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B-1 Operations Management Linear Programming Module B - Part 2.

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1 B-1 Operations Management Linear Programming Module B - Part 2

2 B-2 Problem B 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.

3 B-3 Problem B.23 Container Profit Packing material (lbs.) Packing time (hrs.) A $8 2 1 Amount available K T $6 $  120 =240

4 B-4 Problem B.23 : Maximize: 8x A + 6x K + 14x T 2x A + x K + 3x T  120 (lbs.) 2x A + 6x K + 4x T = 240 (hours) x A, x K, x T  0 x i = Number of class i containers to pack each week. i=A, K, T

5 B-5 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

6 B-6 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.

7 B-7 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.

8 B-8 Computer Solutions - Spreadsheet

9 B-9 Computer Solutions - Spreadsheet

10 B-10 Computer Solutions - Spreadsheet

11 B-11 Computer Solutions - Solver

12 B-12 Computer Solutions - Solver

13 B-13 Computer Solutions - Solver Parameters

14 B-14 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

15 B-15 Computer Solutions - Adding Constraints  Cell Reference: LHS location  Select sign : =  Constraint: RHS location

16 B-16 Computer Solutions - Adding Constraints  1st constraint.  Click Add.  Repeat for second constraint.

17 B-17 Computer Solutions  Click Options to set up Solver for LP.

18 B-18 Computer Solutions - Solver Options  Check ‘on’ Assume Linear Model and Assume Non-Negative.

19 B-19 Computer Solutions  Click Solve to find the optimal solution.

20 B-20 Computer Solutions - Solver Results

21 B-21 Computer Solutions - Optimal Solution  Optimal solution is to use:  0 A containers  K containers  T containers  Maximum profit is $583 per week.  Actually $ … in Excel values are rounded.

22 B-22 Computer Solutions  Optimal solution is to use:  0 class A containers.  class K containers.  class T containers.  Maximum profit is $ per week.  Select Answer and Sensitivity Reports and click OK.  New pages appear in Excel.

23 B-23 Computer Solution - Answer Report

24 B-24 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.

25 B-25 Computer Solution - Sensitivity Report

26 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 FinalReducedObjectiveAllowable CellNameValueCostCoefficientIncreaseDecrease $B$7Sol'n values A cont E+30 $C$7Sol'n values K cont E+30 $D$7Sol'n values T cont E Optimal solution: 0 class A containers … class K containers … class T containers Profit = 0(8) (6) (14) = $

27 B-27 Computer Solution - Sensitivity Report

28 B-28 Sensitivity for Objective Coefficients  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 x A is between -infinity and Coefficient of x K is between -infinity and 14 Coefficient of x T is between 12.4 and infinity

29 B-29 Sensitivity for Objective Coefficients  If profit for class K container was 12 (not 6), what is optimal solution?

30 B-30 Sensitivity for Objective Coefficients  If profit for class K container was 12 (not 6), what is optimal solution?  x A =0, x K =17.14, x T =34.29 (same as before)  profit = (more than before!)

31 B-31 Sensitivity for Objective Coefficients  If profit for class K container was 16 (not 6), what is optimal solution?

32 B-32 Sensitivity for Objective Coefficients  If profit for class K container was 16 (not 6), what is optimal solution?  Different!  Resolve problem to get solution.

33 B-33 Computer Solution - Sensitivity Report

34 B-34 Sensitivity for RHS values  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 $ for up to 60 additional lbs.  Each additional hour of packing time will increase profit by $ for up to 480 additional hours.

35 B-35 Sensitivity for RHS values  Suppose you can buy 50 more lbs. of packing material for $250. Should you buy it?

36 B-36 Sensitivity for RHS values  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 $ per lb.

37 B-37 Sensitivity for RHS values  How much would you pay for 50 more lbs. of packing material?

38 B-38 Sensitivity for RHS values  How much would you pay for 50 more lbs. of packing material?  $ lbs.  $4.2857/lb. = $

39 B-39 Sensitivity for RHS values  If change in RHS is outside range (from allowable increase or decrease), then we can not tell how the objective value will change.

40 B-40 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).

41 B-41 Integer Programming 1 if we build a factory in St. Louis 0 otherwise. { { 1 if we build a factory in Chicago 0 otherwise. We will build one factory in Chicago or St. Louis. x 1 + x 2  1 We will build one factory in either Chicago or St. Louis. x 1 + x 2 = 1 If we build in Chicago, then we will not build in St. Louis. x 2  1 - x 1

42 B-42 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. Harder Formulation Example

43 B-43 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. Another Formulation Example Midnight - 4 am 4 am - 8 am am - noon 13 Noon - 4 pm 15 4 pm - 8 pm 12 8 pm - midnight 9


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