Talk Title: Pitfalls with Linear Programming Optimization of Supply chain Networks
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Talk Title: Pitfalls with Linear Programming Optimization of Supply chain Networks Speaker Name: Robert Stawicki Speaker Title: Assistant Professor Ramapo College of NJ
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Talk Title: Pitfalls with Linear Programming Optimization of Supply chain Networks. 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 - PowerPoint PPT Presentation
Transcript of Talk Title: Pitfalls with Linear Programming Optimization of Supply chain Networks
Talk Title: Pitfalls with Linear Programming Optimization of Supply chain Networks
Speaker Name: Robert StawickiSpeaker Title: Assistant Professor Ramapo College of NJ
Background
• Based on 20 Years Experience Implementing Supply Chain Models for Fortune 100 Companies
• Formulating Models from ScratchOr
• 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 AFixed Cost $2.00Variable Cost / Unit $5.00Total Cost / Unit $7.00
Plant BFixed Cost $0.50Variable Cost / Unit $6.00Total Cost / Unit $6.50
Assume Demand = 10,000 unitsAll Other Costs Equal
Using Full Costing (continued)
Full Costing Based ModelPlant B Produces all 10,000 Units
Plant AFixed Cost $20,000Variable Cost 0Total Cost $20,000
Plant BFixed Cost $5,000Variable Cost 60,000Total Cost $65,000
Total Cost $85,000
Marginal Cost Based ModelPlant A Produces all 10,000 Units
Plant AFixed Cost $20,000Variable Cost 50,000Total Cost $70,000
Plant BFixed Cost $5,000Variable Cost 0Total Cost $5,000
Total Cost $75,000
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Time Frame Too Short
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1 2 3 4 5 6 7 8 9 10 11 12 13-100
0
100
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500
12 Month Model Underestimates Production Requirements
DemandProduceInventory
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24-50
0
50
100
150
200
250
300
350
400
450
A 24 Month Model Adequately Calculates the First 12 Months
DemandProduceInventory
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:+ 𝐿𝐸𝑉𝐸𝐿𝐶𝑂𝑆𝑇∗𝑡∈𝑇𝐼𝑀𝑙∈𝐿𝑂𝐶 𝐿𝑙,𝑡+ + 𝐿𝐸𝑉𝐸𝐿𝐶𝑂𝑆𝑇∗𝑡∈𝑇𝐼𝑀𝑙∈𝐿𝑂𝐶 𝐿𝑙,𝑡−
And the following set of constraints:
Where:LEVELCOST = A large penalty L+, L- = Change in Production Level
∀𝑙,𝑡 ∈𝐿𝑂𝐶,𝑇𝐼𝑀 𝑚∈𝑀𝐴𝐶 𝐾𝑈𝑆𝐸𝑝,𝑚,𝑙,𝑡𝑝∈𝑃𝑅𝑂 ∗ 𝑃𝑝,𝑚,𝑙,𝑡 − 𝑚∈𝑀𝐴𝐶 𝐾𝑈𝑆𝐸𝑝,𝑚,𝑙,𝑡+1𝑝∈𝑃𝑅𝑂 ∗ 𝑃𝑝,𝑚,𝑙,𝑡+1
− 𝐿𝑙,𝑡− + 𝐿𝑙,𝑡+ = 0
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)
D0 D1 D2 D3 D4 D5 D6Demand 5 5 8 10 10 10Production Level 5 8 8 8 8 8 8Delta Level 3 0 0 0 0 0Inventory 3 6 6 4 2 0
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:
+ 𝐿𝐸𝑉𝐸𝐿𝐶𝑂𝑆𝑇𝑙∈𝐿𝑂𝐶 𝐿𝐿𝑙+ + 𝐿𝐸𝑉𝐸𝐿𝐶𝑂𝑆𝑇𝑙∈𝐿𝑂𝐶 𝐿𝐿𝑙−
In addition to the previous set of constraints, add the following two sets of constraints:
∀𝑙 ,𝑡∈𝐿𝑂𝐶 ,𝑇𝐼𝑀 𝐿𝑙 , 𝑡+¿ ≤𝐿𝐿𝑙
+¿ ¿¿
∀𝑙 ,𝑡∈𝐿𝑂𝐶 ,𝑇𝐼𝑀 𝐿𝑙 , 𝑡− ≤ 𝐿𝐿𝑙
−
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.
D0 D1 D2 D3 D4 D5 D6Demand 5 5 8 10 10 10Production Level 5 6 7 8 9 9 9Delta Level 1 1 1 1 0 0Inventory 1 3 3 2 1 0
Loosely defined as, “Product must be <= k periods old.”
