Linear Programming: Basic Concepts

Transcription

1 Linear Programming: Basic Concepts Irwin/McGraw-Hill 1.١ The McGraw-Hill Companies, Inc., 2003

2 Introduction The management of any organization make Decision about how to allocate its resources to various activities to best meet organizational objectives Linear Programming (LP) is a powerful problem-solving tool Applicable to both profit-making and not-for-profit organization Wide variety of resources must be allocated simultaneously to activities Resources: money, different kind of personnel, machinery and equipment Activities: production, marketing, financial Applications of LP go beyond the allocation of resources, however activities are always involved Find the best mix of activities (which one to pursue and at what level) LP uses a Mathematical Model Linear Programming means the planning of activities represented by a linear mathematical model ٢

3 Three Classic Applications of LP Standard for future applications in Linear Programming Product Mix at Ponderosa Industrial Considered limited resources, and determined optimal mix of products sold in a competitive environment. Increased overall profitability of company by 20%. Personnel Scheduling at United Airlines Designed work schedules for all employees at a location to meet service requirements most efficiently (4000 reservations sales, 11 reservations offices, 1000 customer service agents at 10 largest airports). Saved $6 million annually. Planning Supply, Distribution, and Marketing at Citgo Petroleum Corporation The SDM system uses LP to coordinate the Supply, Distribution, and Marketing of each of Citgo s major products throughout the United States. The resulting reduction in inventory added $14 million annually to Citgo s profits. ٣

4 Wyndor Glass Co. Product Mix Problem Wyndor has developed the following new products: An 8-foot glass door with aluminum framing. A 4-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 produced per week) for the two new products that would maximize the total profit? ٤

5 The Management Science Group Begins its Work Should Consider all possible combination of production rates Identifies the needed information to conduct this study Available production capacity in each of the plants How much of the production capacity in each plant would be needed by each product Profitability of each product Production Time Used for Each Unit Produced Plant Doors Windows Available per week 1 1 hour 0 4 hours hours 12 hours 3 3 hours 2 hours 18 hours Unit Profit $300 $500 ٥

6 Formulating the Wyndor Problem on a Spreadsheet Transfer the data onto a spreadsheet Need to answer 3 questions What are the decisions to be made (Decision Variable)? What are the constraints on these decisions (Constraints)? What is the overall measure of performance for these decisions (Objective Function)? Answer Variables: Number of units produced per week for the two new products Constraints: Hours of production time used per week cannot exceed the number of hours available Objective Function: Maximizing the total profit per week from the two products ٦

7 Developing a Spreadsheet Model Step #1: Data Cells Enter all of the data for the problem on the spreadsheet. The cells showing the data are the Data cells shaded light blue Data cells are given range name: profit(c4:d4), HoursUsedPerUnitProduced (C7:D9), and HoursAvailable (G7:G9) Doors Windows Unit Profit $300 $500 Hours Available Hours Used Per Unit Produced Plant Plant Plant ٧

8 Developing a Spreadsheet Model Step #2: Changing Cells Add a cell in the spreadsheet for every decision that needs to be made. If you don t have any particular initial values, just enter 0 in each. It is a good idea to color code these changing cells (e.g., yellow with border). Doors Windows Unit Profit $300 $500 Hours Hours Used Per Unit Produced Available Plant Plant Plant Doors Windows Units Produced 0 0 ٨

9 Developing a Spreadsheet Model Step #3: Constraints (Output cells) For any resource that is restricted, calculate the amount of that resource used in a cell on the spreadsheet (an output cell). Define the constraint in three consecutive cells. For example, if Quantity A Quantity B, put these three items (Quantity A,, Quantity B) in consecutive cells. Doors Windows Unit Profit $300 $500 Hours Hours Hours Used Per Unit Produced Used Available Plant <= 4 Plant <= 12 Plant <= 18 Doors Windows Total Profit Units Produced 1 1 $800 Cells providing output that depends on the changing cells are called output cells E Hours Used =SUMPRODUCT(C7:D7,UnitsProduced) =SUMPRODUCT(C8:D8,UnitsProduced) =SUMPRODUCT(C9:D9,UnitsProduced) ٩

10 Developing a Spreadsheet Model Step #4: Target Cell Develop an equation that defines the objective of the model. Typically this equation involves the data cells and the changing cells in order to determine a quantity of interest (e.g., total profit or total cost). It is a good idea to color code this cell (e.g., orange with heavy border). Doors Windows Unit Profit $300 $500 Hours Hours Used Per Unit Produced Available Plant Plant Plant Doors Windows Total Profit Units Produced 1 1 $800 Target cell is made as large as possible when making decisions G Total Profit =SUMPRODUCT(UnitProfit,UnitsProduced) ١٠

