Question

OBC processes insurance applications for large insurance companies. Insurance application processing is done by a large number of workers. There are permanent staff as well as contractual staff for the processing of these applications. A permanent staff can process 16 applications per day, whereas a temporary staff can process 12 applications per day. On the average, OBC processes at least 450 applications each day. OBC has 40 desktop workstations. The permanent staff generates 0.5 applications with errors each day, whereas a temporary staff averages about 1.4 defective applications per day. The company wants to limit the application processing with errors to 25 per day. The permanent staff is paid $64 per day and the temporary staff is paid $42 per day. Each staff uses 1 desktop. The company wants to determine the number of permanent and temporary staff to deploy in order to minimize costs.

(a) Formulate a linear programming model for this problem

(b) Solve this model by using graphical analysis

Answer 1

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Let the number of permanent staff be X

Number of contractual staff be Y

Permanent staff can process 16 applications per day

Contractual staff can process 12 applications per day.

OBC is processing at least 450 applications a day

Number of applications processed by permanent staff is 16 per staff and total X staffs = 16X

Number of applications processed by contractual staff is 12 per staff and total Y staffs = 12Y

According to the minimum number of applications processed a day

Applications processed a day is let Z.

Max Z = 16X + 12Y ………………………………………………………………………………………………………………..(0)

16X + 12Y >= 450 ……………………………………………………………………………………………………………….(1)

There are 40 desktop work stations.

Hence X+Y <= 40 ………………………………………………………………………………………………………………(2)

Permanent staff generates 0.5 applications with defect. Contractual staff generates 1.4 applications with defect each day.

The maximum total defect should not exceed 25 per day

Hence 0.5X + 1.4Y <= 25 ………………………………………………………………………………………………….(3)

The number of staffs can’t be negative.

Hence, X>=0, Y>=0 …………………………………………………………………………………………………………..(4)

Solving using graphical method:

x>=0 and y>=0 suggests the solution in first quadrant.

Let’s modify the equation a little for the sake of convenience in solving. Y1, Y2 and Y both represent Y only, i.e., number of temporary staffs.

16X + 12Y >= 450

X+Y1 <= 40

0.5X + 1.4Y2 <= 25

Plotting these equations in graph, we will consider the parts above Y line, and below Y1 and Y2 line. Hence,

Graphical Method 5 10 15 20 25 30 35 40 45 — y — 1 — 12

Here, x axis represents number of permanent staffs and y axis represents number of contractual staffs. Hence

Permanent staff -> between 22 and 40

Contractual staff -> between 0 and 10

When we use 40 permanent staff and 0 contractual staff, the productivity will be maximum. Combination of 22 permanent staff and 10 contractual staff will also give the maximum productivity.