular Time Regular Time Catch-Up Catch-Up R T T+R 2T

Economic Production Quota Notation

Simple “Sell-All-You-Can-Make” Model Objective Function: Average weekly profit Reasonability Test: We want the probability of not being able to catch up on overtime to be small (i.e., a): If this is not true, another (lost sales) model should be used.

Simple “Sell-All-You-Can-Make” Model (cont.) Normal Approximation: Express Q = m - ks, so the objective and reasonability test can be written: Solution: The objective function is maximized by: buffer capacity

Intuition from Model Optimal production quota depends on both mean and variance of regular time production (Q* increases with m and decreases with s). Increasing capacity increases profit, since Decreasing variance increases profit, since Model is valid (i.e., has a solution 0 < k* < ) only if since otherwise the term in the  becomes negative. If this occurs, then OT cost does not exceed revenue lost to make-up period and a different model is required.

Other Quota Setting Models Model 2: Lost Sales Run continuously. Choose periodic production quota Q. Demand above Q is lost (or vendored) at a cost. Solution looks like that to the Newsboy problem Model 3: Fixed plus Variable Cost of Overtime Same as Model 1, except that cost of overtime has a fixed component, COT, and a component proportional to the amount of the shortage Solution looks like that to Model 1 except term under  is more complex

Other Quota Setting Models (cont.) Model 4: Backlogging Fixed plus variable cost of overtime. Decision maker can choose to carry shortage to next period at a cost Dependence between periods requires more sophisticated solution techniques (e.g., dynamic programming). Solution consists of Q*, optimal quota, plus S*, an “overtime trigger” such that we use overtime only if the shortage is at least S.

Quota Setting Implementation Iteration between quota setting and aggregate planning may be necessary for consistency. Motivation (setting the “bar”) vs. Prediction (quoting due dates). MPS smoothing – necessary to keep steady quota. Gross capacity control through shift addition/deletion, rather than production slow-down.

Setting WIP Levels Basic Problem: establish WIP levels (card counts) in pull system. Issues: Mean regular time production increases with WIP level. Variance of regular time production also affected by WIP level. WIP levels should be set to facilitate desired throughput. Adjustment may be necessary as system evolves (feedback). Easy method: 1. Specify feasible cycle time, CT, and identify practical production rate, rP. 2. Set WIP from WIP = rP  CT

Demand Management Basic Problem: establish an interface between the customer and the plant floor, that supports both competitive customer service and workable production schedules. Issues: Customer Lead Times: shorter is more competitive. Customer Service: on-time delivery. Batching: grouping like product families can reduce lost capacity due to setups. Interface with Scheduling: customer due dates are are an enormously important control in the overall scheduling process.

Sequencing and Scheduling Basic Problem: develop a plan to guide the release of work into the system and coordination with needed resources (e.g., machines, staffing, materials). Methods: Sequencing: Gives order of releases but not times. Adequate for simple CONWIP lines where FISFO is maintained. The “CONWIP backlog.” Scheduling: Gives detailed release times. Attractive where complex routings make simple sequence impractical. MRP-C.

Sequencing CONWIP Lines Objectives: Maximize profit. No late jobs. All firm jobs selected. Job Sequencing System: Sequences bottleneck line. Uses Quota to explicitly consider capacity. Tries to group like families of jobs to reduce setups. Identifies the “offensive” jobs in an infeasible schedule. Suggests when more work could start in a lightly loaded schedule. Provides sequence for other lines. Work Backlog PN Quant –— ––––– LAN . . .

Real-Time Simulation Basic Problem: anticipate problems in schedule execution and provide vehicle for exploring solutions. Approaches: Deterministic Simulation: Given release schedule and dispatching rules, predict output times. Commercial packages (e.g., FACTOR). Conveyor Model: Allow hot jobs to pass in buffers, not in the lines. Use simplified simulation based on conveyor model. to predict output times.

Shop Floor Control Basic Problem: control flow of work through plant and coordinate with other activities (e.g., quality control, preventive maintenance, etc.) Issues: Customization: SFC is often the most highly customized activity in a plant. Information Collection: SFC represents the interface with the actual production processes and is therefore a good place to collect data. Simplicity: departures from simple mechanisms must be carefully justified.

Tracking and Feedback Basic Problems: Functions: Signal quota shortfall. Update capacity data. Quote delivery dates. Functions: Statistical Throughput Control: Monitored at critical tools. Like SPC, only measuring throughput. Problems are apparent with time to act. Workers aware of situation. Feedback: Collect capacity data. Measure continual improvement.

Conclusions Pull Environment Provides: Less WIP and thereby earlier detection of quality problems. Shorter lead times allowing increased customer response and less reliance on forecasts. Less buffer stock and therefore less exposure to schedule and engineering changes. CONWIP Provides: a pull environment that Has greater throughput for equivalent WIP than kanban. Can accommodate a changing product mix. Can be used with setups. Is suitable for short runs of small lots. Is predictable.

Conclusions (cont.) Planning Hierarchy Provides: Consistent framework for planning. Links between levels. Feedback.

Forecasting The future is made of the same stuff as the present. – Simone Weil

Forecasting “Laws” 1) Forecasts are always wrong! 2) Forecasts always change! 3) The further into the future, the less reliable the forecast! Start of season 20% 40% +10% -10% 16 weeks 26 weeks Trumpet of Doom

Quantitative Forecasting Goals: Predict future from past Smooth out “noise” Standardize forecasting procedure Methodologies: Causal Forecasting: regression analysis other approaches Time Series Forecasting: moving average exponential smoothing seasonal models many others

Time Series Forecasting Historical Data Forecast Time series model A(i), i = 1, … ,t f(t+t), i = 1, 2, …

Time Series Approach Notation:

Time Series Approach (cont.) Procedure: 1. Select model that computes f(t+t) from A(i), i = 1, … , t 2. Forecast existing data and evaluate quality of fit by using: 3. Stop if fit is acceptable. Otherwise, adjust model constants and go to (2) or reject model and go to (1).

Moving Average Assumptions: Model: No trend Equal weight to last m observations Model:

Moving Average (cont.) Example: Moving Average with m = 3 and m = 5. Note: bigger m makes forecast more stable, but less responsive.

Moving Average: m=3,5

Exponential Smoothing Assumptions: No trend Exponentially declining weight given to past observations Model:

Exponential Smoothing, a=0.2

Exponential Smoothing with a Trend Assumptions: Linear trend Exponentially declining weights to past observations/trends Model: Note: these calculations are easy, but there is some “art” in choosing F(0) and T(0) to start the time series.

