Question

Below given is the linear programming model at a manufacturing firm which produces and sells

for different bags: small bags, medium bags, standard bags, and deluxe bags.

DECISION VARIABLES:

xi- Number of bags for group i to produce, i=1(small bag), 2(medium bag), 3(standard bag),

4(deluxe bag).

OBJECTIVE FUNCTION:

Maximize profit, z = 6.5x1 + 7.5x 2 +10x3 + 9x4

CONSTRAINTS:

0.55x1 + 0.6x 2 + 0.7x3 + x4 ≤ 630 (Cutting and dyeing)

0.425x1 + 0.45x 2 + 0.5x3 + 0.833 x4 ≤ 600 (Sewing)

0.55x1 + 0.6x 2 + x3 + 0.67x4 ≤ 708 (Finishing)

0.78x1 + 0.8x 2 + 0.1 x3 + 0.25x4 ≤ 135 (Inspection and Packing)

x1 , x 2 , x3, x4 ≥ 0

The Excel sensitivity report for this linear model is provided below:

Microsoft Excel 12.0 Sensitivity Report

Adjustable Cells

Final Reduced Objective Allowable Allowable

Cell Name Value Cost Coefficient Increase Decrease

$B$26 # Small bag 0.000 ‐0.470 6.5 0.470 1E+30

$C$26 # Medium bag 79.508 0 7.5 7.918 0.480

$D$26 # Standard bag 640.789 0 10 2.479 8.702

$E$26 # Deluxe bag 29.259 0 9 77.667 1.962

Constraints

Final Shadow Constraint Allowable Allowable

Cell Name Value Price R.H. Side Increase Decrease

$B$32 Cutting and dyeing 525.516 0 630 1E+30 104.484

$B$33 Sewing 600 0.246 600 1702.313 232.986

$B$34 Finishing 708 9.679 708 161.682 576.099

$B$3 Inspection &Packaging 135 1.977 135 452.699 58.563

Answer the following questions using the above sensitivity report:

NOTE:

1. EACH QUESTION THAT FOLLOWS REFERS TO THE ORIGINAL PROBLEM. THAT IS, EACH QUESTION IS INDEPENDENT OF THE OTHER QUESTIONS.

2. IF IT IS IMPOSSIBLE TO ANSWER THE QUESTION WITHOUT RESOLVING THE

PROBLEM, YOU MUST STATE THAT IN YOUR ANSWER AND BRIEFLY EXPLAIN

WHY. NO MARKS WILL BE GIVEN FOR UNSUPPORTED ANSWERS. SHOW ALL YOUR CALCULATIONS CLEARLY.

a. Identify the optimal solution and its objective function value?

b. Find the hours used for each process (cutting and sewing, finishing, inspection and

packing):

c. If the profit from small bag increases to $7.25, what will be the new optimal solution and

new objective function value?

d. If the profit from the standard bag increases to $12.00, what will be the new optimal

solution and new objective function value?

e. Due to the expected maintenance work in the cutting and dyeing department, the hours

available in cutting and dyeing will decrease to 550 hours. What is the new optimal

solution and new objective function value?

f. The firm can get 50 more hours in cutting dyeing department, find its new optimal

solution and new objective function value.

g. The firm can produce another bag called “School bags” at a profit of $5.5. One school

required 0.5 hours for cutting and dyeing, 0.42 hours for finishing, and 0.35 hours for

inspection and packing. Find the new optimal solution and new objective function value.

h. The firm wants to produce small bags at least as the number of medium bags. What will

be the new optimal solution and new objective function value?

i. If the profit from deluxe bag is increased to $15 and the hours available in the inspection

and packing is increased to 200 hours, find the new optimal solution and the new

objective function value.

j. If the available hours for finishing is decreased to 500 hours, find the new optimal

solution and the new objective function value.

Answer 1

(a) The objective function is

and the final optimal solution is given by the entries in the Final Value column in the Adjustable Cells table in the reports, which gives us 0.000 small bags, 79.508 medium bags, 640.789 standard bags, and 29.259 deluxe bags

and the corresponding objective function value is 7267.531

(b) Now the number of hours used for each process is given by the Final value column of the Constraints table. So, 526.516 hours are used for Cutting and dyeing, 600 hours are used for Sewing, 708 hours are used for Finishing and 135 hours are used for Inspection and Packaging.

(c) The Allowable Increase in the Adjustable Cells table tells us the maximum amounts by which we can increase each coefficient in the objective function without changing the optimal solution. The allowable increase on small bags is 0.470, hence, the maximum profit from each small bag for which our optimal solution will not change is 6.5+0.470 = 6.97

However, the profit of the small bags changes to 7.25, which is higher than 6.97, hence we have to resolve the problem again, and we cannot give an answer using just the sensitivity reports.

(d) The allowable increase on a standard bag is 2.479, hence the maximum amount till which the profit for a standard bag can become without changing the optimal solution is 10+2.479=12.479

Now, the new profit given is 12, which is less than our maximum allowable profit for the same optimal solution. Hence, even when the profit per standard bag is 12, the optimal solution is 0.000 small bags, 79.508 medium bags, 640.789 standard bags, and 29.259 deluxe bags, and the objective function in this case is 8549.109