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1 IE&M Shanghai Jiao Tong University Supply Chain Management One's work may be finished some day, but one's education never. – Alexandre Dumas

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2 IE&M Shanghai Jiao Tong University 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…

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3 IE&M Shanghai Jiao Tong University Plan of Attack Classification: raw materials work-in-process (WIP) finished goods inventory (FGI) spare parts Justification: Why is inventory being held? benchmarking

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4 IE&M Shanghai Jiao Tong University Plan of Attack (cont.) Structural Changes: 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.

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5 IE&M Shanghai Jiao Tong University Raw Materials Reasons for Inventory: batching (quantity discounts, purchasing capacity, … ) safety stock (buffer against randomness in supply/production) obsolescence (changes in demand or design, also called obsolete inventory) Improvement Policies: ABC classification (stratify parts management) JIT deliveries (expensive and/or bulky items) vendor monitoring/management

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6 IE&M Shanghai Jiao Tong University Raw Materials (cont.) Models: EOQ power-of-two service constrained optimization model

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7 IE&M Shanghai Jiao Tong University ABC Analysis (1) A Parts The first 5% to 10% of the parts, accounting for 75% to 80% of the total annual cost B Parts The next 10% to 15% of the parts, accounting for 10% to 15% of the total annual cost C Parts The bottom 80% or so of the parts, accounting for only 10% of the total annual cost

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8 IE&M Shanghai Jiao Tong University ABC Analysis (2) Basic Idea classify the parts according to their values per year Conclusion: we should differentiate the parts in inventory and use different inventory control strategies for different parts. Class Value per year (%) Percentage to the total inventory A Parts B Parts C parts 75%-80% 10%-15% 10% 5%-10% 10%~15% 80%

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9 IE&M Shanghai Jiao Tong University ABC Analysis (3) Example A semiconductor manufacturer Part NumberPercentageAnnual Value($) Ratio Classes % 30% 50% 90,000 77,000 26,350 15,001 12,500 8, % 23% 5% A B C

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10 IE&M Shanghai Jiao Tong University ABC Analysis (4)

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11 IE&M Shanghai Jiao Tong University Multiproduct EOQ Models Notation: N = total number of distinct part numbers in the system D i = demand rate (units per year) for part i c i = unit production cost of part i A i = fixed cost to place an order for part i h i = cost to hold one unit of part i for one year Q i = the size of the order or lot size for part i (decision variable)

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12 IE&M Shanghai Jiao Tong University Multiproduct EOQ Models (cont.) Cost-Based EOQ Model: For part i, Frequency Constrained EOQ Model: minInventory holding cost subject to: Average order frequency F

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13 IE&M Shanghai Jiao Tong University Multiproduct EOQ Solution Approach Constraint Formulation: Cost Formulation:

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14 IE&M Shanghai Jiao Tong University Multiproduct EOQ Solution Approach (cont.) Cost Solution: Differentiate Y(Q) with respect to Q i, set equal to zero, and solve: Constraint Solution: For a given A we can find Q i (A) using the above formula. The resulting average order frequency is: If F(A) F then penalty is too low and needs to be increased. No surprise - regular EOQ formula

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15 IE&M Shanghai Jiao Tong University 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 Q i (A) for i = 1, …, N. Step (2) Compute the resulting order frequency: If |F(A) - F| <, STOP; Q i * = Q i (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.

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16 IE&M Shanghai Jiao Tong University Multiproduct EOQ Example Input Data: Target F value: F = 12

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17 IE&M Shanghai Jiao Tong University Multiproduct EOQ Example (cont.) Calculations:

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18 IE&M Shanghai Jiao Tong University Powers-of-Two Adjustment Rounding Order Intervals: T 1 * = Q 1 * /D 1 = 36.09/1000 = yrs = days T 2 * = Q 2 * /D 2 = /1000 = yrs = days T 3 * = Q 3 * /D 3 = 11.41/100 = yrs = days T 4 * = Q 4 * /D 4 = 36.09/100 = yrs = days Rounded Order Quantities: Q 1 ' = D 1 T 1 '/365 = /365 = Q 2 ' = D 2 T 2 ' /365 = /365 = Q 3 ' = D 3 T 3 ' /365 = /365 = 8.77 Q 4 ' = D 4 T 4 ' /365 = /365 = 35.07

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19 IE&M Shanghai Jiao Tong University Powers-of-Two Adjustment (cont.) Resulting Inventory and Order Frequency: Optimal inventory investment is $3, and order frequency is 12. After rounding to nearest powers-of-two, we get:

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20 IE&M Shanghai Jiao Tong University 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?

