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Operations Management Linear Programming Module B - Part 2 B-1 Problem B.23 1. Gross Distributors packages and distributes industrial supplies. A standard shipment can be packaged in a class A container, a class K container, or a class T container. The profit from using each type of container is: $8 for each class A container, $6 for each class K container, and $14 for each class T container. The amount of packing material required by each A, K and T container is 2, 1 and 3 lbs., respectively. The amount of packing time required by each A, K, and T container is 2, 6, and 4 hours, respectively. There is 120 lbs of packing material available each week. Six packers must be employed full time (40 hours per week each). Determine how many containers to pack each week. B-2 Problem B.23 Container A K T Profit $8 $6 $14 Amount available Packing material (lbs.) 2 1 3 Packing time (hrs.) 2 6 4 120 =240 B-3 Problem B.23 xi = Number of class i containers to pack each week. i=A, K, T Maximize: 8xA + 6xK + 14xT 2xA + xK + 2xA + 6xK + 3xT 120 (lbs.) 4xT = 240 (hours) xA, xK, xT 0 B-4 Linear Programming Solutions Unique Optimal Solution. Multiple Optimal Solutions. Infeasible (no solution). x + y 800 x 1000 x, y 0 Unbounded (infinite solution). Maximize 3x + 2y x + y 1000 B-5 Computer Solutions Optimal values of decision variables and objective function. Sensitivity information for objective function coefficients. Sensitivity information for RHS (right-hand side) of constraints and shadow price. B-6 Computer Solutions Enter data from formulation in Excel. 1 row for the coefficients of objective. 1 row for coefficients & RHS of each constraint. 1 final row for solution (decision variable) values. Select Solver from the Tools Menu. B-7 Computer Solutions - Spreadsheet B-8 Computer Solutions - Spreadsheet B-9 Computer Solutions - Spreadsheet B-10 Computer Solutions - Solver B-11 Computer Solutions - Solver B-12 Computer Solutions - Solver Parameters B-13 Computer Solutions Set Target Cell: to value of objective function. E3 Equal To: Max or Min By Changing Cells: = Sol’n values (decision variable values). B7:D7 Subject to the Constraints: Click Add to add each constraint: LHS =, , RHS B-14 Computer Solutions - Adding Constraints Cell Reference: LHS location Select sign : <=, =, >= Constraint: RHS location B-15 Computer Solutions - Adding Constraints 1st constraint. Click Add. Repeat for second constraint. B-16 Computer Solutions Click Options to set up Solver for LP. B-17 Computer Solutions - Solver Options Check ‘on’ Assume Linear Model and Assume Non-Negative. B-18 Computer Solutions Click Solve to find the optimal solution. B-19 Computer Solutions - Solver Results B-20 Computer Solutions - Optimal Solution Optimal solution is to use: 0 A containers 17.14 K containers 34.29 T containers Maximum profit is $583 per week. Actually $582.857… in Excel values are rounded. B-21 Computer Solutions Optimal solution is to use: 0 class A containers. 17.14 class K containers. 34.29 class T containers. Maximum profit is $582.857 per week. Select Answer and Sensitivity Reports and click OK. New pages appear in Excel. B-22 Computer Solution - Answer Report Microsoft Excel 8.0e Answer Report Worksheet: [probb.23.xls]Sheet1 Report Created: 1/31/01 9:53:27 PM Target Cell (Max) Cell Name $E$3 Objective LHS Original Value Final Value 28 582.8571429 Adjustable Cells Cell Name Original Value Final Value $B$7 Sol'n values A cont. 1 0 $C$7 Sol'n values K cont. 1 17.14285714 $D$7 Sol'n values T cont 1 34.28571429 Constraints Cell Name $E$4 lbs. LHS $E$5 hours LHS Cell Value Formula Status Slack 120 $E$4<=$F$4 Binding 0 240 $E$5=$F$5 Binding 0 B-23 Sensitivity Analysis Projects