Inventory Being Reborn
Typically modeled as:
∀𝑝,𝑙,𝑡 ∈𝑃𝑅𝑂,𝐿𝑂𝐶,𝑇𝐼𝑀 𝐼𝑝,𝑙,𝑡 − 𝑇𝑝,𝑙,𝑙′ ,𝑡 𝑡+𝑘𝑡+1 𝑙′∈𝐿𝑂𝐶′ − 𝑇𝑝,𝑙,𝑐,𝑡
𝑡+𝑘𝑡+1𝑐∈𝐶𝑈𝑆 ≤ 0
Problem: LP will use the Tp,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)
Period 1 2 3 4Demand 0 0 0 7
Prod Loc (1) 1 1 1 1T(1,2) 1T(1,CUS) 3Inventory (1) 1 2 2
Prod Loc (2) 0 1 1 1Inventory (2) 1 3T(2,CUS) 4
Inventory is Re-bornI
Solutions:
Inventory Being Reborn (continued)
Easiest - Eliminate the Tp,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: ∀𝑝 ,𝑙 ,𝑡∈𝑃𝑅𝑂 ,𝐿𝑂𝐶 ,𝑇𝐼𝑀 𝐼𝑝 , 𝑙 , 𝑡+𝑆𝑉 𝑝 , 𝑙 ,𝑡 ≥𝑆𝑆𝑝 ,𝑙 ,𝑡
Refresher:
Standard Practice:SOCOST = M SVCOST = 0.5*M
Where: SO= Stockout Amount SV= Safety Stock Violation Amount
Honoring Safety Stocks Over CustomersScenario
Period 1 2 3 4Demand 10 10 10 10Capacity 5 5 5 20SS Target 5 5 5 5
Honoring Safety Stocks Over CustomersHonor Safety Stock
Period 1 2 3 4Production 5 5 5 10Sales 0 5 5 10Stockout 10 5 5 0End. Inv. 5 5 5 5SO Penalty 10 5 5 0SS Penalty 0 0 0 0
Total Penalty---> 20
Note: There is an alternate solution with the same total penalty cost in which you ship 10 in Period 3.
Honoring Safety Stocks Over CustomersSatisfy Customer Demand over Safety Stock
Period 1 2 3 4Production 5 5 5 15Sales 5 5 5 10Stockout 5 5 5 0End. Inv. 0 0 0 5SO Penalty 5 5 5 0SS Penalty 2.5 2.5 2.5 0
Total Penalty --> 22.5
Honoring Safety Stocks Over CustomersSolution
𝑆𝑂𝐶𝑂𝑆𝑇𝑡=𝑛 ≫ 𝑆𝑉𝐶𝑂𝑆𝑇𝑡=𝑛
∀𝑡∈𝑇𝐼𝑀 𝑆𝑂𝐶𝑂𝑆𝑇 𝑡=𝑆𝑂𝐶𝑂𝑆𝑇 𝑡+1+𝑆𝑉𝐶𝑂𝑆𝑇 𝑡
Honoring Safety Stocks Over CustomersSolution
Maintain Safety Stock
Period 1 2 3 4SO Units 10 5 5 0SV Units 0 0 0 0SOCOST 2.5 2 1.5 1SVCOST 0.5 0.5 0.5 0.5SO Total 25 10 7.5 0SV Total 0 0 0 0
Total Penalty --> 42.5
Honoring Safety Stocks Over CustomersSolution
Satisfy Customers
Period 1 2 3 4SO Units 5 5 5 0SV Units 5 5 5 0SOCOST 2.5 2 1.5 1SVCOST 0.5 0.5 0.5 0.5SO Total 12.5 10 7.5 0SV Total 2.5 2.5 2.5 0
Total Penalty --> 37.5
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 BPlant A
No issue if inventory is consumed elsewhere during the model horizon.
Starting Inventory is “Free”Solution
Add to the objective function:
+ ∑𝑝∈𝑃𝑅𝑂
∑𝑙 ∈𝐿𝑂𝐶
𝑃𝐶𝑂𝑆𝑇 ∗𝐼𝐶𝑝 , 𝑙
Add a new set of constraints:∀𝑝,𝑙 ∈𝑃𝑅𝑂,𝐿𝑂𝐶 𝐼𝐶𝑝,𝑙 + 𝐼𝑝,𝑙,𝑡=𝑛 − 𝐼𝑝,𝑙,𝑡=1 ≥ 0
Note: Easily modified if you wish to capture increases in inventory as well.
Questions?
Thank you!