11 A Trial Solution Doors Windows Unit Profit $300 $500 Hours Hours Hours Used Per Unit Produced Used Available Plant <= 4 Plant <= 12 Plant <= 18 Doors Windows Total Profit Units Produced 4 3 $2,700 The spreadsheet for the Wyndor problem with a trial solution (4 doors and 3 windows) entered into the changing cells. Does this trial solution provide the best mix production (Optimal Solution)? Not necessarily. ١١

12 This Spreadsheet Model is a LP Model (see p.34) Summary of the formulation procedure 1. Gather the data for the problem 2. Enter the data into data cells on a spreadsheet 3. Identify the decisions to be made on the levels of activities and designate changing cells for displaying these decisions 4. Identify the constraints on these decisions and introduce output cells as needed to specify these constraints 5. Choose the overall measure of performance to be entered into the target cell 6. Use a SUMPRODUCT function to enter the appropriate value into each output cell (including the target cell) 3 and 6 are key for differentiating a LP model from other kinds of mathematical models on spreadsheet ١٢

13 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 4 2W 12 3D + 2W 18 and D 0, W 0. ١٣

14 The Graphical Method for Solving Two-Variable Problems The graphical method provides helpful intuition about linear programming ١٤

15 Displaying Solutions as Points on a Graph: Graphing the Product Mix W Production rate (units per week) for windows A product mix of D = 4 and W = 6 (4, 6) A product mix of D = 2 and W = 3 (2, 3) 1 Origin Production rate (units per week) for doors -1 D -2 ١٥

16 Graph Showing Constraints: D 0 and W 0 W 8 Production rate for windows Production rate for doors D ١٦

17 Nonnegative Solutions Permitted by D 4 W Production rate for windows D = 4 The constraint boundary equation is obtained by replacing the inequality sign by the equality sign Production rate for doors D ١٧

18 Nonnegative Solutions Permitted by 2W 12 Production rate for windows W W = Production rate for doors D ١٨

19 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 4 (3, 4 1_ ) 2 (4, 3) 2 (5, 1 1_ ) Production rate for doors (6, 0) D ١٩

20 Changing Right-Hand Side Creates Parallel Constraint Boundary Lines Production rate for windows W D + 2W = D + 2W = D + 2W = Production rate for doors D ٢٠

21 Nonnegative Solutions Permitted by 3D + 2W 18 Production rate for windows W D + 2W = Production rate for doors D ٢١

22 Graph of Feasible Region Production rate for windows W D + 2 W = 18 D = W =12 4 Feasible 2 region Production rate for doors D ٢٢

23 Objective Function (P = 1,500) Production rate for windows W 8 6 P = 1500 = 300D + 500W 4 Feasible region Production rate for doors D ٢٣

24 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 4 Feasible region Production rate for doors 10 D ٢٤

25 Summary of the Graphical Method Draw the constraint boundary line for each constraint. Use the origin (or any point not on the line) to determine which side of the line is permitted by the constraint. Find the feasible region by determining where all constraints are satisfied simultaneously. Determine the slope of one objective function line. All other objective function lines will have the same slope. Move a straight edge with this slope through the feasible region in the direction of improving values of the objective function. Stop at the last instant that the straight edge still passes through a point in the feasible region. This line given by the straight edge is the optimal objective function line. A feasible point on the optimal objective function line is an optimal solution. ٢٥

26 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 ٢٦

27 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 ٢٧

28 The Complete Solver Dialogue Box ٢٨

29 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. ٢٩

30 The Solver Results Dialogue Box ٣٠

31 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 ٣١

32 Summary of the Graphical Method Draw the constraint boundary line for each constraint. Use the origin (or any point not on the line) to determine which side of the line is permitted by the constraint. Find the feasible region by determining where all constraints are satisfied simultaneously. Determine the slope of one objective function line. All other objective function lines will have the same slope. Move a straight edge with this slope through the feasible region in the direction of improving values of the objective function. Stop at the last instant that the straight edge still passes through a point in the feasible region. This line given by the straight edge is the optimal objective function line. A feasible point on the optimal objective function line is an optimal solution. ٣٢

33 Components of a Linear Program Data Cells Changing Cells ( Decision Variables ) Target Cell ( Objective Function ) Constraints ٣٣

34 Four Assumptions of Linear Programming Linearity Divisibility Certainty Nonnegativity ٣٤

35 Why Use Linear Programming? Linear programs are easy (efficient) to solve The best (optimal) solution is guaranteed to be found (if it exists) Useful sensitivity analysis information is generated Many problems are essentially linear ٣٥

36 The Graphical Method for Solving LP s Formulate the problem as a linear program Plot the constraints Identify the feasible region Draw an imaginary line parallel to the objective function (Z = a) Find the optimal solution ٣٦

37 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. ٣٧