Question

Applicant	Carding	Spinning	Weaving	Inspection	Shipping
Roger Acuff	68	75	72	86	78
Melissa Ball	73	82	66	78	85
Angela Coe	92	101	90	79	74
Maureen Davis	87	98	75	90	92
Fred Evans	58	62	93	81	75
Bob Frank	93	79	94	92	96
Ellen Gantry	77	92	90	81	93
David Harper	79	66	90	85	86
Mary Inchavelia	91	102	95	90	88
Marilu Jones	72	75	67	93	93

Assignment problem with 10 applicants and 5 positions. Use the data to determine the optimal assignments

Formulate a linear programming model for this problem.(List the objective function (minimize or maximize) and all model constraints)

Solve problem using MS Excel’s Solver (Hint: Use the “integer” constraint)

This is a maximization problem, In what circumstances would you (or could you) use the Assignment Model to minimize?

Answer 1

Let the applicants be denoted by 1 to 10

The applicant can be either selected or rejected for a particular position.

Let x_ij denote whether the applicant is selected or rejected, where is is the applicant number and j is the position

x_ij can be either 0 or 1

Let A_ij denote the score of applicant i for position j

Applicant	Carding(1)	Spinning(2)	Weaving(3)	Inspection(4)	Shipping(5)
Roger Acuff(1)	x11	x12	x13	x14	x15
Melissa Ball(2)	x21	x22	x23	x24	x25
Angela Coe(3)	x31	x32	x33	x34	x35
Maureen Davis(4)	x41	x42	x43	x44	x45
Fred Evans(5)	x51	x52	x53	x54	x55
Bob Frank(6)	x61	x62	x63	x64	x65
Ellen Gantry(7)	x71	x72	x73	x74	x75
David Harper(8)	x81	x82	x83	x84	x85
Mary Inchavelia(9)	x91	x92	x93	x94	x95
Marilu Jones(10)	x101	x102	x103	x104	x105

Let

Objective Function Max Z = Σx_ijA_ij

Constraints -

Σx_1j <=1 ...... applicant 1 can be selected for one role only
Σx_2j <=1 ...... applicant 2 can be selected for one role only
Σx_3j <=1 ...... applicant 3 can be selected for one role only
Σx_4j <=1 ...... applicant 4 can be selected for one role only
Σx_5j <=1 ...... applicant 5 can be selected for one role only
Σx_6j <=1 ...... applicant 6 can be selected for one role only
Σx_7j <=1 ...... applicant 7 can be selected for one role only
Σx_8j <=1 ...... applicant 8 can be selected for one role only
Σx_9j <=1 ...... applicant 9 can be selected for one role only
Σx_10j <=1 ...... applicant 10 can be selected for one role only

Σx_i1 <=1 ...... only one applicant can be selected for Carding(1)
Σx_i2 <=1 ...... only one applicant can be selected for Spinning(2)
Σx_i3 <=1 ...... only one applicant can be selected for Weaving(3)
Σx_i4 <=1 ...... only one applicant can be selected for Inspection(4)
Σx_i5 <=1 ...... only one applicant can be selected for Shipping(5)

x_ij = 0, 1

Configure the solver as below -->

Run the solver to get -->

Hence,

For Carding, Angela Joe should be selected
For Spinning, Mary Inchavella should be selected
For Weaving, Fred Evans should be selected
For Inspection, Marilu Jones should be selected
For Shipping, Bob Frank should be selected.

Since we are looking at the scores in assignments here, we took the max objective function.
In the scenario where we look at the time taken to complete the assignment, we would take the minimum objective function, since the applicant who finishes an activity first will be more suitable for the position.