Using Excel s Solver

Transcription

1 Using Excel s Solver How to get the computer to do the work. A Profit Maximization Problem. Lecture 8 Slide 1

2 Is the Solver Installed If your Tools pulldown menu in Excel looks like this, without a Solver option then you need to add the Solver Add-In to your Excel configuration!!! Lecture 8 Slide 2

3 To Install the Solver Select Add-Ins from the Tools pulldown menu! The dialogue box looks like this! Lecture 8 Slide 3

4 To Install the Solver cont Pull down the slider and click against Solver Add-In, then click on OK! The tools pulldown menu should now look like this! Lecture 8 Slide 4

5 To Install the Solver cont You may be asked by Excel to supply the Excel or Office disks which came with your copy of Excel or Microsoft Office. If you are and it doesn t seem to work for you, let me know and we will fix the problem together!! Lecture 8 Slide 5

6 Our Problem, part a. The management of Artificial Limb Company is trying to determine the amount of each of three products to produce over the coming planning period. The following information concerns labor availability, labor utilization, and product profitability. Department Product (hours/unit) Labor- Hours Ankles Hips Knees Available Molding Finishing Quality Profit contribution/unit $30.00 $15.00 $25.00 Lecture 8 Slide 6

7 The Decision Variables a. Develop a linear programming model of the Artificial Limb Company problem. Solve the model to determine the optimal production quantities of products Ankles, Hips and Knees. We are to decide how many ankles, hips and knees to produce, so ankles, hips and knees are the decision variables in this problem. Excel calls them the Adjustable Cells!! Lecture 8 Slide 7

8 The Objective Function The Objective is to Maximize profit (or contribution to profit). The Cell in Excel in which the Objective is calculated is called the Target Cell by Excel!! Contribution in this case is ($30 * Ankles) + ($15 * Hips) + ($25 * Knees). Lecture 8 Slide 8

9 Constraints Molding Hours = (Ankles * 1) + (Hips * 0.35) +(Knees * 0.5) <= 100 hours. Finishing Hours = (Ankles * 0.3) + (Hips * 0.2) +(Knees * 0.4) <= 70 hours. Quality Hours = (Ankles * 0.2) + (Hips * 0.5) +(Knees * 0.2) <= 50 hours. Lecture 8 Slide 9

10 Bounds Ankles => 0 Hips => 0 Knees => 0 Lecture 8 Slide 10

11 A Flow Chart Invoke Excel Create New Worksheet Set Up Objective, Decision Variable, Constraints and Bounds Areas in Spreadsheet Name Objective, Decision Variable, Contraint and Bounds Cells in Excel Label and Enter Parameters into Spreadsheet Enter Objective Function Calculation in Objective Cell Enter Constraint and Bound Calculations Invoke Solver Lecture 8 Slide 11

12 Invoke Solver Set Target Cell (Objective Function) Set Adjustable Cells, in "By Changing Cells Entry Box" (Decision Variables) Add Constraints and Bounds Select Minimization or Maximization Radio Button A Flow Chart cont Set Options Hit Solve Button Select Report Options Review Results Do Again if Appropriate Lecture 8 Slide 12

13 Set Up Areas in Spreadsheet Give yourself enough space for: The Objective Function, Decision Variables, Constraints, and Bounds. Lecture 8 Slide 13

14 Label Variables We will need calculations for: The Objective Function Value. The LHS s of the Constraints. We will need variables for: Each Decision Variable. Each Bound. Lecture 8 Slide 14

15 Label and Enter Parameters The Objective Function Coefficients (Contribution from each product) The Bill of Materials which describes how much of each constrained resource is used to produce one of each product. Lecture 8 Slide 15