Exponential Smoothing with a Trend (cont.) Example: Exponential Smoothing with Trend, a = 0.2, b = 0.5. Note: we start with trend equal to zero.

Exponential Smoothing with Trend, a=0.2, b=0.5

Effects of Altering Smoothing Constants Exponential Smoothing with Trend: various values of a and b Note: these assume we start with trend equal diff between first two demands.

Effects of Altering Smoothing Constants Exponential Smoothing with Trend: various values of a and b Note: these assume we start with trend equal to zero.

Effects of Altering Smoothing Constants (cont.) Observations: assuming we start with zero trend a = 0.3, b = 0.5 work well for MAD and MSD a = 0.6, b = 0.6 work better for BIAS Our original choice of a = 0.2, b = 0.5 had MAD = 3.73, MSD = 22.32, BIAS = -2.02, which is pretty good, although a = 0.3, b = 0.5, with MAD = 3.65, MSD=21.78, BIAS = -1.52 is better.

Winters Method for Seasonal Series Seasonal series: a series that has a pattern that repeats every N periods for some value of N (which is at least 3). Seasonal factors: a set of multipliers ct , representing the average amount that the demand in the tth period of the season is above or below the overall average. Winter’s Method: The series: The trend: The seasonal factors: The forecast:

Winters Method - Sample Calculations Initially we set: smoothed estimate = first season average smoothed trend = zero (T(N)=T(12) = 0) seasonality factor = ratio of actual to average demand From period 13 on we can use initial values and standard formulas...

Winters Method Example

Winters Method Example 25 20 15 Demand 10 5 A(t) f(t) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Month

Conclusions Sensitivity: Lower values of m or higher values of a will make moving average and exponential smoothing models (without trend) more sensitive to data changes (and hence less stable). Trends: Models without a trend will underestimate observations in time series with an increasing trend and overestimate observations in time series with a decreasing trend. Smoothing Constants: Choosing smoothing constants is an art; the best we can do is choose constants that fit past data reasonably well. Seasonality: Methods exist for fitting time series with seasonal behavior (e.g., Winters method), but require more past data to fit than the simpler models. Judgement: No time series model can anticipate structural changes not signaled by past observations; these require judicious overriding of the model by the user.

Shop Floor Control Even a journey of one thousand li begins with a single step. – Lao Tze It is a melancholy thing to see how zeal for a good thing abates when the novelty is over, and when there is no pecuniary reward attending the service. – Earl of Egmont

What is Shop Floor Control? Definition: Shop Floor Control (SFC) is the process by which decisions directly affecting the flow of material through the factory are made. Functions: WIP Tracking Throughput Status Monitoring Work Forecasting Capacity Feedback Quality Control Material Flow

Planning for SFC Gross Capacity Control: Match line to demand via: Varying staffing (no. shifts or no. workers/shift) Varying length of work week (or work day) Using outside vendors to augment capacity Bottleneck Planning: Bottlenecks can be designed Cost of capacity is key Stable bottlenecks are easier to manage Span of Control: Physically or logically decompose system Span of labor management (10 subordinates) Span of process management (related technology?)

Basic CONWIP Rationale: Requirements: Design Issues: Simple starting point Can be effective Requirements: Constant routings Similar processing times (stable bottleneck) No significant setups No assemblies Design Issues: Work backlog – how to maintain and display Line discipline – FIFO, limited passing Card counts – WIP = CT  rP initially, then conservative adjustments Card deficits – violate WIP-cap in special circumstances Work ahead – how far ahead relative to due date? . . .

CONWIP Line Using Cards CONWIP Cards Production Line Inbound Stock Outbound Stock

Card Deficits Jobs without Cards Jobs with Cards B Bottleneck Process Failed Machine

Tandem CONWIP Lines Links to Kanban: when “loops” become single process centers Bottleneck Treatment: Nonbottleneck loops coupled to buffer inventories (cards are released on departure from buffer) Bottleneck loops uncoupled from buffer inventories (cards are released on entry into buffer) Shared Resources: Sequencing policy is needed Upstream buffer facilitates sequencing (and batching if necessary)

Tandem CONWIP Loops Basic CONWIP Multi-Loop CONWIP Kanban Workstation Buffer Card Flow

Coupled and Uncoupled CONWIP Loops Bottleneck CONWIP Loop Buffer Job CONWIP Card Material Flow Card Flow

Splitting Loops at Shared Resource Routing A Routing A Routing B Routing B CONWIP Loop Card Flow Material Flow Buffer

Modifications of Basic CONWIP Multiple Product Families: Capacity-adjusted WIP CONWIP Controller Assembly Systems: CONWIP achieves synchronization naturally (unless passing is allowed) WIP levels must be sensitive to “length” of fabrication lines

CONWIP Controller . . . LAN PC PC Work Backlog Indicator Lights PN Quant –— ––––– LAN Indicator Lights R G PC PC . . . Workstations

CONWIP Assembly Processing Times for Line A 2 1 4 1 Processing Times for Line B 3 3 2 3 Assembly Buffer Card Flow Material Flow

Kanban Advantages: Disadvantages: improved communication control of shared resources Disadvantages: complexity – setting WIP levels tighter pacing – pressure on workers, less opportunity for work ahead part-specific cards – can’t accommodate many active part numbers inflexible to product mix changes handles small, infrequent orders poorly

Kanban with Work Backlog —— Backlog Material Flow Standard Container Card Flow Card

Pull From the Bottleneck Problems with CONWIP/Kanban: Bottleneck starvation due to downstream failures Premature releases due to CONWIP requirements PFB Remedies: PFB ignores WIP downstream of bottleneck PFB launches orders when bottleneck can accommodate them PFB Problem: Floating bottlenecks

Simple Pull From the Bottleneck Material Flow Card Flow

Routings in a Jobshop ASSEMBLY BOTTLENECK 1 2 3 4 Backlog 1 ---------- 2 ---------- 3 ---------- 4 ---------- 5 ---------- . ………. m ---------- ASSEMBLY BOTTLENECK 1 2 3 4

Implementing PFB Notation: Work at Bottleneck: total hours of work ahead of job j is Job Release Mechanism: Release job j whenever Enhancement: establish due date window, before which jobs are not released.

Production Tracking Short Term: Long Term: Statistical Throughput Control (STC) Progress toward quota Overtime decisions Long Term: Long range tracking Capacity feedback Synchronize planning models to reality

STC Notation Note: we might have these instead of m and s, if we stop when quota is made.