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21 IE&M Shanghai Jiao Tong University Work-in-Process Despite the JIT goal of zero inventories, we can never operate a manufacturing system with zero WIP, since zero WIP implies zero throughput. WIP Inventory will be in one of 5 states: queueing (if it is waiting for a resource) processing (if it is being worked on by a resource) waiting to move (if it has to wait for other jobs to arrive in order to form a batch) moving (if it is actually being transported between resources) waiting to match (synchronization)

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22 IE&M Shanghai Jiao Tong University Work-in-Process (Improvement Policies) pull systems (will achieve the same level of throughput with a lower average WIP – reduce ququeing and wait- for-match) synchronization schemes (– reduce wait-for-match) lot splitting (process lots and move lots do not have to be the same – reduce wait-for-batch) flow-oriented layout (more frequent moves can be facilitated by the plant layout – reduce wait-for-batch) floating work (cross-trained workers can increase the effective capacity of the production line – reduce ququeing)

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23 IE&M Shanghai Jiao Tong University Work-in-Process (Improvement Policies) setup reduction (reducing setups increases effective capacity and more frequent setups will decrease effective variability at machines – reduce ququeing) reliability/maintainability upgrades (reduce effective variability at machines – reduce ququeing) focused factories (Not all WIP need be treated equally: low-volume parts could be produced in a job shop environment; high-volume parts could be assigned to lines with few setups and steady flow by using high efficient pull system – reduce ququeing) improved yield/rework (enhanced quality– reduce ququeing) better finite-capacity scheduling (– reduce ququeing)

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24 IE&M Shanghai Jiao Tong University Work-in-Process (cont.) Benchmarks: WIP below 10 times critical WIP (smallest WIP level required by a line to achieve full throughput under the best conditions) relative benchmarks depend on position in supply chain Models: queueing models simulation

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25 IE&M Shanghai Jiao Tong University Science Behind WIP Reduction Cycle Time (can be approximated as): WIP (by Littles Law): Conclusion: CT and WIP can be reduced by reducing utilization, variability, or both. t e : mean processing time; c e : coefficient of variation of processing time; c a : coefficient of variation of arrivals; u: utilization rate

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26 IE&M Shanghai Jiao Tong University 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)?

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27 IE&M Shanghai Jiao Tong University 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 (instead of fixed lead time) cycle time reduction cycle time variability reduction (more variability in cycle times, the more safety stock we must build) late customization (Semi-finished inventory is more flexible, if can be used to produce more than one finished product, which makes it possible to carry less total inventory) balancing labor, capacity and inventory (product is produced during low-demand periods and held as FGI to meet demand during peak periods) improved forecasting

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28 IE&M Shanghai Jiao Tong University Finished Goods Inventory (cont.) Benchmarks: seasonal products make-to-order products make-to-stock products Models: reorder point models queueing models simulation

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

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30 IE&M Shanghai Jiao Tong University Spare Parts Inventory Reasons for Inventory: customer service purchasing/production lead times batch replenishment Improvement Policies: separate scheduled/unscheduled repairing 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

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31 IE&M Shanghai Jiao Tong University Spare Parts Inventory (cont.) 2 distinct types of spare parts Scheduled preventive maintenance Unscheduled emergency repairs Scheduled maintenance represents a predictable demand source This demand is much more stable than customer demand MRP logic is applicable to these parts Unscheduled emergency repairs are unpredictable (Q, r) model can be used

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32 IE&M Shanghai Jiao Tong University Spare Parts Inventory (cont.) Benchmarks: scheduled demand parts unscheduled demand parts Models: (Q,r) distribution requirements planning (DRP) multi-echelon models

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33 IE&M Shanghai Jiao Tong University 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)

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

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35 IE&M Shanghai Jiao Tong University 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 (or Stockout model) min Inventory investment subject to: Average order frequency F Average fill rate S Two different ways to represent customer service.

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36 IE&M Shanghai Jiao Tong University Multi Product (Q,r) Notation

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37 IE&M Shanghai Jiao Tong University Multi-Product (Q,r) Notation (cont.) Decision Variables: Performance Measures:

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38 IE&M Shanghai Jiao Tong University 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.

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39 IE&M Shanghai Jiao Tong University Backorder Cost Formulation Verbal Formulation: min Ordering Cost + Backorder Cost + Holding Cost Mathematical Formulation: Coupling of Q and r makes this hard to solve.

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40 IE&M Shanghai Jiao Tong University 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.

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41 IE&M Shanghai Jiao Tong University 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.