how much a solution will change if there are changes in variables or input data. Shadow price (dual) - Value of one additional unit of a resource. B-24 Computer Solution - Sensitivity Report Microsoft Excel 8.0e Sensitivity Report Worksheet: [probb.23.xls]Sheet1 Report Created: 1/31/01 9:53:27 PM Adjustable Cells Cell $B$7 $C$7 $D$7 Final Value Reduced Objective Allowable Allowable Name Cost Coefficient Increase Decrease Sol'n values A cont. 0 -1.142857143 8 1.142857143 1E+30 Sol'n values K cont. 17.14285714 0 6 8 1E+30 Sol'n values T cont 34.28571429 0 14 1E+30 1.6 Constraints Cell Name $E$4 lbs. LHS $E$5 hours LHS Final Value 120 240 Shadow Constraint Price R.H. Side 4.285714286 120 0.285714286 240 B-25 Allowable Allowable Increase Decrease 60 80 480 80 Computer Solution - Sensitivity Report Microsoft Excel 8.0e Sensitivity Report Worksheet: [probb.23.xls]Sheet1 Report Created: 1/31/01 9:53:27 PM Adjustable Cells Cell $B$7 $C$7 $D$7 Final Value Reduced Objective Allowable Allowable Name Cost Coefficient Increase Decrease Sol'n values A cont. 0 -1.142857143 8 1.142857143 1E+30 Sol'n values K cont. 17.14285714 0 6 8 1E+30 Sol'n values T cont 34.28571429 0 14 1E+30 1.6 Optimal solution: 0 class A containers 17.14285… class K containers 34.28571… class T containers Profit = 0(8) + 17.14285(6) + 34.28571(14) = $582.857 B-26 Computer Solution - Sensitivity Report Microsoft Excel 8.0e Sensitivity Report Worksheet: [probb.23.xls]Sheet1 Report Created: 1/31/01 9:53:27 PM Adjustable Cells Cell $B$7 $C$7 $D$7 Final Value Reduced Objective Allowable Allowable Name Cost Coefficient Increase Decrease Sol'n values A cont. 0 -1.142857143 8 1.142857143 1E+30 Sol'n values K cont. 17.14285714 0 6 8 1E+30 Sol'n values T cont 34.28571429 0 14 1E+30 1.6 Constraints Cell Name $E$4 lbs. LHS $E$5 hours LHS Final Value 120 240 Shadow Constraint Price R.H. Side 4.285714286 120 0.285714286 240 B-27 Allowable Allowable Increase Decrease 60 80 480 80 Sensitivity for Objective Coefficients Objective Allowable Allowable Coefficient Increase Decrease 8 1.142857143 1E+30 6 8 1E+30 14 1E+30 1.6 As long as coefficients are in range indicated, then current solution is still optimal, but profit may change! Current solution is optimal as long as: Coefficient of xA is between -infinity and 9.142857 Coefficient of xK is between -infinity and 14 Coefficient of xT is between 12.4 and infinity B-28 Sensitivity for Objective Coefficients Objective Allowable Allowable Coefficient Increase Decrease 8 1.142857143 1E+30 6 8 1E+30 14 1E+30 1.6 If profit for class K container was 12 (not 6), what is optimal solution? B-29 Sensitivity for Objective Coefficients Objective Allowable Allowable Coefficient Increase Decrease 8 1.142857143 1E+30 6 8 1E+30 14 1E+30 1.6 If profit for class K container was 12 (not 6), what is optimal solution? xA=0, xK=17.14, xT=34.29 (same as before) profit = 685.71 (more than before!) B-30 Sensitivity for Objective Coefficients Objective Allowable Allowable Coefficient Increase Decrease 8 1.142857143 1E+30 6 8 1E+30 14 1E+30 1.6 If profit for class K container was 16 (not 6), what is optimal solution? B-31 Sensitivity for Objective Coefficients Objective Allowable Allowable Coefficient Increase Decrease 8 1.142857143 1E+30 6 8 1E+30 14 1E+30 1.6 If profit for class K container was 16 (not 6), what is optimal solution? Different! Resolve problem to get solution. B-32 Computer Solution - Sensitivity Report Microsoft Excel 8.0e Sensitivity Report Worksheet: [probb.23.xls]Sheet1 Report Created: 1/31/01 9:53:27 PM Adjustable Cells Cell $B$7 $C$7 $D$7 Final Value Reduced Objective Allowable Allowable Name Cost Coefficient Increase Decrease Sol'n values A cont. 0 -1.142857143 8 1.142857143 1E+30 Sol'n values