Universal Claims Processors processes insurance claims for large national
insurance companies. Most claim processing is done by a large pool of
computer operators, some of whom are permanent and some of whom are
temporary. A permanent operator can process 16 claims per day, whereas a
temporary operator can process 12 per day, and on average the company
processes at least 450 claims each day. The company has 40 computer
workstations. A permanent operator generates about 0.5 claim with errors each
day, whereas a temporary operator average about 1.4 defective claims per day.
The company wants to limit claims with errors to 25 per day. A permanent
operator is paid $64 per day, and a temporary operator is paid $42 per day.
The company wants to determine the number of permanent and temporary
operators to hire to minimise costs.
a) Formulate a linear programming model for this problem.
b) Solve this model by using graphical analysis.

The optimal solution is to hire 22 permanent operators and 19 temporary operators, with a total cost of $2272.50.

a) Formulating the linear programming model:

Let x represent the number of full-time employees and y represent the number of temporary employees to be hired.

The objective function is to minimize the cost of hiring operators, which is given by:

Cost = 64x + 42y

The constraints are:

The total number of claims processed per day cannot be less than 450:

16x + 12y ≥ 450

The total number of defective claims per day cannot be more than 25:

0.5x + 1.4y ≤ 25

The number of workstations available is limited to 40:

x + y ≤ 40

The number of operators cannot be negative:

x ≥ 0, y ≥ 0

Following are some ways to express the linear programming model:

Minimize Cost = 64x + 42y

Subject to:

16x + 12y ≥ 450

0.5x + 1.4y ≤ 25

x + y ≤ 40

x ≥ 0, y ≥ 0

b) Solving the model using graphical analysis:

We can graph the constraints to find the feasible region and then use the corner-point method to find the optimal solution.

First, we graph the constraint 16x + 12y ≥ 450:

12y ≥ -16x + 450

y ≥ (-4/3)x + 37.5

Next, we graph the constraint 0.5x + 1.4y ≤ 25:

1.4y ≤ -0.5x + 25

y ≤ (-5/14)x + 25/1.4

y ≤ (-5/14)x + 17.86

Finally, we graph the constraint x + y ≤ 40:

y ≤ -x + 40

Combining these three constraints, we get the following feasible region:

16x + 12y = 450

y ≤ (-4/3)x + 37.5

y ≤ (-5/14)x + 17.86

y ≤ -x + 40

We now evaluate the objective function at the corner points of the feasible region to find the optimal solution.

The corner points are:

(0, 25/1.4) = (0, 17.86)

(0, 37.5)

(21.875, 18.75)

(27.27, 12.73)

(28.125, 11.25)

When we assess the objective function at these intervals, we obtain:

(0, 17.86): Cost = 42y = 42(17.86) = 750.12

(0, 37.5): Cost = 42y = 42(37.5) = 1575

(21.875, 18.75): Cost = 64x + 42y = 64(21.875) + 42(18.75) = 2272.5

(27.27, 12.73): Cost = 64x + 42y = 64(27.27) + 42(12.73) = 2536.08

(28.125, 11.25): Cost = 64x + 42y = 64(28.125) + 42(11.25) = 2587.5

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10). You are meeting with your new remote team for the first time. In order to best encourage their individual performance development, what should you do first?

A) Set clear expectations for each team member, with SMART goals.

B) Ensure that team members understand the larger goals of the company

C) Have regularly scheduled performance reviews.

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