Linear Programming Problems and Solutions

BUsiness decision analysis
maths
<student name>
<UNIVERSITY NAME>
APRIL 25, 2019

Individual Problem 3:
Question 10:
For the linear program SELF test
Max 2A + 3B
s.t.
1A+2B ≤6
5A + 3B ≤ 15
A, B ≤ 0
Find the optimal solution using the graphical solution procedure. What is the value of the objective
function at the optimal solution?
Sol.
On plotting the constraints, we get the following graphical output:
Thus, the requires area has only one extreme point, i.e., (0,0)
We can check the objective function only on this point
Putting P(0,0) in 2A + 3B
Max(z) = 0
Checking objective function for any other value in the range, say, (-1,-1)
z= 2(-1) +3(-1)
z = -5
Hence for all other values other than (0,0), z will be negative,
1A+2B ≤6
5A + 3B ≤ 15
A<=0
B<=0

So max value of z will be at P(0,0)
Question 13:
Consider the following linear program:
Max 1A + 2B
s.t.
1A ≤5
1B ≤5
2A + 2B =12
A, B ≤ 0
a. Show the feasible region
Sol.
b. what are the extreme points of the feasible region?
Sol.
Extreme points of the feasible region:
P(5,1) and Q(1,5)
c. Find the optimal solution using the graphical procedure.
Sol. Checking the objective function at P(5,1)
z= 5 + 2(1)
B<=5 A<=5
2A + 2B = 12
P(5,1)
Q(1,5)

z= 5+2=7
Checking objective function at Q(1,5)
z=1+2(5)
z=11
Optimal solution = Q(1,5)
Question 14:
Par, Inc., is a small manufacturer of golf equipment and supplies. Par’s distributor be- lieves a
market exists for both a medium-priced golf bag, referred to as a standard model, and a high-priced
golf bag, referred to as a deluxe model. The distributor is so confident of the market that, if Par can
make the bags at a competitive price, the distributor will purchase all the bags that Par can
manufacture over the next three months. A careful analysis of the manufacturing requirements
resulted in the following table, which shows the production time requirements for the four required
manufacturing operations and the accounting department’s estimate of the profit contribution per
bag:
The director of manufacturing estimates that 630 hours of cutting and dyeing time, 600 hours of
sewing time, 708 hours of finishing time, and 135 hours of inspection and packaging time will be
available for the production of golf bags during the next three months.
a. If the company wants to maximize total profit contribution, how many bags of each model should
it manufacture?
Sol. Let the company manufacture x no Standard of bags and y no of Deluxe bags
Max(z)=10x + 9y
s.t:
7
10 x +1 y ≤ 630
1
2 x+ 5
6 y ≤ 600