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-114- HMP654/EXECMAS Linear Programming Linear programming is a mathematical technique that allows the decision maker to allocate scarce resources in such.

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Presentation on theme: "-114- HMP654/EXECMAS Linear Programming Linear programming is a mathematical technique that allows the decision maker to allocate scarce resources in such."— Presentation transcript:

1 -114- HMP654/EXECMAS Linear Programming Linear programming is a mathematical technique that allows the decision maker to allocate scarce resources in such a way as to optimize an objective of interest. It is linear because the relationships involved are linear. Problem Formulation Graphical Analysis Spreadsheet Solution Sensitivity Analysis

2 -115- HMP654/EXECMAS Linear Programming Case Problem - (A) p. 99

3 -116- HMP654/EXECMAS Identify problem’s objective Identify decision variables Express objective as a linear combination of decision variables Express constraints as linear combinations of decision variables Identify upper or lower bounds on the decision variables Linear Programming - Formulation

4 -117- HMP654/EXECMAS Linear Programming - Formulation Identify problem’s objective –Maximize Total Net Revenue Identify Decision Variables –X1 = Number of DRG-1 procedures performed in a month –X2 = Number of DRG-2 procedures performed in a month

5 -118- HMP654/EXECMAS Linear Programming - Formulation Express objective as a linear combination of decision variables

6 -119- HMP654/EXECMAS Linear Programming - Formulation The objective function is: Max Z = 300X1 + 400X2 Express constraints as linear combinations of decision variables 2X1 + X2 < 120 (Inpatient days) 10X1 + 30X2 < 900 (Nursing hours) 3X1 + 4X2 < 360 (Diagnostic procs.)

7 -120- HMP654/EXECMAS Linear Programming - Formulation Identify upper or lower bounds on the decision variables X1 > 0 (non-negativity X2 > 0 constraints) Complete l.p. formulation: Max Z = 300X1 + 400X2 subject to 2X1 + X2 < 120 10X1 + 30X2 < 900 3X1 + 4X2 < 360 X1, X2 > 0

8 -121- HMP654/EXECMAS Linear Programming Graphical Analysis

9 -122- HMP654/EXECMAS Linear Programming Graphical Analysis

10 -123- HMP654/EXECMAS Linear Programming Graphical Analysis

11 -124- HMP654/EXECMAS Linear Programming Graphical Analysis Unique optimal solution

12 -125- HMP654/EXECMAS Linear Programming Graphical Analysis Multiple Optimal Solutions

13 -126- HMP654/EXECMAS Linear Programming Graphical Analysis Infeasible Problem

14 -127- HMP654/EXECMAS Linear Programming Graphical Analysis Unbounded Problem

15 -128- HMP654/EXECMAS Linear Programming Spreadsheet Modeling Organize the data for the model on the spreadsheet. Reserve separate cells in the spreadsheet to represent each decision variable in the algebraic model. Create a formula in a cell in the spreadsheet that corresponds to the objective function in the algebraic model. For each constraint in the algebraic model, create a formula in a cell in the spreadsheet that corresponds to the left-hand-side (LHS) of the constraint.

16 -129- HMP654/EXECMAS Linear Programming Spreadsheet Modeling Decision Variables (V) Objective Coefficients (C) Objective Function (F) Constraints Coefficients (C) Constraints LHS (F) Constraints RHS (C) (V) - Variables (C) - Constants (F) - Formulas

17 -130- HMP654/EXECMAS Linear Programming Spreadsheet Modeling Feasible Solution

18 -131- HMP654/EXECMAS Linear Programming Spreadsheet Modeling Infeasible Solution

19 -132- HMP654/EXECMAS Linear Programming Spreadsheet Modeling

20 -133- HMP654/EXECMAS Linear Programming Spreadsheet Modeling Optimal Solution

21 -134- HMP654/EXECMAS Linear Programming

22 -135- HMP654/EXECMAS Linear Programming Identify problem’s objective –Minimize total cost of diet. Identify the decision variables Xi = No. of lbs. of food item i to be included in the optimal diet.

23 -136- HMP654/EXECMAS Linear Programming State the objective function as a linear combination of the decision variables. Min Z = 0.40X1 + 0.35X2 + 2.50X3 + 0.80X4 + 0.80X5 + 0.90X6 + 3.50X7 + 0.40X8

24 -137- HMP654/EXECMAS Linear Programming State the constraints as linear combinations of the decision variables. calories 860X1 + 440X2 + 500X3 + 310X4 + 280X5 + 140X6 + 402X7 + 613X8 > 2,600 860X1 + 440X2 + 500X3 + 310X4 + 280X5 + 140X6 + 402X7 + 613X8 < 4,000 protein 15X1 + 31X2 + 60X3 + 63X4 + 51X5 + 41X6 + 78X7 + 30X8 > 72

25 -138- HMP654/EXECMAS Linear Programming fat 16X1 + 39X2 + 110X3 + 84X4 + 12X5 + 4X6 + 34X7 + 20X8 < 100 iron 3X1 + 6X2 + 10X3 + 14X4 + 8X5 + 12X6 + 18X7 + 4X8 > 12 Identify any upper or lower bounds on the decision variables. Xi > 0i = 1,....,8

26 -139- HMP654/EXECMAS Linear Programming

27 -140- HMP654/EXECMAS Linear Programming

28 -141- HMP654/EXECMAS Linear Programming

29 -142- HMP654/EXECMAS Linear Programming - Transportation Problem Case Problem (A) p. 123

