Question

Your new client is an electronics giant called TVTitans which sells three different categories of televisions: small, medium, and large. The store can sell up to 200 sets a month.

• Each of the small, medium, and large televisions requires 3, 6, and 6 cubic feet of storage space respectively
  • A maximum of 1,020 cubic feet of storage space is available.
• Each product of the three categories, small, medium, and large, requires 2, 2, and 4 sales hours of labor respectively.The profit made from each of these product categories is $40 for small, $80 for medium, and $100 for large.  
  • A maximum of 600 hours of sales labor is available.

1. Mathematically model the problem (i.e. list inequalities conveying the situation)
2. Include 2–3 Excel screenshots showing the data setup, problem-solving process, and final result

Answer 1

ANSWER:-

INCLUDING EXCEL FILES ALSO INSTEAD

MATHEMATICAL MODEL:

Let S, M, and L be the number of small, medium and large television sets.

Objective function:

Maximise Profit, Z = 40S + 80M + 100L

The constraints are:

S + M + L \leq 200 (Max 200 sets a month)
3S + 6M + 6M \leq 1020 (Max 1020 cubic feet of space available)
2S + 2M + 4S \leq 600 (Max 600 labour hours)
and

S, M, L \geq 0 (Non - negativity constraint)

To convert the constraints into its canonical form and since all equations are '\leq' type, slack variables S1, S2 and S3are introduced into the equations so that the problem takes the form of the standard mathematical model :

Maximise Z = 40S + 80M + 100L

Subject to the constraints:

S + M + L + S1 = 200
3S + 6M + 6M + S2 = 1020
2S + 2M + 4S + S3 = 600
and S, M, L, S1, S2 and S3\leq 0

The solution process using simplex method is illustrated below:

Entering variable : L

Leaving variable: S1

Pivot Element: 4

Row Operations used:

R3\ R3/4
R1 R1 - R3
R2 R2 - 6R3

Entering variable: M

Leaving variable: S2

Pivot Element: 3

Row Operations used:

R2 R2/3
R1R1 - (1/2)R2
R3 R3 - (1/2)R2

Here all, Zj - Cj\geq 0.

Hence, the optimal solution is reached with the solution as:

S = 0

M = 40

L = 130

and Z = 16200.

So, to maximize profit, the company should sell zero small size TVs, 40 medium size TVs, and 130 large size TVs in a month. This will give a maximum profit of $16200/month.