STC Mechanics Assumption: Nt is normally distributed with mean mt/R and variance s2t/R. Implications: Nt - Qt/R is normally distributed with mean (m - Q)t/R and variance s2t/R. NR-t is normally distributed with mean m(R - t)/R and variance s2(R - t)/R. If Nt = nt, where nt - Qt/R = x, we will miss quota only if NR-t < Q - nt. Formula: The probability of missing quota by time R given an overage of x is

STC Charts which yields where za is chosen such that F(za) = a. Motivation: information “at a glance” Computations: Pre-compute the overage levels that cause the probability of missing quota to be a specified level a: which yields where za is chosen such that F(za) = a.

STC Chart (Q=)

STC Chart (Q<)

Long-Range Tracking Statistics of Interest: m, mean production during regular time s2, variance of regular time production Observable Statistics: if we stop when quota is achieved, then instead of m and s we observe mS, mean time to make quota s2S, variance of time to make quota Conversion Formulas: If he have mS and sS, then we can smooth these (as shown later) and then convert to m and s by using

Smoothing Capacity Parameters Mean Production: where a and b are smoothing constants. Production Variance: where g is a smoothing constant.

LR Tracking - Mean Production

Smoothed Trend in Mean Production

LR Tracking - Std Dev of Production

Shop Floor Control Takeaways General: SFC is more than material flow control (WIP tracking, QC, status monitoring, … ) good SFC requires planning (workforce policies, bottlenecks, management, … ) CONWIP: simple starting point reduces variability due to WIP fluctuations many modifications possible (kanban, pull-from-bottleneck)

Shop Floor Control Takeaways (cont.) Statistical Throughput Control (STC); tool for OT planning/prediction intuitive graphical display Long Range Tracking: feedback for other planning/control modules exponential smoothing approach

TM 663 Operations Planning November 28,2011 Paula Jensen Chapter 15: Production Scheduling

Agenda Factory Physics Chapter 15: Production Scheduling (New Assignment Chapter 15 problems 1-3)

Production Scheduling Let all things be done decently and in order. – I Corinthians

Goals of Production Scheduling High Customer Service: on-time delivery Low Inventory Levels: WIP and FGI High Utilization: of machines UTILIZATION SERVICE INVENTORY

Meeting Due Dates – Measures Service Level: Used typically in make to order systems. Fraction of orders which are filled on or before their due dates. Fill Rate: Used typically in make to stock systems. Fraction of demands met from stock. Lateness: Used in shop floor control. Difference between order due date and completion date. Average lateness has little meaning. Better measure is lateness variance. Tardiness: Is equal to the lateness of a job if it is late and zero, otherwise. Average tardiness is meaningful but unintuitive.

Reducing WIP and Cycle Time Less WIP Equals Shorter Cycle Times (Little’s Law) Benefits of shorter cycle times: Lower Cost: less money in inventory More Flexibility: less disruptive to change backlog that work in process Better Quality: faster defect detection Less Reliance on Forecasts: cycle times below frozen zone allow make to order Better Forecasts: distant (inaccurate) forecasts are no longer needed

Classic Scheduling – Assumptions MRP/ERP: Benefits – Simple paradigm, hierarchical approach. Problems – MRP assumes that lead times are an attribute of the part, independent of the status of the shop. MRP uses pessimistic lead time estimates.

Classic Scheduling – Assumptions (cont.) Classic Scheduling: (only classic in academia) Benefits – “Optimal” schedules Problems – Bad assumptions. All jobs available at the start of the problem. Deterministic processing times. No setups. No machine breakdowns. No preemption. No cancellation.

Classic Single Machine Results Minimizing Average Cycle Time: Minimize by performing in “shortest process time” (SPT) order. Makespan is not affected. Minimizing Maximum Lateness (or Tardiness): Minimize by performing in “earliest due date” (EDD) order. If there exists a sequence with no tardy jobs, EDD will do it. Minimizing Average Tardiness: No simple sequencing rule will work. Problem is NP Hard.

Classic Multi Machine Results Minimizing “Makespan” on Two Machines: given a set of jobs that must go through a sequence of two machines, what sequence will yield the minimum makespan? Makespan is sequence dependent. Simple algorithm (Johnson 1954): 1. Sort the times of the jobs on the two machines in two lists. 2. Find the shortest time in either list and remove job from both lists. If time came from first list, place job in first available position. If time came from second list, place job in last available position in sequence. 3. Repeat until lists are exhausted. The resulting sequence will minimize makespan.

Johnson’s Algorithm Example Data: Iteration 1: min time is 4 (job 1 on M1); place this job first and remove from lists:

Johnson’s Algorithm Example (cont.) Iteration 2: min time is 5 (job 3 on M2); place this job last and remove from lists: Iteration 3: only job left is job 2; place in remaining position (middle). Final Sequence: 1-2-3 Makespan: 28

Gantt Chart for Johnson’s Algorithm Example Short task on M2 to “clear out” quickly. Short task on M1 to “load up” quickly.

Classic Dispatching Results Optimal Schedules: Impossible to find for most real problems. Dispatching: sorts jobs as they arrive at a machine. Dispatching rules: FIFO – simplest, seems “fair”. SPT – Actually works quite well with tight due dates. (shortest process time) EDD – Works well when jobs are mostly the same size. Many (100?) others. Problems with Dispatching: Cannot be optimal (can be bad). Tends to be myopic.

The Difficulty of Scheduling Problems Dilemma: Too hard for optimal solutions. Need something anyway. Classifying “Hardness”: Class P: has a polynomial solution. Class NP: has no polynomial solution. Example: Sequencing problems grow as n!. Compare en/10000 and 10000n10. At n = 40, en/10000 = 2.4  1013, 10000n10 = 1.0  1020 At n = 80, en/10000 = 5.5  1030, 10000n10 = 1.1  1023 3! = 6, 4! = 24, 5! = 120, 6! = 720, … 10! =3,628,800, while 13! = 6,227,020,800 25!= 15,511,210,043,330,985,984,000,000 en/10000 10000n10

Computation Times Current situation: computer can examine 1,000,000 sequences per second and we wish to build a scheduling system that has response time of no longer than one minute. How many jobs can we sequence optimally?

Effect of Faster Computers Future Situation: New computer is 1,000 times faster, i.e. it can do 1 billion comparisons per second. How many jobs can we sequence optimally now?

Polynomial vs. Non-Polynomial Algorithms Polynomial Example: dispatching (sorting) time goes up as n log n. Suppose, for comparison, that is takes the same amount of time to sort 10 jobs as it does to examine the 10! sequences. How large a problem can we solve using dispatching?

