Inventory Models for Special Cases: Multiple Items & Locations

Transcription

1 CTL.SC1x -Supply Chain & Logistics Fundamentals Inventory Models for Special Cases: Multiple Items & Locations MIT Center for Transportation & Logistics

2 Agenda Inventory Policies for Multiple Items Grouping Like Items Exchange Curves Inventory Policies for Multiple Locations Location Pooling 2

3 Model Assumptions: >1 Items Demand Constant vs Variable Known vs Random Continuous vs Discrete Lead Time Instantaneous Constant vs Variable Deterministic vs Stochastic Internally Replenished Dependence of Items Independent Correlated Indentured Review Time Continuous vs Periodic Number of Locations One vs Multi vs Multi-Echelon Capacity / Resources Unlimited Limited / Constrained Discounts None All Units vs Incremental vs One Time Excess Demand None All orders are backordered Lost orders Substitution Perishability None Uniform with time Non-linear with time Planning Horizon Single Period Finite Period Infinite Number of Items One vs Many Form of Product Single Stage Multi-Stage 3

4 Managing Multiple Items What are the problems with managing items independently? Lack of coordination constantly ordering items Ignores common constraints (e.g., financial budget or space) Missed opportunities for consolidation / synergies Waste of management time Two Issues to Solve Can we aggregate SKUs to use similar operating policies? w Group using common cost characteristics or break points w Group using Power of Two Policies How do we manage inventory under common constraints? w Exchange Curves for Cycle Stock w Exchange Curves for Safety Stock 4

5 Grouping Like Items Break Points 5

6 Grouping Like Items Break Points Basic Idea: Replenish higher value items faster Used for situations with multiple items that have: Relatively stable demand Common ordering costs, c t, and holding charges, h Different annual demands, D i, and purchase costs, c i Approach Pick a base time period, w 0, (typically a week) Create a set of candidate ordering periods (w 1, w 2, etc.) Find D i c i values where TRC(w j )=TRC(w j+1 ) Group SKUs with that fall in common value (D i c i ) buckets 6

7 Grouping Like Items - Example Selected w 0 = 1 week Number of weeks of supply (WOS) to order for item i ordering at time period j = Q ij = D i (w j /52) Selecting between options w 1 & w 2 (where w 1 <w 2 <w 3 etc.) becomes: c t D i /Q i1 + (c i hq i1 )/2 = c t D i /Q i2 + (c i hq i2 )/2 52c t D i /D i w 1 + c i hd i w 1 /104 = 52c t D i /D i w 2 + c i hd i w 2 /104 (c i hd i /104)(w 1 -w 2 ) = (52c t )(1/w 2 1/w 1 ) D i c i = [(104)(52c t )/(h(w 1 -w 2 ))] (1/w 2 1/w 1 ) D i c i = 5408c t / (hw 1 w 2 ) Rule if D i c i 5408c t /(hw 1 w 2 ) then select w 1 Else: if D i c i 5408c t /(hw 2 w 3 ) then select w 2 Else: if D i c i 5408c t /(hw 3 w 4 ) then select w 3 Else:

8 Grouping Like Items - Example Problem: Suppose you need to set up replenishment schedules for several hundred parts that have relatively stable (yet not necessarily the same) demand. They all have similar order costs (c t = $5) and holding charge (h = 0.20). You have the following potential ordering periods (in weeks): w 1 =1, w 2 =2, w 3 =4, w 4 =13, w 5 =26, and w 6 =52. What break-even ordering points should you establish? Solution: Break-point for selecting between 1 week or 2 weeks is: w D i c i = 5408 t / (hw 1 w 2 ) = 5408(5) / (.2)(1)(2) = $67,600 w If D i c i $67,600 then order 1 week s worth each week Break-point for selecting between 2 weeks or 4 weeks is: w D i c i = 5408c t / (hw 2 w 3 ) = 5408(5) / (.2)(2)(4) = $16,900 w If $67,600 > D i c i $16,900 then order 2 week s worth every 2 weeks 8

9 Grouping Like Items - Example Annual Value of each SKU (Dc) $80,000 $70,000 $60,000 $50,000 $40,000 $30,000 $20,000 $10,000 Final Ordering Break Points: Order every 1 week if D i c i $67,600 Order every 2 weeks if D i c i $16,900 Order every 4 weeks if D i c i $2,600 Order every 13 weeks if D i c i $400 Order every 26 weeks if D i c i $100 Order every 52 weeks otherwise $ Proposed Replenishment Cycle Length (weeks) 9