16 Name Variables and Parameters Ankles =Sheet3!$B$7 Ankles_Bound =Sheet3!$B$19 Cont_A =Sheet3!$E$3 Cont_H =Sheet3!$F$3 Cont_K =Sheet3!$G$3 F_Ankles =Sheet3!$E$14 F_Hips =Sheet3!$F$14 F_Knees =Sheet3!$G$14 Finishing_Available=Sheet3!$H$14 Finishing_Hours =Sheet3!$B$14 Hips =Sheet3!$B$8 Hips_Bound =Sheet3!$B$20 Knees =Sheet3!$B$9 Knees_Bound =Sheet3!$B$21 M_Ankles =Sheet3!$E$13 M_Hips =Sheet3!$F$13 M_Knees =Sheet3!$G$13 Molding_Available =Sheet3!$H$13 Molding_Hours =Sheet3!$B$13 Objective =Sheet3!$B$3 Q_Ankles =Sheet3!$E$15 Q_Hips =Sheet3!$F$15 Q_Knees =Sheet3!$G$15 Quality_Available =Sheet3!$H$15 Quality_Hours =Sheet3!$B$15 Excel allows you to give names to Cells. (Insert, Name, Define!!). This is optional, but it will help with the demonstration. Names on far left have been given to Cells on the left. Lecture 8 Slide 16

17 Enter the Objective Function Calculation Enter the equation for Contribution using named cell names or as (B7*E3) +(B8*F3) + (B9*G3) Lecture 8 Slide 17

18 Enter Constraints and Bounds Calculations Enter the equation for # of hours of Quality used, using named cell names or as (B7*E15) + (B8*F15) + (B9*G15) Enter cell names or references for Decision Variables, i.e. =Ankles or =B7, etc Lecture 8 Slide 18

19 Invoke Solver Using the Tools, Solver options from the pulldown menus get the following dialogue box onto the screen. Problem Type Name or Reference of Cell with the Objective Function Calculation in it! Range of Names or References for the Cells into which the values of the Decision Variables will be inserted. Lecture 8 Slide 19

20 Set Target Cell, Problem Type and Adjustable Cells Set two important Options by clicking the Options Button. We will add constraints and bounds into here next using the dialogue brought up by the Add button. Lecture 8 Slide 20

21 Setting the Options Check these two boxes to get Sensitivity Analysis geared to Linear Programs and to automate the bounds. Lecture 8 Slide 21

22 Enter Constraints and Bounds Explicit Bounds are not needed if the Options Assume Nonnegativity box has been checked! Check Values and Constraints, then click on the Solve button. Lecture 8 Slide 22

23 Solve and Select Reports Click on OK button to generate reports highlighted on the right. Lecture 8 Slide 23

24 A Simplex Tableau of the Optimal Solution Ankles Hips Knees s1 s2 s3 Cj Ankles Knees Hips Zj Cj-Zj The Optimal Solution is Ankles = , Hips = , Knees = , s1 = s2 = s3 = 0, Z =4,625 Molding, Finishing and Quality constraints are all binding. The bounds constraints are not binding. Lecture 8 Slide 24

25 Microsoft Excel 9.0 Answer Report Worksheet: [LP1a.xls]Sheet3 Report Created: 1/24/2002 8:27:26 PM Target Cell (Max) Cell Name Original Value Final Value $B$3 Objective $0.00 $4, Adjustable Cells Cell Name Original Value Final Value $B$7 Ankles $B$8 Hips $B$9 Knees Answer Report Constraints Cell Name Cell Value Formula Status Slack $B$13 Molding_Hours 100 $B$13<=$H$13 Binding 0 $B$14 Finishing_Hours 70 $B$14<=$H$14 Binding 0 $B$15 Quality_Hours $B$15<=$H$15 Binding 0 $B$19 Ankles_Bound $B$19>=0 Not Binding $B$20 Hips_Bound $B$20>=0 Not Binding $B$21 Knees_Bound $B$21>=0 Not Binding The Answer Report gives us: Z = 4,625 Ankles = Hips = Knees = All Constraints are Binding. No Bound is binding. Lecture 8 Slide 25

26 Sensitivity Report Microsoft Excel 9.0 Sensitivity Report Worksheet: [LP1a.xls]Sheet3 Report Created: 1/24/2002 9:11:30 PM Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$7 Ankles $B$8 Hips $B$9 Knees Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $B$13 Molding_Hours $B$14 Finishing_Hours $B$15 Quality_Hours $B$19 Ankles_Bound E+30 $B$20 Hips_Bound E+30 $B$21 Knees_Bound E+30 Lecture 8 Slide 26