Effect of Faster Computer Situation: New computer 1000 times faster. How much does the size of dispatching problem we can solve increase?

Implications for Real Problems Computation: NP algorithms are slow to use. No Technology Fix: Faster computers don’t help on NP algorithm. Scheduling is Hard: Real scheduling problems tend to be NP Hard. Scheduling is Big: Real scheduling problems also tend to be quite large; impossible to solve optimally.

Implications for Real Problems (cont.) Robustness? NP hard problems have many solutions, and presumably many “good” ones. Example: 25 job sequencing problem. Suppose that only one in a trillion of the possible solutions is “good”. This still leaves 15 trillion “good” solutions. Our task is to find one of these. Role of Heuristics: Polynomial algorithms can be used to obtain “good” solutions. Example heuristics include: Simulated Annealing Tabu Search Genetic Algorithms

The Bad News Violation of Assumptions: Most “real-world” scheduling problems violate the assumptions made in the classic literature: There are always more than two machines. Process times are not deterministic. All jobs are not ready at the beginning of the problem. Process time are sequence dependent. Problem Difficulty: Most “real-world” production scheduling problems are NP-hard. We cannot hope to find optimal solutions of realistic sized scheduling problems. Polynomial approaches, like dispatching, may not work well.

The Good News Due Dates: We can set the due dates. Job Splitting: We can get smaller jobs by splitting larger ones. Single machine SPT results imply small jobs “clear out” more quickly than larger jobs. Mechanics of Johnson’s algorithm implies we should start with a small job and end with a small job. Small jobs make for small “move” batches and can be combined to form larger “process” batches.

The Good News (cont.) Feasible Schedules: We do not need to find an optimal schedule, only a good feasible one. Focus on Bottleneck: We can often concentrate on scheduling the bottleneck process, which simplifies problem closer to single machine case. Capacity: Capacity can be adjusted dynamically (overtime, floating workers, use of vendors, etc.) to adapt facility (somewhat) to schedule.

Due Date Quoting New Job (c) “Emergency” Positions Rate Out WIP (W) rP, s Backlog (b) Completed m = w + b +c

Dynamic Due Date Quoting – Analytic Approach Notation: Required Condition: Due Date Quote: prob of not completing m units of work within lead time of l safety lead time

Dynamic Due Date Quoting – Observations If s = 0 then l = m/m, regardless of s (i.e., quoting m/m days will achieve 100% service in a system with no variability). The due date is increasing in s (i.e., more variability will require more safety lead time). In order to achieve service of at least s, we must round l up to the next integer to arrive at the due date to quote in units of full days.

Dynamic Due Date Quoting Example Data:

Dynamic Due Date Quoting Example (cont.) Solution: Interpretation: Note that n/m = 1500/100 = 15 days, so to accommodate the variability we are building in 2 days of safety lead time into our quote.

Dynamic Due Date Quoting – Empirical Approach Basic Model: where Methodology: use control chart approach to adjust FF to attain desired service level. Advantages: very few modeling assumptions simple to implement adapts to continuous improvement

Scheduling via Lot Sizing Case: TJ International – Parallam Product Issues: Press is bottleneck 12 hour changeover time to switch widths (12”, 14”, 16”, 19”) Many products made from each width, but less 14”/16” than 16”/19” Currently try to run at least a week between changeovers Seasonal demand: inventory build up in off-season Problem: determine run sequence each month Lathe/Clip Press Saw

Scheduling via Lot Sizing Notation: Problem: choose lot sizes for the month to meet demand as “efficiently” as possible.

Cyclic Schedule Approach: fix lot sizes for all products and compute number of cycles that can be run in the month as: If the horizon is short, N should be rounded down to the nearest integer. Otherwise, a non-integer N is ok since you will be rescheduling anyway. Lot Sizes: The lot sizes will be: These, of course, need to be rounded or ceilinged. Problem: You don’t necessarily want to make the same number of runs of a low demand product as of a high demand product. spare capacity setup/cycle

Non-Cyclic Schedules Fundamental Issue: What is your objective? Minimize Sum of Run Lengths? If we write out the Lagrangian for the nonlinear optimization problem and optimize assuming continuous lot sizes, we get where Problems: Since we solved for lot sizes, we wind up with a non-integer number of setups for the month. How do we allocate the setup time? What order do we run the products in the non-cyclic schedule? Conclusion: We should solve for the number of run cycles for each product. But what objective do we use?

MiniMax Schedule Objective: minimize the maximum run length. We are never too far away from changing over to another product. Perhaps a good surrogate for maximizing flexibility. Notation: Let decision variable

MiniMax Schedule (cont.) IP Formulation:

Example Data: H = 22 days  18 hrs/day = 396 hrs u = 0.9 Minimal Run Schedule: Suppose the plant has a policy of never making less than 1500 so their current lot sizes are 15000, 12000, 1500, 1500. This takes 19.2 days, close to the 22 days available, but it means that some product won’t be produced until pretty late in the month. Total Prod Time Req = 281.67 hrs

Example (cont.) Cyclic Schedule: The number of cycles per month is yielding lot sizes of But N = 2.87 means that the first two cycles in the month use the above lot sizes while the last one uses reduced lot sizes (87% of base).

Example (cont.) MiniMax Schedule: We can implement the above formulation as follows: 1. Start with ni = 1 for all i. Note that the product with max run length is product 1 with a run of 158 hours. 2. Add setup for product 1, so n1 = 2, and its max run length becomes 83 and now product 2 has the longest run length at 128 hours. 3. Add a setup for product 2, so n2 = 2, and its max run length becomes 68 and now product 1 has longest run length at 83 hours. 4. Continue adding setups to product with longest run until we run out of capacity. The optimal solution is:

Example (cont.) Results: The schedule goes as follows: 1. We set up for the 4th product and get it out of the way. 2. Likewise for the 3rd. 3. Now we run 3750 of product 1 and 3000 of product 2 in succession for 4 times. The longest anyone would ever have to wait would be for product 2 at the beginning of the month at 3.73 days. After that the longest wait would be 2.53 days. Conclusion: Minimax schedule results in shorter wait for product than either minimal run schedule or cyclic schedule. And it is simple to compute.

TJ International Case Non-Cyclic Scheduling? Fewer runs of 12”, 14” But long setups make running anything twice in the month difficult. Skip 12”, 14” some months? How to Handle Seasonality? Could run high volume products in off-season (sure to sell) But running 12”, 14” products would help skip setups and increase capacity in peak season Need to choose fractile of demand to stock in off-season (75%?)

