Question

1. A young couple, Eve and Steven, want to divide their main household chores (budgeting, cooking, dishwashing, and laundering) between them so that each has two tasks but the total time they spend on household duties is kept to a minimum. Their efficiencies on these tasks differ, where the time each would need to perform the task is given by the following table:

	Budgeting (1)	Cooking(2)	Dishwashing(3)	Laundry(4)
Eve	4.5 hrs	7.8 hrs	3.6 hrs	2.9 hrs
Steven	4.9 hrs	7.2 hrs	4.3 hrs	3.1 hrs

Formulate this problem as a linear programming problem and solve it using MS Excel Solver.

Answer 1

Formulation:

Let Y_jk be the binary integer such that the j-th person is appointed to the k-th job for j=1,2 and k=1,2,3,4

Minimize Z = total time = 4.5 Y₁₁ + 7.8 Y₁₂ + 3.6 Y₁₃ + 2.9 Y₁₄ + 4.9 Y₂₁ + 7.2 Y₂₂ + 4.3 Y₂₃ + 3.1 Y₂₄

Subject to,

Y₁₁ + Y₁₂ + Y₁₃ + Y₁₄ = 2

Y₂₁ + Y₂₂ + Y₂₃ + Y₂₄ = 2

Y_1k + Y_2k = 1 for k=1,2,3,4

Y_jk = {0,1}

Excel formulation:

A D E G 1 Dishwashing (3) Laundry (4) Budgeting (1) Cooking (2) Total 2 Eve -SUM(C3F3) 3 Steven |-SUM(C4F4) 4 Total E3+E4 —С3+C4 =D3+D4 F3+F4 6 Dishwashing (3) Laundry (4) Budgeting (1) Cooking (2) 7 Eve 4.5 7.8 3.6 2.9 8 Steven 4.9 7.2 4.3 3.1 9 10 Total time SUMPRODUCT(C3 F4,C8 F9) 11 n

Set Objective: $C$11 To: OValue Of: C Min Мax By Changing Variable Cells: $C$3:$F$4 Subject to the Constraints: $C$3:$F$4 binary Add $C$5:$F$5= 1 $G$3:$G$4 2 Change Delete Reset All Load/Save Make Unconstrained Variables Non-Negative Select a Solving Method: Simplex LP Options Solving Method Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP Simplex engine for linear Solver Problems, and select the Evolutionary engine for Solver problems that are non-smooth Solve Help Close

Solution:

	Budgeting (1)	Cooking (2)	Dishwashing (3)	Laundry (4)	Total
Eve	1	0	1	0	2
Steven	0	1	0	1	2
Total	1	1	1	1
	Budgeting (1)	Cooking (2)	Dishwashing (3)	Laundry (4)
Eve	4.5	7.8	3.6	2.9
Steven	4.9	7.2	4.3	3.1
Total time	18.4