# LP Model Problem

## LP Model Problem

﻿Question
M/S Vijay Enterprises will manufacture a product for the next four months: March, April, May and June. The demand for each month is 520, 720, 520 and 620 units respectively. The company has a steady workforce of 10 employees but can meet fluctuating production needs by hiring and firing temporary workers, if necessary. The extra cost of hiring and firing in any month are \$200 and \$400 per worker, respectively. A permanent worker can produce 12 units per month, and a temporary worker, lacking experience; only produce 10 units per month. The company can produce more than needed in any month and can carry the surplus over to next month at a holding cost of \$50 per month. Develop an optimal hiring/firing policy for the company over the next four months.

Solution
Mathematical Model
As permanent workers cannot be fired their impact on production can be accounted for by subtracting the amount that they produce. The remaining demand, if any, is satisfied through hiring and firing of temporary workers. Thus the net demand is
Demand for March = 520 – (12*10) = 400 unit
Demand for April = 720 – (12*10) = 600 unit
Demand for May = 520 – (12*10) = 400 unit
Demand for June = 620 – (12*10) = 500 unit

For i = 1, 2, 3, 4 the variables can be defined as
Xi = Net number of temporary workers at the start of month i after any hiring or firing
Si = Number of temporary hired or fired at the start of month i
Ii = Ending inventory for month i
Here Si can be positive when workers are hired and it can be negative when the workers are fired. It can be zero if no workers are hired.
Objective Function
The objective function is to minimize the sum of the cost of hiring and firing plus the cost of holding inventory from one month to the next.
Inventory holding cost = 50 (I1 + I2 + I3)
Here I4 = 0 as to get optimal solution.
Cost of hiring and firing = 200 (No of employees hired at the start of four months)
+
400 (No of...