Scheduling in Pull Environments Simple CONWIP Lines: Simple sequence sufficient. No setups  EDD sequence. CONWIP Lines with Shared Resources: No setups  EDD within lines. Need to augment sequence at shared lines. FISFO at shared resources. FISFO sequencing

Scheduling in Pull Environments (cont.) CONWIP Lines with Setups: Sequence still reasonable. Must balance capacity loss with due date requirements. Exact solution impossible. Heuristics starting with EDD and rearranging sequence can work well. More General Environments: Many routings make simple sequence insufficient. Backward scheduling – due dates may be infeasible. Forward scheduling – may be suboptimal.

A Sequencing Example Problem Description: 16 jobs Each job takes 1 hour on single machine (bottleneck resource) 4 hour setup to change families Fixed due dates Find feasible solution if possible

A Sequencing Example (cont.) EDD Sequence: Average Tardiness: 10.375

Sequencing Example (cont.) Greedy Approach: Consider all pairwise interchanges Choose one that reduces average tardiness by maximum amount Continue until no further improvement is possible

Sequencing Example (cont.) First Interchange: Exchange jobs 4 and 5. Average Tardiness: 5.0 (reduction of 5.375!)

Sequencing Example (cont.) Configuration After Greedy Search: Average Tardiness: 0.5 (9.875 lower than EDD)

Sequencing Example (cont.) A Better (Feasible) Sequence: Average Tardiness: 0

Diagnostic Scheduling Goals: Schedule at appropriate level of detail for environment. Make use of realistic, obtainable data. Accommodate “intangibles” in decision-support mode. Approach: Deterministic model. Based on conveyor model. Release as late as possible subject to due dates. Provide diagnostics on infeasibilities. Types of Infeasibility: WIP (must move out due dates). Capacity (can move due dates or increase capacity).

Scheduling Infeasibilities Example Problem Description: Capacity = 100/day Minimum Practical Lead Time = 3 days WIP in system with 1, 2, 3 days to go = 95, 90, 115

Scheduling Infeasibilities Example (cont.) * WIP infeasibility ** capacity infeasibility

Cumulative Demand vs. Cumulative Capacity

Surplus WIP Infeasibility Capacity Infeasibility

MRP-C Notation

MRP-C Notation (cont.)

MRP-C Example How should we resolve infeasibility?

MRP-C Example: Correcting Infeasibility Delay 40 units of demand in period 12: 10 to period 13, 20 to period 14, 10 to period 14

Multi-Stage MRP-C Calculations End Items Stage 1 Stage 2 Stage 3 Requirements Requirements Requirements Starts Starts Starts Issues must work backward from end items (like in MRP) disaggregation can be tricky (poor tie-breaking can cause WIP infeasibility)

Scheduling Software Approaches Fixed leadtime backward scheduling (MRP) Rule based forward scheduling (FACTOR) AI/Expert System approaches (MIMI) Bottleneck scheduling (OPT) Heuristics (MADEMA/PROMIS) Diagnostic (backward) scheduling (MRP-C) Perturbation scheduling (developmental)

Perturbation Scheduling Simulation Engine: input lead times, WIP positions, demands, priority rules fast simulator based on conveyor model predict range of completion times Due Date Quoting: given a set of demands find due dates that ensure target service level useful in coordinating multi-level system Lead Time Optimization: specify objective (e.g., average tardiness) use perturbation analysis to compute lead time gradients optimize lead times during simulation lead times specify release schedule

Linking Scheduling with Releases Approach: Use Scheduling Module to plan. Use SFC Module to launch orders. Track progress relative to schedule with Production Tracking Module. Scheduling Modules: Finite capacity scheduler (e.g., MRP-C) MRP Ad hoc method

Linking Scheduling with Release (cont.) SFC Modules: CONWIP Kanban Pull-From-the Bottleneck Arbitrary WIP-Cap Mechanism Benefits: Prevent WIP explosions Stabilize cycle times  facilitates better scheduling Clear opportunity for feedback from floor to scheduler.

Scheduling Takeaways Scheduling is hard! Even simple toy problems generate complicated mathematics. Scheduling can be simplified through the environment: due date quoting flow simplification  sequencing in place of scheduling Finite capacity scheduling is coming. But in what form is unclear (shifting bottleneck, genetic algorithms, rule based systems, etc.). Diagnostics are important in scheduling. pure optimization generally impossible need good interface to allow human intervention

TM 663 Operations Planning December 4, 2011 Paula Jensen Chapter 16: Aggregate and Workforce Planning Chapter 17: Supply Chain Management

Agenda Factory Physics Chapter 16: Aggregate and Workforce Planning Chapter 17: Supply Chain Management (New Assignment Chapter 16 problems 1-4 Chapter 17 problem 1)

Aggregate Planning And I remember misinformation followed us like a plague, Nobody knew from time to time if the plans were changed. – Paul Simon

Aggregate Planning Issues Role of Aggregate Planning Long-term planning function Strategic preparation for tactical actions Aggregate Planning Issues Production Smoothing: inventory build-ahead Product Mix Planning: best use of resources Staffing: hiring, firing, training Procurement: supplier contracts for materials, components Sub-Contracting: capacity vendoring Marketing: promotional activities

Hierarchical Production Planning Marketing Parameters FORECASTING Product/Process Parameters CAPACITY/FACILITY PLANNING WORKFORCE PLANNING Labor Policies Capacity Plan Personnel Plan AGGREGATE PLANNING Aggregate Plan Strategy WIP/QUOTA SETTING Customer Demands Master Production Schedule DEMAND MANAGEMENT WIP Position SEQUENCING & SCHEDULING Tactics REAL-TIME SIMULATION Work Schedule Work Forecast SHOP FLOOR CONTROL Control PRODUCTION TRACKING

Basic Aggregate Planning Problem: project production of single product over planning horizon. Motivation for Study: mechanics and value of LP as a tool intuition of production smoothing Inputs: demand forecast (over planning horizon) capacity constraints unit profit inventory carrying cost rate

A Simple AP Model Notation:

A Simple AP Model (cont.) summed over planning horizon Formulation sales revenue - holding cost demand capacity inventory balance non-negativity

A Simple AP Example Data: Optimal Solution:

A Simple AP Example (cont.) Interpretation solution shadow prices allowable increases / decreases

Product Mix Planning Problem: determine most profitable mix over planning horizon Motivation for Study: linking marketing/promotion to logistics. Bottleneck identification. Inputs: demand forecast by product (family?); may be ranges unit hour data capacity constraints unit profit by product holding cost

Basic Verbal Formulation maximize profit subject to: production  capacity, at all workstations in all periods sales  demand, for all products Note: we will need some technical constraints to ensure that variables represent reality.

