A young couple, Eve and Steven, want to divide their main household chores (mark

Question

A young couple, Eve and Steven, want to divide their main household chores (marketing, cooking, dish washing, and laundering) between them so that the difference between time each spend on household duties is less than or equal to 2 hours. Marketing and cooking are mutually exclusive, i.e., both tasks cannot be assigned to same person. They want to assign tasks to minimize the total time spend on all chores. Their efficiencies on these tasks differ, where the time each would need to perform the task is given by the following table

4.1   

Formulate the problem (define decision variables clearly and formulate objective function and constraints) and solve by excel solver

Time needed per week (hours) Marketing Cooking Dish washing Laundry Eve 3.5 4.8 3.6 3.9 Steven 3.9 4.2 4.3

4.1   

Explanation / Answer

(a) Let Xij = 1 if person i performs chore j

= 0 if not

fori= 1,2 (Eve, Steven), j= 1,2,3,4 (marketing, cooking, dishwashing, laundering)

Minz= 4.5X11+ 7.8X12+ 3.6X13+ 2.9X14 + + 4.9X21+ 7.2X22+ 4.3X23+ 3.1X24

s.t.

X11+X12+X13+X14 = 2 (each person does 2 chores)

X21+X22+X23+X24= 2

X11+X21= 1 (each chore is done by 1 person)

X21+X22= 1

X13+X23= 1

X14+X24= 1

with Xij = either 1 or 0

(b) The optimal binary integer solution is X11, X13, X22, X24= 1, all others = 0 and

minz= 18.4 hours.

Alternatively, we can formulate as an assignment problemas follows:

"Let Xij= person i to be assigned to chore j "the same objective function and constraints"withXij >=0"

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