Appendix
Basic Formulation𝑀𝑖𝑛 𝑍= 𝑃𝐶𝑂𝑆𝑇𝑝,𝑚,𝑙,𝑡𝑡∈𝑇𝐼𝑀𝑙∈𝐿𝑂𝐶𝑚∈𝑀𝐴𝐶𝑝∈𝑃𝑅𝑂 ∗ 𝑃𝑝,𝑚,𝑙,𝑡 + 𝐼𝐶𝑂𝑆𝑇𝑝,𝑙,𝑡𝑡∈𝑇𝐼𝑀𝑙∈𝐿𝑂𝐶𝑝∈𝑃𝑅𝑂 ∗ 𝐼𝑝,𝑙,𝑡
+ 𝑇𝐶𝑂𝑆𝑇𝑝,𝑙,𝑙′ ,𝑡𝑡∈𝑇𝐼𝑀𝑙′ ∈𝐿𝑂𝐶′𝑙∈𝐿𝑂𝐶𝑝∈𝑃𝑅𝑂 ∗ 𝑇𝑝,𝑙,𝑙′ ,𝑡 + 𝑇𝐶𝐶𝑂𝑆𝑇𝑝,𝑙,𝑐,𝑡 𝑡∈𝑇𝐼𝑀𝑐∈𝐶𝑈𝑆𝑙∈𝐿𝑂𝐶𝑝∈𝑃𝑅𝑂 ∗ 𝑇𝐶𝑝,𝑙,𝑐,𝑡
+ 𝑆𝑂𝐶𝑂𝑆𝑇𝑝,𝑐,𝑡 𝑡∈𝑇𝐼𝑀𝑐∈𝐶𝑈𝑆𝑝∈𝑃𝑅𝑂 ∗ 𝑆𝑂𝑝,𝑐,𝑡 + 𝑆𝑉𝐶𝑂𝑆𝑇𝑝,𝑙,𝑡 𝑡∈𝑇𝐼𝑀𝑙∈𝐿𝑂𝐶𝑝∈𝑃𝑅𝑂 ∗ 𝑆𝑉𝑝,𝑙,𝑡
Where:PRO = Set of All ProductsMAC = Set of All MachinesLOC = Set of All LocationsTIM = Set of All Time PeriodsCUS = Set of All CustomersPCOST= Cost to ProduceICOST = Cost to Hold InventoryTCOST = Inter LOC Transportation Cost SOCOST = Stockout CostTCCOST = LOC to CUS Transportation Cost
SVCOST = Safety Stock Violation CostP = Amount to ProduceI = Inventory at the END of the PeriodT = Amount to Move Between LOC’sTC = Amount to Move Between LOC– CUSSO = Demand not FulfilledSV = Amount of Safety Stock ViolationK = CapacitySS = Safety Stock D = Demand
Basic Formulation (Continued)Subject To:Capacity Constraint:
Safety Stock:
Material Balance:
Demand:
∀𝑚 , 𝑙 , 𝑡∈𝑀𝐴𝐶 ,𝐿𝑂𝐶 ,𝑇𝐼𝑀 ∑𝑝∈𝑃𝑅𝑂
𝐾𝑈𝑆𝐸𝑝 ,𝑚 , 𝑙 ,𝑡∗𝑃𝑝 ,𝑚 , 𝑙 ,𝑡 ≤𝐾𝑚 , 𝑙 ,𝑡
∀𝑝 ,𝑙 ,𝑡∈𝑃𝑅𝑂 ,𝐿𝑂𝐶 ,𝑇𝐼𝑀 ∑𝑚∈𝑀𝐴𝐶
𝑃𝑝 ,𝑚 , 𝑙 , 𝑡+𝐼𝑝 ,𝑙 , 𝑡− 1− 𝐼𝑝 ,𝑙 , 𝑡+ ∑𝑙′∈𝐿𝑂𝐶′
𝑇 𝑝 ,𝑙′ , 𝑙 ,𝑡
∀𝑝 ,𝑐 , 𝑡∈𝑃𝑅𝑂 ,𝐶𝑈𝑆 ,𝑇𝐼𝑀 ∑𝑙∈𝐿𝑂𝐶
𝑇𝐶𝑝 ,𝑙 ,𝑐 ,𝑡+𝑆𝑂𝑝 ,𝑐 , 𝑡=𝐷𝑝 ,𝑐 ,𝑡
∀𝑝 ,𝑙 ,𝑡∈𝑃𝑅𝑂 ,𝐿𝑂𝐶 ,𝑇𝐼𝑀 𝐼𝑝 , 𝑙 , 𝑡+𝑆𝑉 𝑝 , 𝑙 ,𝑡 ≥𝑆𝑆𝑝 ,𝑙 ,𝑡
− ∑𝑙 ′∈𝐿𝑂𝐶′
𝑇 𝑝 ,𝑙 , 𝑙′ ,𝑡− ∑𝑐∈𝐶𝑈𝑆
𝑇 𝑝 ,𝑙 , 𝑐 ,𝑡=0
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 - 100 * 10 * 52 = 52,000
Total 574,000