Product Mix Notation

Product Mix Formulation sales revenue - holding cost demand capacity inventory balance non-negativity

Product Mix (Goldratt) Example Assumptions: two products, P and Q constant weekly demand, cost, capacity, etc. Objective: maximize weekly profit Data:

A Cost Approach Unit Profit Maximum Production of Q : 50 units Product P : $45 Product Q : $60 Maximum Production of Q : 50 units Available Capacity for Producing P 2400 - 10 (50) = 1,900 minutes on Workcenter A 2400 - 30 (50) = 900 minutes on Workcenter B 2400 - 5 (50) = 2,150 minutes on Workcenter C 2400 - 5 (50) = 2,150 minutes on Workcenter D Maximum Production of P: 900/15 =60 units Net Weekly Profit: $45  60 +$60  50 -$5,000 = $700

A Bottleneck Approach Identifying the Bottleneck:Workcenter B, because 15 (100) + 10 (50) = 2,000 minutes on workcenter A 15 (100) + 30 (50) = 3,000 minutes on workcenter B 15 (100) + 5 (50) = 1,750 minutes on workcenter C 15 (100) + 5 (50) = 1,750 minutes on workcenter D Profit per Minute of Bottleneck Time used: $45/15 = $3 per minute spent processing P $60/30 = $2 per minute spent processing Q Maximum Production of P: 100 units Maximum Production of Q: 900/30=30 units Net Weekly Profit: $45100 + $60 30 -$5,000 = $1,300

A Modified Example Changes: in processing times on workcenters B and D. Data:

A Bottleneck Approach Identifying the Bottleneck:Workcenter B, because 15 (100) + 10 (50) = 2,000 minutes on workcenter A 15 (100) + 35 (50) = 3,250 minutes on workcenter B 15 (100) + 5 (50) = 1,750 minutes on workcenter C 25 (100) + 14 (50) = 3,200 minutes on workcenter D Bottleneck at B: $45/15 = $3 per minute spent processing P $60/35 = $1.71 per minute spent processing Q Maximum Production of P: 2400/25 = 96 units Maximum Production of Q: 0 units Net Weekly Profit: $4596 -$5,000 = -$680

A Bottleneck Approach (cont.) Bottleneck at D: $45/25 = $1.80 per minute spent processing P $60/14 = $4.29 per minute spent processing Q Maximum Production of Q: 2400/35 = 68.57>50, produce 50 Available time on Bottleneck: 2400 - 14(50) = 1,700 minutes on workcenter D Maximum Production of P: 1700/25= 68 units Net Weekly Profit: $4543+$60 50-$5000= -$65

An LP Approach Formulation: Solution: Net Weekly Profit : Round solution down (still feasible) to: To get $45 75 + $60 36 - $5,000 = $535.

Extensions to Basic Product Mix Model Other Resource Constraints: Notation: Constraint for Resource j: Utilization Matching: Let q represent fraction of rated capacity we are willing to run on resource j.

Extensions to Basic Product Mix Model (cont.) Backorders: Substitute Allow to become positive or negative Penalize differently in objective if desired Overtime: Define as hours of OT used on resource j in period t Add to in capacity constraint. Penalize in objective if desired

Workforce Planning Problem: determine most profitable production and hiring/firing policy over planning horizon. Motivation for Study: hiring/firing vs. overtime vs. Inventory Build tradeoff iterative nature of optimization modeling. Inputs: demand forecast (assume single product for simplicity) unit hour data labor content data capacity constraints hiring/ firing costs overtime costs holding costs unit profit

Workforce Planning Notation

Workforce Planning Notation (cont.)

Workforce Planning Formulation

Workforce Planning Example Problem Description 12 month planning horizon 168 hours per month 15 workers currently in system regular time labor at $35 per hour overtime labor at $52.50 per hour $2,500 to hire and train new worker $2,500/168=$14.88  $15/hour $1,500 to lay off worker $1,500/168=$8.93  $9/hour 12 hours labor per unit demand assumed met (St=dt, so St variables are unnecessary)

Workforce Planning Example (cont.) Solutions: “Chase” Solution: infeasible LP optimal Solution: layoff 9.5 workers Add constraint: Ft=0 results in 48 hours/worker/week of overtime Add constraint: Ot  0.2Wt Reasonable solution?

Conclusions No single AP model is right for every situation Simplicity promotes understanding Linear programming is a useful AP tool Robustness matters more than precision Formulation and Solution are not separate activities.

Inventory Management One's work may be finished some day, but one's education never. – Alexandre Dumas

Hierarchical Planning – Roles of Inventory Personnel Plan FORECASTING CAPACITY/FACILITY PLANNING WORKFORCE Marketing Parameters Product/Process Labor Policies Capacity AGGREGATE Aggregate Strategy Work Schedule WIP/QUOTA SETTING DEMAND MANAGEMENT SEQUENCING & SCHEDULING Customer Demands Master Production SHOP FLOOR CONTROL WIP Position Tactics REAL-TIME SIMULATION PRODUCTION TRACKING Forecast Control flow control - push/pull, etc. WIP tracking build-ahead,batching, safety stock, etc. prediction of WIP movement buffer sizes seasonal build, inv/OT tradeoff flexibility, teaming aggregation,postponement,etc. capacity vs. inventory to assure service

Inventory is the Lifeblood of Manufacturing Plays a role in almost all operations decisions shop floor control scheduling aggregate planning capacity planning, … Links to most other major strategic decisions quality assurance product design facility design marketing organizational management, … Managing inventory is close to managing the entire system…

Plan of Attack Classification: Justification: raw materials work-in-process (WIP) finished goods inventory (FGI) spare parts Justification: Why is inventory being held? benchmarking

Plan of Attack (cont.) Structural Changes: Modeling: major reorganization (e.g., eliminate stockpoints, change purchasing contracts, alter product mix, focused factories, etc.) reconsider objectives (e.g., make-to-stock vs. make-to-order, capacity strategy, time-based-competition, etc.) Modeling: What to model – identify key tradeoffs. How to model – EOQ, (Q,r), optimization, simulation, etc.

