Question

A small firm makes three similar products, which all follow the same three-step process, consisting of milling,
inspection, and drilling. Product A requires 12 minutes of milling, 5 minutes for inspection, and 10 minutes of
drilling per unit; product B requires 10 minutes of milling, 4 minutes for inspection, and 8 minutes of drilling per
unit; and product C requires 8 minutes of milling, 4 minutes for inspection, and 16 minutes of drilling. The
department has 20 hours available during the next period for milling, 15 hours for inspection, and 24 hours for
drilling. Product A contributes $2.40 per unit to profit, product B contributes $2.50 per unit, and product C
contributes $3.00 per unit. Determine the optimal mix of products in terms of maximizing contribution to profits
for the period.
Requirements:
a) Define the decision variables;
b) Show the business objective and objective function;
c) List all constraint equations;
d) Create an EXCEL working sheet for the linear programming model;
e) Provide both Answer and Sentivity reports;
f) Find the range of optimality for the profit coefficient of each variable using the reoprts generated.

Answer 1

1. Define the Decision Variables:

A is the production quantity of Product A- In Units;

B is the production quantity of Product B- In Units;

C is the production quantity of Product C- In Units.

b. Business Objective is to Maximize this which is the Objective Function = 2.4A + 2.5B + 3C

c. Constraints are Subject to: 12A + 10B + 8C ≤ 1200    (for milling)

                  5A   + 4B   + 4C ≤ 900      (for inspection)

                  10A + 8B   + 6C ≤ 1440    (for drilling)

Also note that; A ≥ 0, B ≥ 0, C ≥ 0

Optimal Solution solved using Excel is:

A=0,

B=80,

C=50 and the maximum profit = 350

Optimality range for variable A is [-∞, 3.042] = [2.4 - ∞, 2.4+0.642]

Optimality range for variable B is [1.95, 3.75] = [2.5-0.55, 2.5+1.25]

Optimality range for variable C is [2, 5] = [3-1, 3+2]

Shadow price for resource of milling hours is 0.16666667 with a feasibility range of [1200-480, 1200+600] = [720, 1800]. Since 25 hours = 25*60=1500 minutes which is within the range of [720, 1800],

new maximum profit will be equal to = 350 + 0.16666667*(1500-1200) = 400.

Shadow price for resource of drilling hours is 0.10416667 with a feasibility range of [1440-480, 1440+960] = [960, 2400]. Since 20 hours = 20*60=1200 minutes which is within the range of [960, 2400],

therefore, the new maximum profit will be equal to = 350 + 0.10416667*(1200-1440) = 325.