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1 5. Linear Programming 1.Introduction to Constrained Optimization –Three elements: objective, constraints, decisions –General formulation –Terminology 2.Linear Programming –Properties of LP –Solving LP problems using Solver –Sensitivity analysis: constraints and shadow prices, objective function coefficients

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2 Portfolio Management A portfolio manager wants to structure a portfolio from several investments: A, B, C, D Decisions: Objective: Constraints: If is more important –Objective: –Constraints:

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3 Constrained Optimization Constrained Optimization: find decisions that Maximize (or minimize): objective function Subject to: constraints (limitations on resources) Applications –Portfolio management-- Distribution –Location planning -- Production planning –Production scheduling-- Workforce planning –Many others Example: Produce and ship 100 products from 20 plants to 50 DCs around the world, to minimize costs. Constraints? Decisions? Objective?

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4 General Formulation LHSRHS

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5 Terminology Decision variables - things we can control Objective function - performance measure Feasible region - The region in which the decision variables satisfy all of the constraints (choice set) Feasible solution - A solution that satisfies all constraints (lies within the feasible region). Optimal solution - the feasible solution that achieves the best (max or min) value for the objective function. Optimal objective function value - The value of the objective function at the optimal solution.

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6 Linear Programs An important tractable special case –Very easy for the computer to solve a large scale problem –Wide applications Linear objective function and constraints The feasible region will be a convex polyhedron –Convex: No holes or indentations –Polyhedron: flat sides Optimal solution will always be at a corner – ignore infinite feasible points on sides and interior points

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7 Product Mix Problem $16.00$6.00 Marginal Profit 121.52M2 2042M1 Available Machine Hours BA Products Machines hours Maximize Subject to:

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8 We must tell Excel’s Solver that: Decision variables are in C3:D3 (Changing Cells) Objective function is in F4 (Target Cell) Constraints are F6:F7 (hours used) E6:E7 (hours available) LP in Excel: Formulation (LP_MILP.xls) It is a Linear Model with Non-Negative decision variables (under Options) In Excel Spreadsheet: For given values of the decision variables in C3:D3, calculate the objective value: F4 = SUMPRODUCT(C4:D4, C3:D3) calculate the LHS of the constraints: F6 = SUMPRODUCT(C6:D6, $C$3:$D$3) F7 = SUMPRODUCT(C7:D7, $C$3:$D$3)

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9 LP in Excel: Instructions for Solver (LP_MILP.xls) Solver Options: Select “Linear Model” & “Non-Negative” Go to Tools and find Solver

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10 What is the solution? ProdA =, ProdB = Profit = How much machine time? Mach1 =, Mach2 = Unused machine time? Mach1 =, Mach2 = Answer Report: The Solution (LP_MILP.xls) Max 6X A + 16X B s.t. 2X A + 4X B 20 2X A + 1.5X B 12 X A 0, X B 0

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11 Reduced Cost Why would we lose $2 to produce 1 unit of ProdA ( Reduced Cost )? Allowable Increase (a) How much must the profit margin of ProdA increase before we will produce it ( Allowable Increase )? Allowable Increase (b) What if the profit margin of ProdA increases by more than $2 ( Allowable Increase )? Sensitivity Report: Sensitivity Analysis of Objective Function

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12 Sensitivity Report: Sensitivity Analysis of Objective Function If you change an Objective Coefficient within its Allowable Increase/Decrease, the “Final Values of the Variables” do not change (i.e. the same corner is optimal) Outside that range, the “Final Values” change (i.e., new optimal solution).

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13 The shadow price is the rate of change of objective function per unit increase of the RHS (constraint limit). If 2 hours of Mach 1 time were unavailable (i.e. 20 decreases to 18), how much would the objective function change? Why is the Shadow Price = zero for Mach 2? Sensitivity Report: Sensitivity Analysis of Constraints Within the Allowable Increase/Decrease for the RHS, the “Shadow Price” is constant. Outside that range, the “Shadow Price” changes

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14 Takeaways 1.Introduction to Constrained Optimization –Three elements: objective (max or min), constraints ( ≤, ≥, =), decision variables –Formulation 2.Linear Programming: linear constraints and objective function –Properties of LP: optimal solution at a corner –Solving LP problems using Solver Answer: optimal objective value, decisions, binding/nonbinding constraints Sensitivity analysis: –Objective function coefficients: reduced cost, allowable increase/decrease (when zero, multiple solutions) –Constraints: shadow prices (if 0, non-bottleneck), allowable increase/decrease on RHS of constraints

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15 1 2 3 4 5 6 7 8 9 10 11 12 2X A + 4X B = 20 (Machine 1) 2X A + 1.5X B = 12 (Machine 2) XAXA XBXB Graph 1: Solution See Output 1: Solution Which constraints are “binding” the optimal solution?

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16 1 2 3 4 5 6 7 8 9 10 11 12 2X A + 4X B = 20 (Machine 1) 2X A + 1.5X B = 12 (Machine 2) 6X A + 16X B = 16 6X A + 16X B = 80 XAXA XBXB Solution See Output 1: Solution Which constraints are “binding” the optimal solution? 6X A + 16X B = 32

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17 1 2 3 4 5 6 7 8 9 10 11 12 2X A + 1.5X B = 12 (Machine 2) 6X A + 16X B = 80 XAXA XBXB 2X A + 4X B = 20 (Machine 1) How many optimal solutions are there in this case? Changing One of the Objective Function Coefficient Allowable Increase = 2 for Objective Coefficient of Product A Let’s add 2 to the objective coefficient of X A : 8X A + 16X B = 80

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18 1 2 3 4 5 6 7 8 9 10 11 12 2X A + 1.5X B = 12 (Machine 2) Now Try: 10X A + 16X B = 80 6X A + 16X B = 80 Changing the Objective Function Coefficient More than the Allowable Increase or Decrease Allowable Increase = 2 for Objective Coefficient of Product A 2X A + 4X B = 20 (Machine 1) New Solution: ( ) XAXA XBXB

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19 1 2 3 4 5 6 7 8 9 10 11 12 2X A + 1.5X B = 12 (Machine 2) Changing the LHS of a Constraint Machine 1 Constraint has: Constraint R.H. Side = 20, Allowable Increase = 12, Allowable Decrease = 20 2X A + 4X B = 20 Now lets try 2X A + 4X B = 16 Do the decision variables change? Does the objective function change? XAXA XBXB

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