Linear Programming. Chapter 2: Basic Concepts. Lee-Anne Johennesse. Advanced Quantitative Methods 7 March 2016

Transcription

1 Linear Programming Chapter 2: Basic Concepts Lee-Anne Johennesse Advanced Quantitative Methods 7 March 2016

2 Linear Programming Chapter 2: Basic Concepts Introduction Part A The Wyndor Glass Company Product Mix Problem Formulating the Wyndor Problem on a Spreadsheet The Algebraic Model for Wyndor The Graphical Method Applied to the Wyndor Problem Using the Excel Solver with the Wyndor Problem Part B A Minimization Example The Profit & Gambit Co. Part C Case Study 2-1: Auto Assembly McGraw-Hill/Irwin

3 Learning Objectives 1. Explain what linear programming is. 2. Identify the three key questions to be addressed in formulating any spreadsheet model. 3. Name and identify the purpose of the four kinds of cells used in linear programming spreadsheet models.. Formulate a basic linear programming model in a spreadsheet from a description of the problem. 5. Present the algebraic form of a linear programming model from its formulation on a spreadsheet. 6. Apply the graphical method to solve a two-variable linear programming problem. 7. Use Excel to solve a linear programming spreadsheet model.

4 Introduction Management are responsible resource allocation decisions to meet organizational objectives money, personnel, machinery and equipment (Production, marketing, advertising, financial activities) Linear programming is a powerful problem-solving tool that aids management in making such decisions. finding the best mix of activities which ones to pursue and at what levels The word linear in the name refers to the form of the mathematical expressions in this model. Programming does not refer to computer programming; rather, it is essentially a synonym for planning. Thus, linear programming means the planning of activities represented by a linear mathematical model

5 Wyndor Glass Co. Product Mix Problem Wyndor s new products: An 8-foot glass door with aluminum framing. A -foot by 6-foot double-hung, wood-framed window. The company has three plants Plant 1 produces aluminum frames and hardware. Plant 2 produces wood frames. Plant 3 produces glass and assembles the windows and doors. Questions: 1. Should they go ahead with launching these two new products? 2. If so, what should be the product mix? (the number of units of each product produced per week for the two new products) McGraw-Hill/Irwin 2.5

6 Developing a Spreadsheet Model Wyndor Glass Co. Product-Mix Problem Doors Windows Range Name Cells Unit Profit $300 $500 HoursAvailable G7:G9 Hours Hours HoursUsed E7:E9 Hours Used Per Unit Produced Used Available HoursUsedPerUnitProduced C7:D9 Plant <= TotalProfit G12 Plant <= 12 UnitProfit C:D Plant <= 18 UnitsProduced C12:D12 Doors Windows Total Profit Units Produced 3 $2,700 McGraw-Hill/Irwin 2.6

7 Algebraic Model for Wyndor Glass Co. Let D = the number of doors to produce W = the number of windows to produce Maximize P = $300D + $500W subject to D 2W 12 3D + 2W 18 and D 0, W 0. McGraw-Hill/Irwin 2.7

8 Terminology for Linear Programming Models 1. D and W (or C12 and D12 in Figure 2.3) are the decision variables D 500W [or SUMPRODUCT (UnitProfit, Units Produced)] is the objective function. 3. P (or G12) is the value of the objective function (objective value).. D 0 and W 0 (or C12 0 and D12 0) are called the nonnegativity constraints (or nonnegativity conditions). 5. The other constraints are referred to as functional constraints (or structural constraints). 6. The parameters of the model are the constants in the algebraic model (the numbers in the data cells). 7. Any choice of values for the decision variables (regardless of how desirable or undesirable the choice) is called a solution for the model. 8. A feasible solution is one that satisfies all the constraints, whereas an infeasible solution violates at least one constraint. 9. The best feasible solution, the one that maximizes P (or G12), is called the optimal solution. McGraw-Hill/Irwin 2.8

9 Graphing the Product Mix 2 Decision Variables W Production rate (units per week) for windows A product mix of D = and W = 6 (, 6) A product mix of D = 2 and W = 3 (2, 3) 1 Origin Production rate (units per week) for doors -1 D -2 McGraw-Hill/Irwin 2.9

10 Graph Showing Constraints: D 0 and W 0 W 8 Production rate for windows Production rate for doors D McGraw-Hill/Irwin 2.10

11 Nonnegative Solutions Permitted by D W 8 D = Production rate for windows Production rate for doors D McGraw-Hill/Irwin 2.11

12 Nonnegative Solutions Permitted by 2W 12 Production rate for windows W W = Production rate for doors D McGraw-Hill/Irwin 2.12

13 Boundary Line for Constraint 3D + 2W 18 Production rate for windows W 10 (0, 9) 8 (1, 7 1_ ) 2 6 (2, 6) 3 D + 2 W = 18 (3, 1_ ) 2 (, 3) 2 (5, 1 1_ ) Production rate for doors (6, 0) D McGraw-Hill/Irwin 2.13