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42 IE&M Shanghai Jiao Tong University Relationship Between Cost and Constraint Formulations Method: 1) Use cost model to find Q i and r i, but keep track of average order frequency (F) and fill rate (S) 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 B i (Q i,r i ), S i (Q i,r i ), I i (Q i,r i ) depend on both Q i and r i, solution will be coupled, so step (2) above wont work without iteration between A and b (or k).

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43 IE&M Shanghai Jiao Tong University Type I (Base Stock) Approximation for Backorder Model Approximation: replace B i (Q i,r i ) with base stock formula for average backorder level, B(r i ) Note that this decouples Q i from r i because F i (Q i,r i ) = D i /Q i depends only on Q i and not r i Resulting Model:

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44 IE&M Shanghai Jiao Tong University Solution of Approximate Backorder Model Taking derivative with respect to Q i and solving yields: Taking derivative with respect to r i and solving yields: if G i is normal ( i, i ), where (z i ) =b/(h i +b) EOQ formula again base stock formula again

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45 IE&M Shanghai Jiao Tong University Using Approximate Cost Solution to Get a Solution to the Constraint Formulation 1) Pick initial A, b values. 2) Solve for Q i, r i using: 3) Compute average order frequency and backorder level: 4) Adjust A until Adjust b until Note: use exact formula for B(Q i,r i ) not approx. Note: search can be automated with Solver in Excel.

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46 IE&M Shanghai Jiao Tong University Type I and II Approximation for Fill Rate (or Stockout) Model Approximation: Use EOQ to compute Q i as before Replace B i (Q i,r i ) with B(r i ) (Type I approx.) in inventory cost term. Replace S i (Q i,r i ) with 1-B(r i )/Q i (Type II approx.) in stockout term Resulting Model: Note: we use this approximate cost function to compute r i only, not Q i

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47 IE&M Shanghai Jiao Tong University Solution of Approximate Fill Rate Model EOQ formula for Q i yields: Taking derivative with respect to r i and solving yields: if G i is normal ( i, i ), where ( z i )= kD i /(kD i +hQ i ) Note: modified version of basestock formula, which involves Q i

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

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49 IE&M Shanghai Jiao Tong University Multi-Product (Q,r) Example jh j ($/unit)D j (units/yr)l j (days) j (units) Cost and demand data

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50 IE&M Shanghai Jiao Tong University Multi-Product (Q,r) Example j Q j (units) kD j /(kD j +h j Q j ) (unitless) r j (units)F j (Order Freq.) S j (Fill Rate) B j (Backo rder Level) I j (Invent ory Level) ($) Results of multipart Stockout model (Q,r) calculations Step 1. Assume a target average order frequency of F=12. Step 2. Assume a target average fill rate of S=0.95. Result: k = Too low!!

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51 IE&M Shanghai Jiao Tong University Conclusions of Multipart Stockout (Q,r) Model This algorithm produce a very high fill rate (99.5%) for inexpensive, low-demand part 3. Intuitively, the algorithm is trying to achieve an average fill rate of 95% as cheaply as possible, so it makes service high where it can do so cheaply (i.e., where the unit holding cost is low) and where it has a big impact on the overall average (i.e., where annual demand is high).

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52 IE&M Shanghai Jiao Tong University Multi-Product (Q,r) Example jQ j (units) b j /(b j +h) (unitless) r j (units)F j (Order Freq.) S j (Fill Rate) B j (Backor der Level) I j (Invento ry Level) ($) Results of multipart Backorder model (Q,r) calculations 1.Using backorder model algorithm to adjust the backorder cost b until the total backorder level achieves a specified target. 2.To make a comparison of stockout model and backorder model, we set B=1.497 units. 3.Use backorder algorithm to find the backorder penalty that causes total backorders to achieve this backorder level (B=1.497 units ), it turns out when b=116.5, B=1.497 units.

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53 IE&M Shanghai Jiao Tong University Multi-Product (Q,r) Example jQ j (units)b j /(b j +h) (unitless) r j (units)F j (Order Freq.) S j (Fill Rate) B j (Backorde r Level) I j (Inventor y Level) ($) Results of multipart Backorder model (Q,r) calculations Conclusions: This algorithm results in low backorder levels for inexpensive parts 2 and 4, but higher backorder levels for expensive parts 1 and 3. Higher backorder levels for high-demand parts (i.e., parts 1 and 2) because higher demand produces more backorders when all other things are equal.