Raw Materials Reasons for Inventory: Improvement Policies: batching (quantity discounts, purchasing capacity, … ) safety stock (buffer against randomness in supply/production) obsolescence Improvement Policies: Pareto analysis (focus on 20% of parts that represent 80% of $ value) ABC classification (stratify parts management) JIT deliveries (expensive and/or bulky items) vendor monitoring/management

Raw Materials (cont.) Benchmarks: Models: small C parts: 4-8 turns A,B parts: 12-25+ turns bulky parts: up to 50+ turns Models: EOQ power-of-two service constrained optimization model

Multiproduct EOQ Models Notation: N = total number of distinct part numbers in the system Di = demand rate (units per year) for part i ci = unit production cost of part i Ai = fixed cost to place an order for part i hi = cost to hold one unit of part i for one year Qi = the size of the order or lot size for part i (decision variable)

Multiproduct EOQ Models (cont.) Cost-Based EOQ Model: For part i, but what is A? Frequency Constrained EOQ Model: min Inventory holding cost subject to: Average order frequency  F

Multiproduct EOQ Solution Approach Constraint Formulation: Cost Formulation:

Multiproduct EOQ Solution Approach (cont.) Cost Solution: Differentiate Y(Q) with respect to Qi, set equal to zero, and solve: Constraint Solution: For a given A we can find Qi(A) using the above formula. The resulting average order frequency is: If F(A) < F then penalty on order frequency is too high and should be decreased. If F(A) > F then penalty is too low and needs to be increased. No surprise - regular EOQ formula

Multiproduct EOQ Procedure – Constrained Case Step (0) Establish a tolerance for satisfying the constraint (i.e., a sufficiently small number that represents “close enough” for the order frequency) and guess a value for A. Step (1) Use A in previous formula to compute Qi(A) for i = 1, … , N. Step (2) Compute the resulting order frequency: If |F(A) - F| < e, STOP; Qi* = Qi(A), i = 1, … , N. ELSE, If F(A) < F, decrease A If F(A) < F, increase A Go to Step (1). Note: The increases and decreases in A can be made by trial and error, or some more sophisticated search technique, such as interval bisection.

Multiproduct EOQ Example Input Data:

Multiproduct EOQ Example (cont.) Calculations:

Powers-of-Two Adjustment Rounding Order Intervals: T1* = Q1*/D1 = 36.09/1000 = 0.03609 yrs = 13.17  16 days T2* = Q2*/D2 = 114.14/1000 = 0.11414 yrs = 41.66  32 days T3* = Q3*/D3 = 11.41/100 = 0.11414 yrs = 41.66  32 days T4* = Q4*/D4 = 36.09/100 = 0.3609 yrs = 131.73  128 days Rounded Order Quantities: Q1' = D1 T1'/365 = 1000  16/365 = 43.84 Q2' = D2 T2' /365 = 1000  32/365 = 87.67 Q3' = D3 T3' /365 = 100  32/365 = 8.77 Q4' = D4 T4' /365 = 100  128/365 = 35.07

Powers-of-Two Adjustment (cont.) Resulting Inventory and Order Frequency: Optimal inventory investment is $3,126.53 and order frequency is 12. After rounding to nearest powers-of-two, we get:

Questions – Raw Materials Do you track vendor performance (i.e., as to variability)? Do you have a vendor certification program? Do your vendor contracts have provisions for varying quantities? Are purchasing procedures different for different part categories? Do you make use of JIT deliveries? Do you have excessive “wait to match” inventory? (May need more safety stock of inexpensive parts.) Do you have too many vendors? Is current order frequency rationalized?

Work-in-Process Reasons for Inventory: queueing (variability) processing waiting to move (batching) moving waiting to match (synchronization)

Work-in-Process (cont.) Improvement Policies: pull systems synchronization schemes lot splitting flow-oriented layout, floating work setup reduction reliability/maintainability upgrades focused factories improved yield/rework better scheduling judicious vendoring

Work-in-Process (cont.) Benchmarks: coefficients of variation below one WIP below 10 times critical WIP relative benchmarks depend on position in supply chain Models: queueing models simulation

Science Behind WIP Reduction Cycle Time: WIP: Conclusion: CT and WIP can be reduced by reducing utilization, variability, or both.

Questions – WIP Are you using production leveling and due date negotiation to smooth releases? Do you have long, infrequent outages on machines? Do you have long setup times on highly utilized machines? Do you move product infrequently in large batches? Do some machines have utilizations in excess of 95%? Do you have significant yield/rework problems? Do you have significant waiting inventory at assembly stations (i.e., synchronization problems)?

Finished Goods Inventory Reasons for Inventory: respond to variable customer demand absorb variability in cycle times build for seasonality forecast errors Improvement Policies: dynamic lead time quoting cycle time reduction cycle time variability reduction late customization balancing labor/inventory improved forecasting

Finished Goods Inventory (cont.) Benchmarks: seasonal products: 6-12 turns make-to-order products: 30-50+ turns make-to-stock products: 12-24 turns Models: reorder point models queueing models simulation

Questions – FGI All the WIP questions apply here as well. Are lead times negotiated dynamically? Have you exploited opportunities for late customization (e.g., bank stocks, product standardization, etc.)? Have you adequately considered variable labor (seasonal hiring, cross-trained workers, overtime)? Have you evaluated your forecasting procedures against past performance?

Spare Parts Inventory Reasons for Inventory: Improvement Policies: customer service purchasing/production lead times batch replenishment Improvement Policies: separate scheduled/unscheduled demand increase order frequency eliminate unnecessary safety stock differentiate parts with respect to fill rate/order frequency forecast life cycle effects on demand balance hierarchical inventories

Spare Parts Inventory (cont.) Benchmarks: scheduled demand parts: 6-24 turns unscheduled demand parts: 1-12+ turns (highly variable!) Wharton survey Models: (Q,r) distribution requirements planning (DRP) multi-echelon models

Multi-Product (Q,r) Systems Many inventory systems (including most spare parts systems) involve multiple products (parts) Products are not always separable because: average service is a function of all products cost of holding inventory is different for different products Different formulations are possible, including: constraint formulation (usually more intuitive) cost formulation (easier to model, can be equivalent to constraint approach)

Model Inputs and Outputs Costs Order (A) Backorder (b) or Stockout (k) Holding (h) Stocking Parameters (by part) Order Quantity (Q) Reorder Point (r) Inputs (by part) Cost (c) Mean LT demand (q) Std Dev of LT demand (s) MODEL Performance Measures (by part and for system) Order Frequency (F) Fill Rate (S) Backorder Level (B) Inventory Level (I)

Multi-Prod (Q,r) Systems – Constraint Formulations Backorder model min Inventory investment subject to: Average order frequency  F Average backorder level  B Fill rate model Average fill rate  S Two different ways to represent customer service.

