Question

Quality Air Conditioning manufactures three home air conditioners: an economy model, a standard model, and a deluxe model. The profits per unit are $63, $95, and $135, respectively. The production requirements per unit are as follows:

	Number of Fans	Number of Cooling Coils	Manufacturing Time (hours) Economy
Economy	1	1	8
Standard	1	2	12
Deluxe	1	4	14

For the coming production period, the company has 200 fan motors, 320 cooling coils, and 2400 hours of manufacturing time available. How many economy models (E), standard models (S), and deluxe models (D) should the company produce in order to maximize profit? The linear programming model for the problem is as follows.

??? 63 ?+95 ?+135 ?

s.t. ?+?+?≤200 Fan motor

?+2?+4?≤320 Cooling coils

8?+12?+14?≤2400 Manufacturing time

?,?,?≥0
a) What is the optimal solution, and what is the value of the objective function? Use Excel Solver to solve this problem.

b) Which constraints are binding?

c) Which constraint shows extra capacity? How much?

d) If the profit for the deluxe model were increased to $150 per unit, would the optimal solution change?

e) Identify the range of optimality for each objective function coefficient.

f) Suppose the profit for the economy model is increased by $6 per unit, the profit for the standard model is decreased by $2 per unit, and the profit for the deluxe model is increased by $4 per unit. What will the new optimal solution be?

g) Identify the range of feasibility for the right-hand-side values.

h) If the number of fan motors available for production is increased by 100, will the shadow price for that constraint change? Explain.

Answer 1

Decision Variable:

Let,

E = units of an economic model to be produced

S = units of the standard model to be produced

D = units of a deluxe model to be produced

Objective Function:

The objective is to maximize the total profit function by producing the models:

Max. Z = $63E + $95S + $135D

Subject To:

Sr. No.	Constraint	Equation
1	Number of fans Available	E + S + D <= 200
2	Number of cooling coils Available	E + 2S + 4D <= 320
3	Manufacturing time available	8E + 12S + 12D <= 2200
4	Non-negative Constraint	E, S, D >= 0

Excel Model and Solver Solution:

a. Optimal Solution

The production mix for Quality Air Conditioning is as follows:

Units of Economy Model = 80 units

Units of Standard Model = 120 units

Units of Deluxe Model = 0 units

Optimal Profit level = $16,440

b. What are the dual values for the constraints? Interpret each?

Sensitivity Analysis report:

The Shadow price or dual values for fan and coil available constraints are 31 and 32. It means if an additional unit of the fan is made available, the optimal solution remains the same but the optimal objective value will increase by $31. Similarly, if an additional unit of the coil is made available, the optimal solution remains the same but the optimal objective value will increase by $32.

c. Which constraints shows extra capacity? How much?

The manufacturing time constraint is not binding. For optimal solution only 2080 hours are consumed, so extra capacity unused = 2200 – 2080 = 120 hours

d. If the profit for the deluxe model were increased to $150 per unit, would the optimal solution change? Explain

The final value of the deluxe model is zero in other words the model is not included in the optimal solution. To be included in the optimal solution the unit profit of deluxe model has to be increased. The reduced cost value for the decision variable deluxe model is -24, it means if the unit profit is increased by $24, the deluxe model will be included in the optimal solution. The target unit profit of deluxe model should be $135+$24 = $159.

As the increase unit profit of deluxe model of $150 is less than the target unit profit, the model will not be included in the optimal solution. Thus, the optimal solution will remain the same.