-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

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

-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

-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

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

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

-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 = 300X X2 subject to 2X1 + X2 < X1 + 30X2 < 900 3X1 + 4X2 < 360 X1, X2 > 0

-121- HMP654/EXECMAS Linear Programming Graphical Analysis

-122- HMP654/EXECMAS Linear Programming Graphical Analysis

-123- HMP654/EXECMAS Linear Programming Graphical Analysis

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

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

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

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

-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.

-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

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

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

-132- HMP654/EXECMAS Linear Programming Spreadsheet Modeling

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

-134- HMP654/EXECMAS Linear Programming

-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.

-136- HMP654/EXECMAS Linear Programming State the objective function as a linear combination of the decision variables. Min Z = 0.40X X X X X X X X8

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

-138- HMP654/EXECMAS Linear Programming fat 16X1 + 39X X3 + 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

-139- HMP654/EXECMAS Linear Programming

-140- HMP654/EXECMAS Linear Programming

-141- HMP654/EXECMAS Linear Programming

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

-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

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

-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 X1B X1C X1D X2A X2B X2C X2D X3A X3B X3C X3D

-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

-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

-148- HMP654/EXECMAS Linear Programming - Transportation Problem Complete Formulation: Min Z = 1.20X1A X1B X1C X1D X2A X2B X2C X2D X3A X3B X3C X3D 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

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

-150- HMP654/EXECMAS Linear Programming - Transportation Problem

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

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

-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

-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 = 3.50XA XA XA XA XB XB XB XB XC XC XC XC XD1 + 10XD XD XD4

-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

-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

-157- HMP654/EXECMAS Linear Programming - Assignment Problem Complete Formulation: Min Z = 3.50XA XA XA XA XB XB XB XB XC XC XC XC XD1 + 10XD XD XD4 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

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

-159- HMP654/EXECMAS Linear Programming - Assignment Problem

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