Talk Title: Pitfalls with Linear Programming Optimization of Supply chain Networks

1. Talk Title: Pitfalls with Linear Programming Optimization of Supply chain Networks Speaker Name: Robert Stawicki Speaker Title: Assistant Professor Ramapo College of NJ

2. Background • Based on 20 Years Experience Implementing Supply Chain Models for Fortune 100 Companies • Formulating Models from Scratch Or • Models Provided by Major Supply Chain Solutions Vendors

3. Outline • LP Formulation for Supply Chain Optimization • Six Common Pitfalls and Their Work Arounds • Using Full Costing • Time Frame Too Short • Production Levelling • Inventory being “Reborn” (Product Aging Constraints) • Honoring Safety Stocks while Stocking Out Customers • Starting Inventory is “Free”

4. Minimize Total Cost • Production Cost • Inventory Carrying Cost • Intra-Company Transportation Cost • Transportation Cost to Customers • Stock Out Costs • Safety Stock Violation Cost • Subject To: • Material Balance Constraints • Capacity Constraints • Satisfy Demand Constraints • Satisfy Safety Stock Constraints Basic Formulation

5. Basic Formulation (Continued) • To Discuss: • Maximize vs. Minimize • Stockout vs. Backorder • Other Constraints • Model Size • See Appendix

6. Plant A Fixed Cost $2.00 Variable Cost / Unit $5.00 Total Cost / Unit $7.00 Plant B Fixed Cost $0.50 Variable Cost / Unit $6.00 Total Cost / Unit $6.50 Using Full Costing Assume Demand = 10,000 units All Other Costs Equal

7. Full Costing Based Model Plant B Produces all 10,000 Units Plant A Fixed Cost $20,000 Variable Cost 0 Total Cost $20,000 Plant B Fixed Cost $5,000 Variable Cost 60,000 Total Cost $65,000 Using Full Costing (continued) Total Cost $85,000 ___________________________________________________________________________________________________________________________________________________________________________________ Marginal Cost Based Model Plant A Produces all 10,000 Units Plant A Fixed Cost $20,000 Variable Cost 50,000 Total Cost $70,000 Plant B Fixed Cost $5,000 Variable Cost 0 Total Cost $5,000 Total Cost $75,000

8. Time Frame Too Short ___________________________________________________________________________________________________________________________________________________________________________________

9. Production Levelling A typical method is to add the following to the objective function: Loosely defined: “Minimize the change in production level from period to period.” And the following set of constraints: Where: LEVELCOST = A large penalty L+, L- = Change in Production Level

10. Production Levelling(Continued) The problem with this formulation is LP sees no difference between several small changes and one large one. It may actually prefer the large one as shown below.

11. Production Levelling(Continued) Instead of the previous change, add the following to the objective function: A better formulation: Notice only one variable per location for all time periods: In addition to the previous set of constraints, add the following two sets of constraints:

12. Production Levelling(Continued) For the same demand pattern, the change in production level from period to period is much smaller. Note: A similar approach works well for minimizing the change in other variables across multiple time periods.

13. Inventory Being Reborn Loosely defined as, “Product must be <= k periods old.” Typically modeled as: Problem: LP will use the Tp,l,l’,tvariables to bypass this constraint by moving inventory between locations. (see example next slide)

14. Inventory Being Reborn (continued) Assume: Production Capacity at Locations 1&2 = 1 unit/period. k=2 Inventory is Re-bornI

15. Inventory Being Reborn (continued) Easiest - Eliminate the Tp,l,l’,tvariables. To discuss: “execution” vs. “planning” Solutions: Harder - Add an additional time based domain to most of the variables and inventory balance rows. This is beyond the scope of this presentation.

16. Honoring Safety Stocks Over Customers Refresher: Min Z = Where: SO= Stockout Amount SV= Safety Stock Violation Amount Safety Stock Constraint: Standard Practice: SOCOST = M SVCOST = 0.5*M

17. Honoring Safety Stocks Over Customers Scenario

18. Honoring Safety Stocks Over Customers Honor Safety Stock Note: There is an alternate solution with the same total penalty cost in which you ship 10 in Period 3.

19. Honoring Safety Stocks Over Customers Satisfy Customer Demand over Safety Stock

20. Honoring Safety Stocks Over Customers Solution

21. Honoring Safety Stocks Over Customers Solution Maintain Safety Stock

22. Honoring Safety Stocks Over Customers Solution Satisfy Customers

23. Starting Inventory is “Free” Objective function does not account for inventory consumption LP may ship to inappropriate locations Reporting Issues

24. Starting Inventory is “Free” P= $15 P= $10 Plant A Plant B T= $10 T= $15 No issue if inventory is consumed elsewhere during the model horizon.

25. Starting Inventory is “Free” Solution Add to the objective function: Add a new set of constraints: Note: Easily modified if you wish to capture increases in inventory as well.

26. Questions?

27. Thank you!

28. Appendix

29. Basic Formulation Where: PRO = Set of All Products MAC = Set of All Machines LOC = Set of All Locations TIM = Set of All Time Periods CUS = Set of All Customers PCOST= Cost to Produce ICOST = Cost to Hold Inventory TCOST = Inter LOC Transportation Cost SOCOST = Stockout Cost TCCOST = LOC to CUS Transportation Cost SVCOST = Safety Stock Violation Cost P = Amount to Produce I = Inventory at the END of the Period T = Amount to Move Between LOC’s TC = Amount to Move Between LOC– CUS SO = Demand not Fulfilled SV = Amount of Safety Stock Violation K = Capacity SS = Safety Stock D = Demand

30. Subject To: Capacity Constraint: Basic Formulation (Continued) Material Balance: Demand: Safety Stock:

31. Assumptions: • 10 Plants • 5 Machines / Plant • 100 Products • 100 Customers (Assume 3 Plants/Customer) • 52 Periods Model Size Variables: P – 100 * 5* 10 * 52 = 260,000 I – 100 * 10 * 52 = 52,000 T – 100 * 10 * 9 * 52 = 468,000 TC – 100 * 3 * 100 *52 = 1,560,000 SO = 100 * 100 * 52 = 520,000 SV = 100 * 10 *52 52,000 Total 2,912,000

32. Constraints: • Capacity – 10 * 5* 52 = 2,000 • Balance – 100 * 10 * 52= 52,000 • Demand – 100 * 100 * 52 = 520,000 • Safety Stock - 100 * 10 * 52 = 52,000 • Total 574,000 Model Size (continued) Note: Real Models tend to be smaller because not every combination exists.