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54 IE&M Shanghai Jiao Tong University Multi-Product (Q,r) Example jQ j (units)b j /(b j +h) (unitless) r j (units)F j (Order Freq.) S j (Fill Rate) B j (Backorde r Level) I j (Inventor y Level) ($) Results of multipart Backorder model (Q,r) calculations Conclusions: 3. Difference between 2 solutions: the fill rates and inventory levels are different. Given same backorder level, the backorder model generates smaller inventory investment ($ vs $3782.7). But it does so at the price of a lower fill rate (93.4% vs 95%). 4. If fill rate is the right measure of service, the stockout model is appropriate. If backorder level (or time delay) is a better representation of service, then the backorder model makes more sense.

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55 IE&M Shanghai Jiao Tong University 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 inexpensive one. Reduction in total inventory investment resulting from use of optimized solution instead of constant service (i.e., same fill rates 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

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56 IE&M Shanghai Jiao Tong University 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?

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57 IE&M Shanghai Jiao Tong University Multi-Echelon Inventory Systems Many supply chains Involve multiple levels as well as multiple parts Because of variability pooling, stocking inventory in a central location, such as a warehouse or distribution center, allow holding less safety stock than holding separate inventories at individual demand sites. The basic challenge in multiechelon supply chains is to balance the efficiency of central inventories with the responsiveness of distributed inventories so as to provide high system performance without excessive investment in inventory. Questions: How much to stock? Where to stock it? How to coordinate levels?

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58 IE&M Shanghai Jiao Tong University Types of Multi-Echelon Systems Level 1Level 2Level 3 Stocking SiteInventory Flow Serial System General Arborescent System

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59 IE&M Shanghai Jiao Tong University System Configurations Feature of a multiechelon supply chain Lower-level locations are supplied by higher-level locations. Some variations Transshipment between locations at the same level For example, regional warehouses can supply one another In short, multiechelon systems can be extremely complex. System configuration itself is a decision variable. Determining the # of inventory levels, the locations of warehouses, and the policies for interconnecting them can be among the most important logistics decisions for a distribution center design. Research indicates that applying directly single-level approaches to multiechelon supply chains may work poorly.

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60 IE&M Shanghai Jiao Tong University Performance Measures Fill rate: the fraction of demands that are met out of stock. This could apply at any level in the supply chain Backorder level: average # of orders waiting to be filled. This measure applies to systems where backordering occurs. Backorder level is closely related to the average backorder delay, by using Littles Law: average backorder delay = average backorder level/ average demand rate Lost sales: # of potential orders lost due to stockout. This measure applies to systems in which customers go elsewhere rather than wait for a backordered item. Lost sales = (1-Fill rate) × average demand rate If fill rate is 95% and annual demand is 100 parts a year, then (1-0.95)(100) =5 parts per year will be lost due to stockout. Probability of delay: likelihood that an activity (e.g., a machine repair) will be delayed for lack of inventory. In general, the probability of delay in a multipart, multilevel system is a function of the fill rates of various parts. In summary, fill rate and average backorder level are key measures, since other measures can be computed from them.

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61 IE&M Shanghai Jiao Tong University Two Echelon System Warehouse 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 W FFF

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62 IE&M Shanghai Jiao Tong University Facility Notation

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63 IE&M Shanghai Jiao Tong University Warehouse Notation

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64 IE&M Shanghai Jiao Tong University Variables and Measures in Two Echelon Model Decision Variables: Performance Measures:

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65 IE&M Shanghai Jiao Tong University Facility Lead Times (mean) Delay due to backordering: Effective lead time for part i to facility m: by Littles law use this in place of in base stock model for facilities Wait is analogous to cycle time, backorder level is analogous to WIP, and demand rate is analogous to throughput.

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66 IE&M Shanghai Jiao Tong University Facility Lead Times (std dev) If y=delay for an order that encounters stockout, then: Variance of L im : Note: S i =S i (Q i,r i ) 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 Suppose there are 2 possibilities: Either order sees no delay and lead time is l jm, or it encounters a stockout delay and has lead time l jm +y (y is a deterministic delay), since the probability of stockout is 1-S j, we have

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67 IE&M Shanghai Jiao Tong University Two Echelon (Single Product) Example D = 14 units per year (Poisson demand) at warehouse l = 45 days Q = 5 r= 3 D m = 7 units per year at a facility l m = 1 day (warehouse to facility) B(Q,r) = S(Q,r) = W = 365B(Q,r)/D = 365(0.0114/14) = days E[L m ] = = days m = D m E[L m ] = (7/365)(1.296) = units from previous example single facility that accounts for half of annual demand

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68 IE&M Shanghai Jiao Tong University 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.

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69 IE&M Shanghai Jiao Tong University Observations on Multi-Echelon Systems 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.)

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