OPTIMIZAÇÃO E DECISÃO 10/11

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1 OPTIMIZAÇÃO E DECISÃO 10/11 PL #1 Linear Programming Alexandra Moutinho (from Hillier & Lieberman Introduction to Operations Research, 8 th edition) The Wyndor Glass Co. Problem Wyndor Glass Co. produces high-quality glass products, including windows and doors. It has 3 plants: Plant 1 makes aluminum frames and hardware; Plant 2 makes wood frames; Plant 3 produces the glass and assembles the products. Production Time per Batch, Hours Product Production Time Available per Plant Doors Windows Week, Hours Profit per batch $3,000 $5,000 Determine what the production rates should be for the two products in order to maximize their total profit, subject to the restrictions imposed by the limited production capacities available in the 3 plants. 1. Formulate linear programming problem; 2. Solve with Graphical Method (freehand manner or using IOR Tutorial); 3. Solve with Excel Solver; 4. Create and analyze the Answer, Sensitivity and Limits reports generated by Excel Solver. Resolution: 1. Formulate linear programming problem: Maximize = subject to the restrictions and 0 1

2 0. 2. Solve with Graphical Method (freehand manner or using IOR Tutorial): 3. Solve with Excel Solver a) Make data table in Excel: b) Select data cells and color them a light blue: Optimização e Decisão 09/10 - PL #1 Linear Programming - Alexandra Moutinho 2

3 c) Name data cells in the Formulas tab / Defined Names section / Define Name: C4:D4 fi Profit_Per_Batch G7:G9 fi Hours_Available C7:D9 fi Hours_Used_Per_Batch_Produced d) Answer 3 questions: 1) What are the decisions to be made? Answer: production rates. 2) What are the constraints of these decisions? Answer: number of hours of production time used per week by the 2 products hours available (functional constraints). 3) What is the overall measure of performance for these decisions? Answer: total profit (maximize). e) Name necessary cells in the Formulas tab / Defined Names section / Define Name. Also, color, border and insert formula: C12:D12 fi Batches_Produced (production rates are changing or adjustable cells, light yellow with border) E7:E9 fi Hours_Used (output cells, ex.: = SUMPRODUCT(C7:D7;Batches_Produced)) G12 fi Total_Profit (target cell, orange with bold border, = SUMPRODUCT(Proit_Per_Batch;Batches_Produced),) f) Try some values for the Batches Produced and check the correspondent Total Profit. Optimização e Decisão 09/10 - PL #1 Linear Programming - Alexandra Moutinho 3

4 g) Use Solver (in Data tab / Analysis section / Solver) to obtain the optimal solution (maximum Total Profit): Optimização e Decisão 09/10 - PL #1 Linear Programming - Alexandra Moutinho 4

5 Optimal solution =2, =6, corresponding to number of batches produced of aluminum doors and wooden windows respectively, corresponds to maximum total profit = $36,000. h) Check other Excel files in the CD for larger problems. i) Check supplementary chapter 21 on CD for The art of modeling in spreadsheets. 4. Create and analyze the Answer, Sensitivity and Limits reports generated by Excel Solver Select the reports you wish Excel Solver to generate: They will appear as new worksheets: a. Answer report Optimização e Decisão 09/10 - PL #1 Linear Programming - Alexandra Moutinho 5

6 We can see that the optimal solution to the LP has value $ and that Plant 1 used 2 hours, Plant 2 used 12 hours, Plant 3 used 18 hours, =2 and =6. Note that we had three constraints for the total number of hours available for each plant in our LP. The number of hours available for Plant 1 constraint is declared to be 'Not Binding' whilst the other two constraints are declared to be 'Binding'. Constraints with a 'Slack' value of zero are said to be tight or binding in that they are satisfied with equality at the LP optimal. Constraints which are not tight are called loose or not binding. b. Sensitivity report This sensitivity report provides us with information relating to: changing the objective function coefficient for a variable; forcing a variable which is currently zero to be nonzero; changing the right-hand side of a constraint. We deal with each of these in turn, and note here that the analysis presented below ONLY applies for a single change, if two or more things change then we effectively need to resolve the LP. Optimização e Decisão 09/10 - PL #1 Linear Programming - Alexandra Moutinho 6

7 Changing the objective function coefficient for a variable To illustrate this suppose we vary the coefficient of in the objective function. How will the LP optimal solution change? Currently =2 and =6. The current solution value for of 2 is in cell C12 and the current objective function coefficient for is The Allowable Increase/Decrease columns tell us that, provided the coefficient of in the objective function lies between = 7500 and = 0, the values of the variables in the optimal LP solution will remain unchanged. Note though that the actual objective function value will change as the objective function coefficient of is changing. In terms of the original problem we are effectively saying that the decision to produce 2 batches of variant 1 and 6 batches of variant 2 remains optimal even if the profit per batch on variant 1 is not actually 3000 (but lies in the range 0 to 7500). Similar conclusions can be drawn about. Forcing a variable which is currently zero to be non-zero For the variables, the Reduced Cost column gives us, for each variable which is currently zero (in our case neither), an estimate of how much the objective function will change if we make (force) that variable to be non-zero. Note here that the value in the Reduced Cost column for a variable is often called the 'opportunity cost' for the variable. The objective function will always get worse (go down if we have a maximization problem, go up if we have a minimization problem) by at least this estimate. Note here than an alternative (and equally valid) interpretation of the reduced cost is the amount by which the objective function coefficient for a variable needs to change before that variable will become non-zero. Changing the right-hand side of a constraint For each constraint the column headed Shadow Price tells us exactly how much the objective function will change if we change the right-hand side of the corresponding constraint within the limits given in the Allowable Increase/Decrease columns. For example for the Plant 2 constraint, provided the right-hand side of that constraint remains between = 18 and 12-6 = 6 the objective function change will be exactly The direction of the change in the objective function (up or down) depends upon the direction of the change in the right-hand side of the constraint and the nature of the objective (maximize or minimize). To decide whether the objective function will go up or down use: Hence: constraint more (less) restrictive after change in right-hand side implies objective function worse (better); if objective is maximize (minimize) then worse means down (up), better means up (down). if you had to take 2 hours away from Plant 2 or Plant 3 which one would you choose? And if you had to take 3 hours? Optimização e Decisão 09/10 - PL #1 Linear Programming - Alexandra Moutinho 7

8 what would the new objective function value be in these two cases? The value in the column headed Shadow Price for a constraint is often called the 'marginal value' or 'dual value' for that constraint. Note that, as would seem logical, if the constraint is loose the shadow price is zero (as if the constraint is loose a small change in the right-hand side cannot alter the optimal solution). c. Limits report This is the most self-explanatory report. Optimização e Decisão 09/10 - PL #1 Linear Programming - Alexandra Moutinho 8