14 Nonnegative Solutions Permitted by 3D + 2W 18 Production rate for windows W D + 2W = Production rate for doors D McGraw-Hill/Irwin 2.1

15 Graph of Feasible Region Production rate for windows W D + 2 W = 18 D = 6 2 W =12 Feasible 2 region Production rate for doors D McGraw-Hill/Irwin 2.15

16 Objective Function (P = 1,500) Production rate for windows W 8 6 P = 1500 = 300D + 500W Feasible region Production rate for doors D McGraw-Hill/Irwin 2.16

17 Finding the Optimal Solution Production rate W for windows 8 P = 3600 = 300D + 500W P = 3000 = 300D + 500W 6 Optimal solution (2, 6) P = 1500 = 300D + 500W Feasible region Production rate for doors 10 D McGraw-Hill/Irwin 2.17

18 Using SOLVER: Identifying the Target Cell and Changing Cells Choose the Solver from the Tools menu. Select the cell you wish to optimize in the Set Target Cell window. Choose Max or Min depending on whether you want to maximize or minimize the target cell. Enter all the changing cells in the By Changing Cells window B C D E F G Doors Windows Unit Profit $300 $500 Hours Hours Hours Used Per Unit Produced Used Available Plant <= 1 Plant <= 12 Plant <= 18 Doors Windows Total Profit Units Produced 1 1 $800 McGraw-Hill/Irwin 2.18

19 Adding Constraints To begin entering constraints, click the Add button to the right of the constraints window. Fill in the entries in the resulting Add Constraint dialogue box B C D E F G Doors Windows Unit Profit $300 $500 Hours Hours Hours Used Per Unit Produced Used Available Plant <= 1 Plant <= 12 Plant <= 18 Doors Windows Total Profit Units Produced 1 1 $800 McGraw-Hill/Irwin 2.19

20 The Complete Solver Dialogue Box McGraw-Hill/Irwin 2.20

21 Some Important Options Click on the Options button, and click in both the Assume Linear Model and the Assume Non-Negative box. Assume Linear Model tells the Solver that this is a linear programming model. Assume Non-Negative adds nonnegativity constraints to all the changing cells. McGraw-Hill/Irwin 2.21

22 The Solver Results Dialogue Box McGraw-Hill/Irwin 2.22

23 The Optimal Solution B C D E F G Doors Windows Unit Profit $300 $500 Hours Hours Hours Used Per Unit Produced Used Available Plant <= 1 Plant <= 12 Plant <= 18 Doors Windows Total Profit Units Produced 2 6 $3,600 McGraw-Hill/Irwin 2.23

24 The Profit & Gambit Co. Management has decided to undertake a major advertising campaign that will focus on the following three key products: A spray prewash stain remover. A liquid laundry detergent. A powder laundry detergent. The campaign will use both television and print media (tv will feature liquid detergent) The general goal is to increase sales of these products. Management has set the following goals for the campaign: Sales of the stain remover should increase by at least 3%. Sales of the liquid detergent should increase by at least 18%. Sales of the powder detergent should increase by at least %. Question: how much should they advertise in each medium to meet the sales goals at a minimum total cost? McGraw-Hill/Irwin 2.2

25 Profit & Gambit Co. Spreadsheet Model B C D E F G Television Print Media Unit Cost ($millions) 1 2 Increased Minimum Increase in Sales per Unit of Advertising Sales Increase Stain Remover 0% 1% 3% >= 3% Liquid Detergent 3% 2% 18% >= 18% Powder Detergent -1% % 8% >= % Total Cost Television Print Media ($millions) Advertising Units 3 10 McGraw-Hill/Irwin 2.25

26 Algebraic Model for Profit & Gambit Let TV = the number of units of advertising on television PM = the number of units of advertising in the print media Minimize Cost = TV + 2PM (in millions of dollars) subject to Stain remover increased sales: PM 3 Liquid detergent increased sales: 3TV + 2PM 18 Powder detergent increased sales: TV + PM and TV 0, PM 0. McGraw-Hill/Irwin 2.26

27 Applying the Graphical Method Amount of print media advertising PM 10 Feasible region 8 6 PM = 3 2 -TV + PM = 3 TV + 2 PM = Amount of TV advertising TV McGraw-Hill/Irwin 2.27

28 The Optimal Solution PM Cost = 15 = TV + 2 PM 10 Feasible region Cost = 10 = TV + 2 PM (,3) optimal solution Amount of TV advertising TV McGraw-Hill/Irwin 2.28

29 Properties of Linear Programming Solutions An optimal solution must lie on the boundary of the feasible region. There are exactly four possible outcomes of linear programming: A unique optimal solution is found. An infinite number of optimal solutions exist. No feasible solutions exist. The objective function is unbounded (there is no optimal solution). If an LP model has one optimal solution, it must be at a corner point. If an LP model has many optimal solutions, at least two of these optimal solutions are at corner points. McGraw-Hill/Irwin 2.29