Speaker Name: Robert Stawicki Speaker Title: Assistant Professor Ramapo College of NJ
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
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”
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
Basic Formulation (Continued) To Discuss: Maximize vs. Minimize Stockout vs. Backorder Other Constraints Model Size See Appendix
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
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 ___________________________________________________________________________________________________________________________________________________________________________________
Time Frame Too Short ___________________________________________________________________________________________________________________________________________________________________________________
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
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 )
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:
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.
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 )
Assume: Production Capacity at Locations 1&2 = 1 unit/period. k=2 Inventory Being Reborn (continued) Inventory is Re-bornI
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.
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
Honoring Safety Stocks Over Customers Scenario
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.
Honoring Safety Stocks Over Customers Satisfy Customer Demand over Safety Stock
Honoring Safety Stocks Over Customers Solution
Honoring Safety Stocks Over Customers Solution Maintain Safety Stock
Honoring Safety Stocks Over Customers Solution Satisfy Customers
Starting Inventory is “Free” Objective function does not account for inventory consumption LP may ship to inappropriate locations Reporting Issues
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.
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.
Questions?
Thank you!
Appendix
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
Basic Formulation (Continued) Subject To: Capacity Constraint: Safety Stock: Material Balance: Demand:
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
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 * 10 * 52 = 52,000 Total 574,000