10 Grouping Using Power of Two Policies 10

11 Power of Two Policies T * 2 2k 2 T * Recall from previous lesson: Order in time intervals of powers of two Select a realistic base period, T Base (day, week, month) Guarantees that TRC will be within 6% of optimal! " ln T * % $ ' # 2 & ln 2 ( ) k ( ) ln T * 2 ( ) ln 2 2c t D Simple Process Create table of SKUs Calculate T * for each SKU T * = Q* D = Calculate T practical for each SKU c e D = 2c t Dc e T practical = 2 '! ln T* $ ) ( # &* ( " 2 %* ( ln( 2 * ( ) * ( * In Spreadsheets: T practical =2^ROUNDUP[LN(T * /SQRT(2)) / LN(2)] 11

12 Grouping Power of Two 18 Practical versus Optimal Cycle Lengths 16 Practical Cycle Length (Tpractical) weeks Optimal Cycle Length (T*) weeks Data of 55 SKUs Various c, c t, h, and D values T Practical T*(weeks) T Practical low high

13 Comparing Both Methods Proposed Cycle Length (T) Optimal Cycle Length (T*) Break-Point Power of Two 13

14 Exchange Curves for Cycle Stock 14

15 Exchange Curves Cycle Stock What if I have a budget for inventory? Find best allocation of inventory budget across multiple SKUs Cost parameters are management policy levers! Relevant Cost Parameters Holding Charge (h) w There is no single correct value w Reflection of management s investment and risk profile Ordering Cost (c t ) w Not known with any precision w Cost allocations for time and systems differ between firms Exchange Curve Trade-off between total annual cycle stock (TACS) and number of replenishments (N) Determine the c t /h value that meets budget constraints n Q TACS = i c i N = i=1 2 n D i i=1q i 15

16 Exchange Curves Cycle Stock " 2c t D % i $ n Q TACS = i c hc ' c i n i # i & n c = = t D i c i = c t i=1 2 i=1 2 i=1 2h h 1 2 n i=1 D i c i n D n N = i D n = i hd = i c i = h 1 i=1q i=1 i=1 i 2c t D 2c i t c t 2 hc i n i=1 D i c i Approach: Create table of SKUs with Annual Value (D i c i ) and (D i c i ) Find the sum of (D i c i ) term for SKUs being analyzed Calculate TACS and N for range of (c t /h) values Chart (N vs TACS) Approach adapted from Silver, Pyke, Peterson (1998), Inventory Management and Production Planning and Scheduling 16

17 Exchange Curves Cycle Stock Total Average Cycle Stock Value $150,000 $140,000 $130,000 $120,000 $110,000 $100,000 $90,000 $80,000 $70,000 $60,000 $50,000 $40,000 $30,000 $20,000 $10,000 $- Data of 65 SKUs from Hospital Ward c t /h N TACS $137, $130, $122, $114,659 c t /h = Current Operations Number of Annual Replenishments c t /h = $106, $96, $86, $75, $61, $43, $41, $38, $36, $33, $30, $27, $23, $19, $13, $13, $12, $11, $10, $9, $8, $7, $6, $4,334 17

18 Exchange Curves Safety Stock 18

19 Exchange Curves Safety Stock What if we have a safety stock budget? Need to trade-off cost of safety stock and level of service Key parameter is safety factor (k) usually set by management Estimate the aggregate service level for different budgets Process: 1. Select an inventory metric to target 2. Starting with a high metric value calculate: w The required k i to meet that target for each SKU w The resulting safety stock cost for each SKU and the total safety stock (TSS) w The other resulting inventory metrics of interest for each SKU and total 3. Lower the metric value, go to step 2 4. Chart resulting TSS versus Inventory Metrics TSS = n i=1 k i σ DLi c i TVIS = n i=1! D $ i # c Q i σ DLi G(k i )& " i % 19