K cont. 17.14285714 0 6 8 1E+30 Sol'n values T cont 34.28571429 0 14 1E+30 1.6 Constraints Cell Name $E$4 lbs. LHS $E$5 hours LHS Final Value 120 240 Shadow Constraint Price R.H. Side 4.285714286 120 0.285714286 240 B-33 Allowable Allowable Increase Decrease 60 80 480 80 Sensitivity for RHS values Shadow Constraint Price R.H. Side 4.285714286 120 0.285714286 240 Allowable Allowable Increase Decrease 60 80 480 80 Shadow price is change in objective value for each unit change in RHS as long as change in RHS is within range. Each additional lb. of packing material will increase profit by $4.2857... for up to 60 additional lbs. Each additional hour of packing time will increase profit by $0.2857... for up to 480 additional hours. B-34 Sensitivity for RHS values Shadow Constraint Price R.H. Side 4.285714286 120 0.285714286 240 Allowable Allowable Increase Decrease 60 80 480 80 Suppose you can buy 50 more lbs. of packing material for $250. Should you buy it? B-35 Sensitivity for RHS values Shadow Constraint Price R.H. Side 4.285714286 120 0.285714286 240 Allowable Allowable Increase Decrease 60 80 480 80 Suppose you can buy 50 more lbs. of packing material for $250. Should you buy it? NO. $250 for 50 lbs. is $5 per lb. Profit increase is only $4.2857 per lb. B-36 Sensitivity for RHS values Shadow Constraint Price R.H. Side 4.285714286 120 0.285714286 240 Allowable Allowable Increase Decrease 60 80 480 80 How much would you pay for 50 more lbs. of packing material? B-37 Sensitivity for RHS values Shadow Constraint Price R.H. Side 4.285714286 120 0.285714286 240 Allowable Allowable Increase Decrease 60 80 480 80 How much would you pay for 50 more lbs. of packing material? $214.28 50 lbs. $4.2857/lb. = $214.2857... B-38 Sensitivity for RHS values Shadow Constraint Price R.H. Side 4.285714286 120 0.285714286 240 Allowable Allowable Increase Decrease 60 80 480 80 If change in RHS is outside range (from allowable increase or decrease), then we can not tell how the objective value will change. B-39 Extensions of Linear Programming Integer programming (IP): Some or all variables are restricted to integer values. Allows “if…then” constraints. Much harder to solve (more computer time). Nonlinear programming: Some constraints or objective are nonlinear functions. Allows wider range of situations to be modeled. Much harder to solve (more computer time). B-40 Integer Programming { 1 { 0 x1 x2 1 if we build a factory in St. Louis 0 otherwise. if we build a factory in Chicago otherwise. We will build one factory in Chicago or St. Louis. x1 + x 2 1 We will build one factory in either Chicago or St. Louis. x1 + x 2 = 1 If we build in Chicago, then we will not build in St. Louis. x2 1 - x1 B-41 Harder Formulation Example You are creating an investment portfolio from 4 investment options: stocks, real estate, T-bills (Treasury-bills), and cash. Stocks have an annual rate of return of 12% and a risk measure of 5. Real estate has an annual rate of return of 10% and a risk measure of 8. T-bills have an annual rate of return of 5% and a risk measure of 1. Cash has an annual rate of return of 0% and a risk measure of 0. The average risk of the portfolio can not exceed 5. At least 15% of the portfolio must be in cash. Formulate an LP to maximize the annual rate of return of the portfolio. B-42 Another Formulation Example A business operates 24 hours a day and employees work 8 hour shifts. Shifts may begin at midnight, 4 am, 8 am, noon, 4 pm or 8 pm. The number of employees needed in each 4 hour period of the day to serve demand is in the table below. Formulate an LP to minimize the number of employees to satisfy the demand. Midnight - 4 am 3 4 am 8 am 6 8 am noon 13 Noon 4 pm 15 B-43 4 pm - 8 pm 8 pm midnight 12 9