30 -143- HMP654/EXECMAS Linear Programming - Transportation Problem Identify problem’s objective –Minimize Transportation Costs for the System. Identify the decision variables –No. of cases of IV fluids shipped from Warehouse i to Hospital j. i = 1,..,3; j = A,...,D

31 -144- HMP654/EXECMAS Linear Programming - Transportation Problem A BCD W3W1W2 X1A X3A X2B

32 -145- HMP654/EXECMAS Linear Programming - Transportation Problem State the objective function as a linear combination of the decision variables. Cost contribution = $0.02 x distance (i,j) Min Z = 1.20X1A + 1.00X1B + 2.20X1C + 0.80X1D + 0.60X2A + 1.60X2B + 2.00X2C + 1.00X2D + 1.00X3A + 1.40X3B + 2.80X3C + 1.80X3D

33 -146- HMP654/EXECMAS Linear Programming - Transportation Problem State the constraints as linear combinations of the decision variables. Demand: Hospital A X1A + X2A + X3A = 1,200 Hospital B X1B + X2B + X3B = 1,500 Hospital C X1C + X2C + X3C = 2,300 Hospital D X1D + X2D + X3D = 2,400

34 -147- HMP654/EXECMAS Linear Programming - Transportation Problem Supply: Warehouse #1 X1A + X1B + X1C + X1D < 1,800 Warehouse #2 X2A + X2B + X2C + X2D < 2,400 Warehouse #3 X3A + X3B + X3C + X3D < 3,200 Identify any upper or lower bounds on the decision variables. Xi,j > 0 i = 1,..,3; j = A,...,D

35 -148- HMP654/EXECMAS Linear Programming - Transportation Problem Complete Formulation: Min Z = 1.20X1A + 1.00X1B + 2.20X1C + 0.80X1D + 0.60X2A + 1.60X2B + 2.00X2C + 1.00X2D + 1.00X3A + 1.40X3B + 2.80X3C + 1.80X3D X1A + X2A + X3A = 1,200 s.t. X1B + X2B + X3B = 1,500 X1C + X2C + X3C = 2,300 X1D + X2D + X3D = 2,400 X1A + X1B + X1C + X1D < 1,800 X2A + X2B + X2C + X2D < 2,400 X3A + X3B + X3C + X3D < 3,200 Xi,j > 0 i = 1,..,3; j = A,...,D

36 -149- HMP654/EXECMAS Linear Programming - Transportation Problem Spreadsheet Model

37 -150- HMP654/EXECMAS Linear Programming - Transportation Problem

38 -151- HMP654/EXECMAS Linear Programming - Transportation Problem Optimal Solution

39 -152- HMP654/EXECMAS Linear Programming - Assignment Problem Case Problem (A) p. 131

40 -153- HMP654/EXECMAS Linear Programming - Assignment Problem Identify problem’s objective –Minimize Total Costs Identify the decision variables Xi,j = 1 if technician i is assigned to test j. Xi,j = 0 if technician i is not assigned to test j. i = A,...,D ; j = 1,...,4

41 -154- HMP654/EXECMAS Linear Programming - Assignment Problem State the objective function as a linear combination of the decision variables. Cost contribution = salary(i) x time (i,j)/60 Min Z = 3.50XA1 + 8.75XA2 + 1.75XA3 + 5.25XA4 + 2.81XB1 + 6.75XB2 + 2.03XB3 + 6.75XB4 + 2.48XC1 + 12.38XC2 + 1.65XC3 + 6.19XC4 + 3.75XD1 + 10XD2 +2.50XD3 + 6.25XD4

42 -155- HMP654/EXECMAS Linear Programming - Assignment Problem State the constraints as linear combinations of the decision variables. –Each technician must be assigned to one, and only one test. XA1 + XA2 +XA3 + XA4 = 1 XB1 + XB2 + XB3 + XB4 = 1 XC1 + XC2 + XC3 + XC4 = 1 XD1 + XD2 + XD3 + XD4 =1

43 -156- HMP654/EXECMAS Linear Programming - Assignment Problem –To each test, one, and only one technician must be assigned XA1 + XB1 + XC1 + XD1 = 1 XA2 + XB2 + XC2 + XD2 = 1 XA3 + XB3 + XC3 + XD3 = 1 XA4 + XB4 + XC4 + XD4 = 1 Identify any upper or lower bounds on the decision variables. Xi,j > 0 i = A,..,D; j = 1,...,4

44 -157- HMP654/EXECMAS Linear Programming - Assignment Problem Complete Formulation: Min Z = 3.50XA1 + 8.75XA2 + 1.75XA3 + 5.25XA4 + 2.81XB1 + 6.75XB2 + 2.03XB3 + 6.75XB4 + 2.48XC1 + 12.38XC2 + 1.65XC3 + 6.19XC4 + 3.75XD1 + 10XD2 + 2.50XD3 + 6.25XD4 XA1 + XA2 +XA3 + XA4 = 1 XB1 + XB2 + XB3 + XB4 = 1 XC1 + XC2 + XC3 + XC4 = 1 XD1 + XD2 + XD3 + XD4 =1 s.t. XA1 + XB1 + XC1 + XD1 = 1 XA2 + XB2 + XC2 + XD2 = 1 XA3 + XB3 + XC3 + XD3 = 1 XA4 + XB4 + XC4 + XD4 = 1 Xi,j > 0 i = A,..,D; j = 1,...,4

45 -158- HMP654/EXECMAS Linear Programming - Assignment Problem Spreadsheet Model

46 -159- HMP654/EXECMAS Linear Programming - Assignment Problem

47 -160- HMP654/EXECMAS Linear Programming - Assignment Problem Optimal Solution

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