20 Exchange Curves Safety Stock Total Cost of Safety Stock ($/year) $9,000 $8,000 $7,000 $6,000 $5,000 $4,000 $3,000 $2,000 $1,000 TSS = n i=1 k i σ DLi c i TVIS = n i=1! D $ i # c Q i σ DLi G(k i )& " i % Data of 65 SKUs from Hospital Ward CSL k TVIS TSS $19 $8, $233 $6, $515 $5, $808 $5, $1,123 $5, $1,468 $4, $1,814 $4, $2,130 $4, $2,491 $4, $2,902 $3, $3,299 $3, $5,358 $3, $7,776 $2, $10,437 $1, $13,313 $1, $16,245 $1, $19,887 $ $23,426 $ $27,707 $- $- $- $2,000 $4,000 $6,000 $8,000 $10,000 $12,000 $14,000 $16,000 $18,000 $20,000 $22,000 $24,000 Total Value of Items Short ($/year) 20

21 Multiple Locations

22 Example: MedEx Situation: MedEx is a medical device manufacturer that delivers products directly to hospitals wards. One item, the X104, is used by three different wards within Northwest Hospital with daily demand ~N(22, 4.6). The purchase cost (c) is $156, the lead time (L) to replenish is 2 days, order cost (c t ) is $40, annual holding charge (h) is 20%, and CSL is set at 99.9%. Assume a 365 day year. Currently each ward manages their own inventory independently using an (s, Q) inventory replenishment policy. Problem: How much cycle and safety stock should each ward hold? Case adapted from DeScioli, D. (2005) Differentiating the Hospital Supply Chain For Enhanced Performance, MIT Supply Chain Management Program Thesis. Image Source: 22

23 Example: MedEx individual wards Solution for each ward: Find Average Cycle Stock w Q* = [(2)(40)(365)(22)/(156)(.2)] = units w Average cycle stock per ward = Q*/2 = (144/2) = 72 units Find Average Safety Stock w µ DL = (22)(2) = 44 units w σ DL = (4.6)( 2) = units w k = 3.09 for CSL = 99.9% (from table or spreadsheet) w Average safety stock per ward = kσ DL = (3.09)(6.5) = 20.1 units Solution - across all three wards Average total cycle stock = 3(72) = 216 units or $6,739 Average safety stock = (3)(20.1) = 60.3 units or $1,881 What if they pool their inventories to a common location? 23

24 Example: MedEx pooled inventory Solution: Find Pooled Demand w Each ward has daily demand ~N(22, 4.6) w E[Daily Pooled Demand] = (3)(22) = 66 units w V[Daily Pooled Demand] = V[Ward 1 Demand]+V[Ward 2 Demand]+V[Ward 3 Demand] Or we could just say σ pooled = σ wardi n = (4.6)( 3) = Find Average Cycle Stock w Q* = [(2)(40) (365)(66)/(156)(.2)] = units w Average cycle stock = Q*/2 = (249/2) = 125 units Find Average Safety Stock w µ DL = (66)(2) = 132 units w σ DL = (8)( 2) = units w k = 3.09 for CSL = 99.9% (from table or spreadsheet) w Average safety stock = kσ DL = (3.09)(11.3) = units Solution - across all three wards Average total cycle stock = 125 units or $3,900 Average safety stock = 35 units or $1,092 24

25 Example: MedEx pooled inventory Strategy Average Cycle Stock Average Safety Stock Average Inventory Independent Pooled Why did the inventory reduce by 3? Coincidence? Independent q i * = Cycle Stock 2c d t i = 2c D t c e c e n * n! q $ IOH = i! # i=1 " 2 & = n Q * $ # & % " 2 % Safety Stock SS independent = kσ di = kσ D n Pooled Q* = 2c D! t IOH = Q * $ # & c e " 2 % SS pooled = kσ D 25

26 Key Points from Lesson 26

27 Key Points Inventory Policies for Multiple Items Grouping Like Items Use common operating policies w Break-even Q points vs. Power of Two vs.????? Exchange Curves Budget constraints w Use c t /h and k as management levers Inventory Policies for Multiple Locations Location Pooling Square Root Law w Pooling inventory from n to 1 location reduces SS & CS by n w Similarly, pooling from n to m locations reduces SS & CS by (n/m)!!!caution!!! This is based on many assumptions... w Evenly distributed demand w Ordering follows EOQ with common c t w Demand distribution in different locations is independent 27

28 CTL.SC1x -Supply Chain & Logistics Fundamentals Questions, Comments, Suggestions? Use the Discussion! MIT Center for Transportation & Logistics