# ABI201 PROBLEMS IN LINEAR PROGRAMMING by nuhman10

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```									PART – 1 SOLVE THE FOLLOWING MATHEMATICAL MODEL
Solve this problem using graphical linear programming and answer the questions that follow.
Use simultaneous equations to determine the optimal values of the decision variables.
Maximize      Z = 6X1 + 10X2

Subject to
Material 3X1 + 6X2 ≤ 48 kg
Labour 4X1 + 8X2 ≤ 80 hr
X1, X2 ≥ 0

PART – 2 WRITE THE LINEAR PROGRAMMING MODEL FOR EACH PROBLEM.

1. An RTW manufacturer produces two types of men’s T-shirt; with collar and without
collar labeled style A and style B respectively. Each unit of style A needs one man-hour;
while each unit of style B requires 2 man-hours. No more than 16 units of each style A
and style B can be sold each day. Manpower capacity is limited to 24 hours a day. The
contribution to profit of each unit of style A and style B is BD4 and BD3 respectively.
Develop linear programming model for this problem?

2. A research team which is composed of a senior researcher, an assistant researcher and a
secretary who receive BD280, BD230 and BD180 per month respectively must finish a
particular project within a period of 100 days. However, it is stipulated that the senior
researcher and the assistant researcher render at least 12 days and 25 days respectively,
and the secretary be on the job no more than 70 days. Set up an LP MODEL that will
optimize the cost of the project.

PART – 3 SOLVING THE LINEAR PROGRAMMING APPLICATIONS (EASY)

PROBLEM#1

Cycle Trends is introducing two new lightweight bicycle frames, the Deluxe and the
Professional, to be made from aluminum and steel alloys. The anticipated unit profits are \$10
for the Deluxe and \$15 for the Professional. The number of pounds of each alloy needed per
frame is summarized on the next slide. A supplier delivers 100 pounds of the aluminum alloy
and 80 pounds of the steel alloy weekly. How many Deluxe and Professional frames should
Cycle Trends produce each week? Pounds of each alloy needed per frame are
Aluminum Alloy           Steel Alloy
Deluxe                    2                       3
Professional               4                      2

PROBLEM#2

An appliance manufacturer produces two models of microwave ovens: H and W. Both models
Require fabrication and assembly work; each H uses four hours of fabrication and two hours
of assembly, and each W uses two hours of fabrication and six hours of assembly. There are
600 fabrication hours available this week and 480 hours of assembly. Each H contributes \$40
to profits, and each W contributes \$30 to profits. What quantities of H and W will maximize
profits?

PROBLEM#3

GM operates a plant that assembles and finishes cars and trucks. It takes 5 man-days to
assemble a truck and 2 man-days to assemble a car. It takes 3 man-days to finish each type of
vehicle. Because of man-power limitations, assembly can take no more than 180 man-days
per week and finishing no more than 135 man-days per week. If the profit on each truck is
\$300 and \$200 on each car, how many of each should they produce to maximize profit?

PROBLEM#4

XYZ Furniture Co. produce two type of table, Classic and Professional, The scarcity material
is high quality Red Wood which is limited to 1500 ft2. Classic chair required 3 ft2 and
Professional required 5 ft2. XYZ Co. obligate to Supply a new establishment by 180
professional Chairs this week. If the Classic Chair and professional Chair contribute a profit
of BD 25 and BD 40 Respectively. What is the best mix that can produce to gain maximum
profit? Solve this problem using Graphical Method?

PROBLEM#5

The Outdoor Furniture Corporation manufacturer's two products, Professional and picnic
tables, for use in offices and parks. The firm has two main resources; its carpenters (labor
force) and a supply of red-wood for use in the furniture. During the next production cycle,
1200 hours of labor are available under union agreement. The firm also has a stock of 3500
feet of good quality red-wood. Each professional that outdoor furniture produces requires 5
labor hours and 10 feet of red-wood; each picnic table take 5 labor hours and 35 feet of red-
wood. Completed Professional will yield a profit of \$9 each, and tables will result in a profit
of \$ 20.
(A) What is the equation of the objective function?
(B) List all the inequality constrains for this problem?
(C) Solve the above Linear Programming Model using graphical method and identify how
many Professional and tables should Outdoor Furniture produce to obtain the largest
possible profit?

PROBLEM#6

A craftsman named Sunil Babar builds two kinds of birdhouses, one for wrens and a second
for bluebirds. Each wren birdhouse takes four hours of labor and four units of lumber. Each
bluebird house requires two hours of labor and twelve units of lumber. The craftsman has
available 60 hours of labor and 120 units of lumber. Wren houses yield a profit of \$6 each and
bluebird houses yield a profit of \$15 each. Write out the objective and constraints and Solve
graphically to find the maximum profit.

PROBLEM#6

A furniture manufacturing company manufactures dining room tables and chairs. A table
requires 1 labor-hour for assembling and 1 labor-hour for finishing. A chair requires 2 labor-
hours for assembling and 1 labor-hour for finishing. The maximum labor-hours available per
day for assembly and finishing are 8 and 10, respectively. The profit made on a table is BD 60
and the profit on a chair is BD 20. How many tables and chairs should be produced to
maximize their profit?

PROBLEM#7

The production planner for a private label soft drink maker is planning the production of two
soft drinks: Root light (R) and Sassafras Soda (S). Two resources are constrained: production
time (T), of which she has at most 720 minute per day; and carbonated water (W), of which
she can get at most 1500 gallons per day. A case of Root Light requires 2 minutes of time and
5 gallons of water to produce, while a case of sassafras soda requires 3 minutes of time and 5
gallons of water. Further more Private label soft drink maker has an obligation to supply
grocer store 80 case per day. Profits for the Root Light are \$6.00 per case, and profits for the
Sassafras soda are \$4.00 per case. Find the best mix that lead to maximum profit.

PROBLEM#8
A small candy shop is preparing for the holiday season. The owner must decide how many
bags of deluxe mix and how many bags of standard mix of Peanut/Raisin Delite to put up.
The deluxe mix has 2/3 kg raisins and 1/3 kg peanuts, and the standard mix has 1/2 kg raisins
and 1/2 kg peanuts per bag. The shop has 90 kg of raisins and 60 kg of peanuts to work with.
Peanuts cost \$.60 per kg and raisins cost \$1.50 per kg. The deluxe mix will sell for \$2.90 per
kg, and the standard mix will sell for \$2.55 per kg. The owner estimates that no more than
110 bags of one type can be sold. If the goal is to maximize profits, how many bags of each
type should be prepared? What is the expected profit?

PART – 3 CHALLENGED LINEAR PROGRAMMING APPLICATION (HARD)

CHALLENGED PROBLEM # 1

The National Credit Union has \$250,000 available to invest in a 12-month commitment. The
money can be placed in Treasury notes yielding an 8% return or in municipal bonds at an
average rate of return of 9%. Credit union regulations require diversification to the extent that
at least 50% of the investment be placed in Treasury notes. Because of defaults in such
municipalities as Cleveland and New York, it is decided that no more than 40% of the
investment be placed in bonds. How much should the National Credit Union invest in each
security so as to maximize its return on investment?

CHALLENGED PROBLEM # 2

A banker has funds available to invest. She can purchase a Type A bond yielding a 5% return
on the amount invested and she can purchase a Type B bond yielding a 10% return on the
amount invested. Her client insists that she invest a t least twice as much in A bonds as in B
founds. How much should be invested in each type of bond to maximize the client’s return if
not more that \$17000 is to be invested in B bonds and at least \$5000 must be invested in A
bonds?