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**Supply Chain Management**

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

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**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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**Plan of Attack Classification: Justification: raw materials**

work-in-process (WIP) finished goods inventory (FGI) spare parts Justification: Why is inventory being held? benchmarking

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**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.

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**Raw Materials Reasons for Inventory: Improvement Policies:**

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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**Raw Materials (cont.) Models: EOQ power-of-two**

service constrained optimization model

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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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**Percentage to the total inventory**

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. Value per year (%) Percentage to the total inventory Class A Parts 75%-80% 5%-10% B Parts 10%-15% 10%~15% 80% C parts 10%

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**ABC Analysis (3) Example：A semiconductor manufacturer 20% 30% 50% 72%**

Part Number Percentage Annual Value($) Ratio Classes 1 2 3 4 5 6 7 8 9 10 20% 30% 50% 90,000 77,000 26,350 15,001 12,500 8,250 1200 850 504 150 72% 23% 5% A B C

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ABC Analysis (4)

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**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)

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**Multiproduct EOQ Models (cont.)**

Cost-Based EOQ Model: For part i, Frequency Constrained EOQ Model: min Inventory holding cost subject to: Average order frequency F

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**Multiproduct EOQ Solution Approach**

Constraint Formulation: Cost Formulation:

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**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

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**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.

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**Multiproduct EOQ Example**

Input Data: Target F value: F = 12

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**Multiproduct EOQ Example (cont.)**

Calculations:

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**Powers-of-Two Adjustment**

Rounding Order Intervals: T1* = Q1*/D1 = 36.09/1000 = yrs = 16 days T2* = Q2*/D2 = /1000 = yrs = 32 days T3* = Q3*/D3 = 11.41/100 = yrs = 32 days T4* = Q4*/D4 = 36.09/100 = yrs = 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

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**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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**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 (防备) 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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**Work-in-Process WIP Inventory will be in one of 5 states:**

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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**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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**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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**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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**Science Behind WIP Reduction**

Cycle Time (can be approximated as): WIP (by Little’s Law): Conclusion: CT and WIP can be reduced by reducing utilization, variability, or both. te: mean processing time; ce: coefficient of variation of processing time; ca: coefficient of variation of arrivals; u: utilization rate

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Questions – WIP Are you using production leveling and due date negotiation to smooth releases? Do you have long, infrequent outages (储运损耗) 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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**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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**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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**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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**Spare Parts Inventory Reasons for Inventory: Improvement Policies:**

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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**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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**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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**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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**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)

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**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) Average fill rate S Two different ways to represent customer service.

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**Multi Product (Q,r) Notation**

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**Multi-Product (Q,r) Notation (cont.)**

Decision Variables: Performance Measures:

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**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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**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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**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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**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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**Relationship Between Cost and Constraint Formulations**

Method: 1) Use cost model to find Qi and ri, 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 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).

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**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:

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**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)

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**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.

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**Type I and II Approximation for Fill Rate (or Stockout) 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

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**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)

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**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.

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**Multi-Product (Q,r) Example**

Cost and demand data j hj ($/unit) Dj (units/yr) lj (days) j (units) j (units) 1 2 3 4 100 10 1000 60 30 15 164.4 82.2 27.4 4.1 12.8 9.1 5.2 2.0

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**Multi-Product (Q,r) Example**

Results of multipart Stockout model (Q,r) calculations j Qj (units) kDj/(kDj+hjQj) (unitless) rj (units) Fj (Order Freq.) Sj (Fill Rate) Bj (Backorder Level) Ij (Inventory Level) ($) 1 2 3 4 36.1 114.1 11.4 0.666 0.863 0.387 169.9 92.1 25.9 5.0 27.7 8.8 2.8 0.922 0.995 0.749 0.988 0.544 0.022 0.918 0.014 2410.6 670.24 512.52 189.33 12.0 0.95 1.497 3782.7 Step 1. Assume a target average order frequency of F=12. Step 2. Assume a target average fill rate of S=0.95. Too low!! Result: k = 7.213

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**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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**Multi-Product (Q,r) Example**

Results of multipart Backorder model (Q,r) calculations j Qj (units) bj/(bj +h) (unitless) rj (units) Fj (Order Freq.) Sj (Fill Rate) Bj (Backorder Level) Ij (Inventory Level) ($) 1 2 3 4 36.1 114.1 11.4 0.538 0.921 165.6 95.0 27.9 7.0 27.7 8.8 2.8 0.875 0.997 0.840 0.998 0.974 0.010 0.511 0.002 2024.8 698.76 671.85 209.1 12.0 0.934 1.497 3604.5 Using backorder model algorithm to adjust the backorder cost b until the total backorder level achieves a specified target. To make a comparison of stockout model and backorder model, we set B=1.497 units. 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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**Multi-Product (Q,r) Example**

Results of multipart Backorder model (Q,r) calculations j Qj (units) bj/(bj +h) (unitless) rj (units) Fj (Order Freq.) Sj (Fill Rate) Bj (Backorder Level) Ij (Inventory Level) ($) 1 2 3 4 36.1 114.1 11.4 0.538 0.921 165.6 95.0 27.9 7.0 27.7 8.8 2.8 0.875 0.997 0.840 0.998 0.974 0.010 0.511 0.002 2024.8 698.76 671.85 209.1 12.0 0.934 1.497 3604.5 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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**Multi-Product (Q,r) Example**

Results of multipart Backorder model (Q,r) calculations j Qj (units) bj/(bj +h) (unitless) rj (units) Fj (Order Freq.) Sj (Fill Rate) Bj (Backorder Level) Ij (Inventory Level) ($) 1 2 3 4 36.1 114.1 11.4 0.538 0.921 165.6 95.0 27.9 7.0 27.7 8.8 2.8 0.875 0.997 0.840 0.998 0.974 0.010 0.511 0.002 2024.8 698.76 671.85 209.1 12.0 0.934 1.497 3604.5 Conclusions: 3. Difference between 2 solutions: the fill rates and inventory levels are different. Given same backorder level, the backorder model generates smaller inventory investment ($ 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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**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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**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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**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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**Types of Multi-Echelon Systems**

Level 1 Level 2 Level 3 Serial System General Arborescent System Stocking Site Inventory Flow

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**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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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 Little’s 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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**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

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Facility Notation

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Warehouse Notation

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**Variables and Measures in Two Echelon Model**

Decision Variables: Performance Measures:

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**Facility Lead Times (mean)**

Delay due to backordering: Effective lead time for part i to facility m: by Little’s law Wait is analogous to cycle time, backorder level is analogous to WIP, and demand rate is analogous to throughput. use this in place of in base stock model for facilities

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**Facility Lead Times (std dev)**

Suppose there are 2 possibilities: Either order sees no delay and lead time is ljm, or it encounters a stockout delay and has lead time ljm+y (y is a deterministic delay), since the probability of stockout is 1-Sj, we have 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

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**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) = S(Q,r) = W = 365B(Q,r)/D = 365(0.0114/14) = days E[Lm] = = days m= DmE[Lm] = (7/365)(1.296) = units single facility that accounts for half of annual demand from previous example

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**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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**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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