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Speaker Name: Robert Stawicki Speaker Title: Assistant Professor Ramapo College of NJ

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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

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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”

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Basic Formulation 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

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

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Using Full Costing 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 Assume Demand = 10,000 units All Other Costs Equal

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Using Full Costing (continued) 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 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 ___________________________________________________________________________________________________________________________________________________________________________________

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Time Frame Too Short ___________________________________________________________________________________________________________________________________________________________________________________

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

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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. Production Levelling ( Continued )

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A better formulation: Production Levelling ( Continued ) Instead of the previous change, add the following to the objective function: 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:

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

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Loosely defined as, “Product must be <= k periods old.” Inventory Being Reborn Typically modeled as: Problem: LP will use the T p,l,l’,t variables to bypass this constraint by moving inventory between locations. ( see example next slide )

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

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Solutions: Inventory Being Reborn (continued) Easiest - Eliminate the T p,l,l’,t variables. To discuss: “execution” vs. “planning” Harder - Add an additional time based domain to most of the variables and inventory balance rows. This is beyond the scope of this presentation.

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

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Honoring Safety Stocks Over Customers Scenario

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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.

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Honoring Safety Stocks Over Customers Satisfy Customer Demand over Safety Stock

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Honoring Safety Stocks Over Customers Solution

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Honoring Safety Stocks Over Customers Solution Maintain Safety Stock

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Honoring Safety Stocks Over Customers Solution Satisfy Customers

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

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

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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.

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Questions?

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Thank you!

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Appendix

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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

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

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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 Assumptions: 10 Plants 5 Machines / Plant 100 Products 100 Customers (Assume 3 Plants/Customer) 52 Periods

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Model Size (continued) Note: Real Models tend to be smaller because not every combination exists. 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

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Inventory Control IME 451, Lecture 3.

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