27 Part b. of Problem b. In computing the profit contribution per unit, management doesn't deduct labor costs because they are considered fixed for the upcoming planning period. However, suppose that overtime can be scheduled in some of the departments. Which departments would you recommend scheduling for overtime? How much would you be willing to pay per hour of overtime in each department? We can answer the question using the Shadow Prices. An extra hour of Molding is worth $17.85, an extra hour of Finishing is worth $39.28, and an extra hour of Quality is worth $1.78. So I would recommend overtime in Molding and Finishing departments provided the cost is below $17.85 and $39.28 per hour respectively. Note that the Solution will change if more than Molding hours are added or if more than 7.77 Finishing hours are added. Lecture 8 Slide 27

28 Part c A Revised Model Suppose that 10, 6, and 8 hours of overtime may be scheduled in Molding, Finishing and Quality departments, respectively. The cost per hour of overtime is $18 in Molding, $22.50 in Finishing, and $12 in Quality. Formulate a linear programming model that can be used to determine the optimal production quantities if overtime is made available. What are the optimal production quantities, and what is the revised total contribution to profit? How much overtime do you recommend using in each department? What is the increase in the total contribution to profit if overtime is used? Lecture 8 Slide 28

29 Decision Variables and Objective Function Not only do we want to know how many Ankles, Hips and Knees to produce, but also how much overtime to schedule for MoldingOT, Finishing OT and QualityOT. The Objective Function must take into account the contribution from products sold and the cost of the overtime:- Z = 30 Ankles + 15 Hips + 25 Knees 18 MoldingOT 22.5 FinishingOT 12 QualityOT Lecture 8 Slide 29

30 New Constraints Molding Hours = (Ankles * 1) + (Hips * 0.35) +(Knees * 0.5) <= MoldingOT hours. Finishing Hours = (Ankles * 0.3) + (Hips * 0.2) +(Knees * 0.4) <= 70 + FinishingOT hours. Quality Hours = (Ankles * 0.2) + (Hips * 0.5) +(Knees * 0.2) <= 50 + QualityOT hours. MoldingOT <= 10 hours. FinishingOT <= 6 hours. QualityOT <= 8 hours. Lecture 8 Slide 30

31 Set Up Spreadsheet Lecture 8 Slide 31

32 Solve and Select Reports Lecture 8 Slide 32

33 Microsoft Excel 9.0 Answer Report Worksheet: [LP1a.xls]LP1c. Report Created: 1/24/ :09:25 PM Answer Report Target Cell (Max) Cell Name Original Value Final Value $B$3 Objective $0.00 $4, Adjustable Cells Cell Name Original Value Final Value $B$7 Ankles $B$8 Hips $B$9 Knees $B$10 MoldingOT 0 0 $B$11 FinishingOT 0 6 $B$12 QualityOT 0 0 Constraints Cell Name Cell Value Formula Status Slack $B$16 Molding Hours $B$16<=$H$16 Binding 0 $B$17 Finishing Hours $B$17<=$H$17 Binding 0 $B$18 Quality Hours $B$18<=$H$18 Binding 0 $B$19 MoldingOThrs 0 $B$19<=$H$19 Not Binding 10 $B$20 FinishingOThrs 6 $B$20<=$H$20 Binding 0 $B$21 QualityOThrs 0 $B$21<=$H$21 Not Binding 8 The Optimal Solution is 3.26 Ankles, Hips and Knees with 6 hours of Finishing Overtime for a Contribution of $4, Molding Hours, Finishing Hours, Quality Hours and FinishingOThrs constraints are binding the other two are not. Lecture 8 Slide 33

34 Sensitivity Report Microsoft Excel 9.0 Sensitivity Report Worksheet: [LP1a.xls]LP1c. Report Created: 1/24/ :09:25 PM Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$7 Ankles $B$8 Hips $B$9 Knees $B$10 MoldingOT E+30 $B$11 FinishingOT E $B$12 QualityOT E+30 Constraints Final Shadow Constraint Allowable Allowable Cell Name Value Price R.H. Side Increase Decrease $B$16 Molding Hours $B$17 Finishing Hours $B$18 Quality Hours $B$19 MoldingOThrs E $B$20 FinishingOThrs $B$21 QualityOThrs E+30 8 Note we would take another 1.77 hours of Finishing Overtime at less than $16.78 per hour before the solution changed. Lecture 8 Slide 34

35 Reading and Homework. Read Chapter 3, Sections 3.1, 3.2 and 3.3 Read LP Example #2 Risk Minimization Handout. Read LP Example #3 Advertising Mix Problem Handout. Lecture 8 Slide 35