Multi Product (Q,r) Notation

Multi-Product (Q,r) Notation (cont.) Decision Variables: Performance Measures:

Backorder Constraint Formulation Verbal Formulation: min Inventory investment subject to: Average order frequency  F Total backorder level  B Mathematical Formulation: “Coupling” of Q and r makes this hard to solve.

Backorder Cost Formulation Verbal Formulation: min Ordering Cost + Backorder Cost + Holding Cost Mathematical Formulation: “Coupling” of Q and r makes this hard to solve.

Fill Rate Constraint Formulation Verbal Formulation: min Inventory investment subject to: Average order frequency  F Average fill rate  S Mathematical Formulation: “Coupling” of Q and r makes this hard to solve.

Fill Rate Cost Formulation Verbal Formulation: min Ordering Cost + Stockout Cost + Holding Cost Mathematical Formulation: Note: a stockout cost penalizes each order not filled from stock by k regardless of the duration of the stockout “Coupling” of Q and r makes this hard to solve.

Relationship Between Cost and Constraint Formulations Method: 1) Use cost model to find Qi and ri, but keep track of average order frequency and fill rate using formulas from constraint model. 2) Vary order cost A until order frequency constraint is satisfied, then vary backorder cost b (stockout cost k) until backorder (fill rate) constraint is satisfied. Problems: Even with cost model, these are often a large-scale integer nonlinear optimization problems, which are hard. Because Bi(Qi,ri), Si(Qi,ri), Ii(Qi,ri) depend on both Qi and ri, solution will be “coupled”, so step (2) above won’t work without iteration between A and b (or k).

Type I (Base Stock) Approximation for Backorder Model replace Bi(Qi,ri) with base stock formula for average backorder level, B(ri) Note that this “decouples” Qi from ri because Fi(Qi,ri) = Di/Qi depends only on Qi and not ri Resulting Model:

Solution of Approximate Backorder Model Taking derivative with respect to Qi and solving yields: Taking derivative with respect to ri and solving yields: EOQ formula again base stock formula again if Gi is normal(i,i), where (zi)=b/(hi+b)

Using Approximate Cost Solution to Get a Solution to the Constraint Formulation 1) Pick initial A, b values. 2) Solve for Qi, ri using: 3) Compute average order frequency and backorder level: 4) Adjust A until Adjust b until Note: use exact formula for B(Qi,ri) not approx. Note: search can be automated with Solver in Excel.

Type I and II Approximation for Fill Rate Model Use EOQ to compute Qi as before Replace Bi(Qi,ri) with B(ri) (Type I approx) in inventory cost term. Replace Si(Qi,ri) with 1-B(ri)/Qi (Type II approx) in stockout term Resulting Model: Note: we use this approximate cost function to compute ri only, not Qi

Solution of Approximate Fill Rate Model EOQ formula for Qi yields: Taking derivative with respect to ri and solving yields: Note: modified version of basestock formula, which involves Qi if Gi is normal(i,i), where (zi)=kDi/(kDi+hQi)

Using Approximate Cost Solution to Get a Solution to the Constraint Formulation 1) Pick initial A, k values. 2) Solve for Qi, ri using: 3) Compute average order frequency and fill rate using: 4) Adjust A until Adjust b until Note: use exact formula for S(Qi,ri) not approx. Note: search can be automated with Solver in Excel.

Multi-Product (Q,r) Insights All other things being equal, an optimal solution will hold less inventory (i.e., smaller Q and r) for an expensive part than for an expensive one. Reduction in total inventory investment resulting from use of “optimized” solution instead of constant service (i.e., same fill rate for all parts) can be substantial. Aggregate service may not always be valid: could lead to undesirable impacts on some customers additional constraints (minimum stock or service) may be appropriate

Questions – Spare Parts Inventory Is scheduled demand handled separately from unscheduled demand? Are stocking rules sensitive to demand, replenishment lead time, and cost? Can you predict life-cycle demand better? Are you relying on historical usage only? Are your replenishment lead times accurate? Is excess distributed inventory returned from regional facilities to central warehouse? How are regional facility managers evaluated against inventory? Frequency of inspection? Are lateral transhipments between regional facilities being used effectively? Officially?

Multi-Echelon Inventory Systems Questions: How much to stock? Where to stock it? How to coordinate levels?

Types of Multi-Echelon Systems Level 1 Level 2 Level 3 Serial System General Arborescent System Stocking Site Inventory Flow

Two Echelon System Warehouse Facilities evaluate with (Q,r) model compute stocking parameters and performance measures Facilities evaluate with base stock model (ensures one-at-a-time demands at warehouse consider delays due to stockouts at warehouse in replenishment lead times F W F F

Facility Notation

Warehouse Notation

Variables and Measures in Two Echelon Model Decision Variables: Performance Measures:

Facility Lead Times (mean) Delay due to backordering: Effective lead time for part i to facility m: by Little’s law use this in place of  in base stock model for facilities

Facility Lead Times (std dev) If y=delay for an order that encounters stockout, then: Variance of Lim: Note: Si=Si(Qi,ri) this just picks y to match mean, which we already know we can use this in place of  in normal base stock model for facilities

Two Echelon (Single Product) Example D = 14 units per year (Poisson demand) at warehouse l = 45 days Q = 5 r = 3 Dm = 7 units per year at a facility lm = 1 day (warehouse to facility) B(Q,r) = 0.0114 S(Q,r) = 0.9721 W = 365B(Q,r)/D = 365(0.0114/14) = 0.296 days E[Lm] = 1 + 0.296 = 1.296 days m= DmE[Lm] = (7/365)(1.296) = 0.0249 units single facility that accounts for half of annual demand from previous example

Two Echelon Example (cont.) Standard deviation of demand during replenishment lead time: Backorder level: computed from basestock model using m and m Conclusion: base stock level of 2 probably reasonable for facility.

Observations on Multi-Echelon Systems Service at central DC is a means to an ends (i.e., service at facilities). Service matters at locations that interface with customers: fill rate (fraction of demands filled from stock) average delay (expected wait for a part) Multi-echelon systems are hard to model/solve exactly, so we try to “decouple” levels. Example: set fill rate at at DC and compute expected delay at facilities, then search over DC service to minimize system cost. Structural changes are an option (e.g., change number of DC's or facilities, allow cross-sharing, have suppliers deliver directly to outlets, etc.)