Question

1. A company wishes to estimate the profit from two new products. The estimated cost of producing the first product is €3 per unit and it is expected to sell for €5 per unit while the second product costs €5 per unit and is expected to sell for €8 per unit. The first product takes 1 hour of packaging time per day while the second one takes 2 hours of packaging time per day. The time taken for delivery of the first product is 2 hours while the second takes just 1 hour to deliver. The total time available in the packaging and delivery sections is 6 and 8 hours per day respectively.   
(a) Formulate the linear programming problem.  

(b) Solve the problem to find the optimum solution. Clearly label and show the feasible solution space on your diagram.

(c) Over what range can the coefficient for the first product vary in the objective function before the current solution is no longer optimal?

(d) Over what range can the coefficient for the second product vary in the objective function before the current solution is no longer optimal?

(e) Compute the shadow prices for the constraints.

Answer 1

Step 1

Solution

a)

Let X1 be the number of units of product 1 to be produced and

X2 be the number of units of product 2 to be produced

Here profit per unit is calculated as a difference between cost of production and selling price.

Therefore profit for product 1 is 3 and for product 2 is 3.

The objective function is to maximize the profit which is given by -

Step 2

b)

The solution space which is satisfying the above criteria is

Therefore from graph the optimum solution is

Product 1 = 3.33 approx 4 units

Product 2 =1.33 approx 2 units

Maximum Profit = €14

c)

By conducting sensitivity analysis it is observed that the coefficient for the first product vary in the objective function by can be increased by 4 and decreased by 0.5.

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