In class Exercise/T3
In-Class Exercises
Linear Programming
1.
Consider the following LP problem:
Maximize profit = 30X1 + 10X2
Subject to: 3X1 + X2 ≤ 300
X1 + X2 ≤ 200
X1 ≤ 100
X2 ≥ 50
X1, X2 ≥ 0
a) Solve the problem graphically.
b) Is there more than one optimal solution? Explain
2.
The Chris Beehner Company manufactures two lines of designer yard
gates, called model A and model B. Every gate requires blending a
certain amount of steel and zinc; the company has available a total of
25,000 lb of steel and 6,000 lb of zinc. Each model A gate requires a
mixture of 125 lb of steel and 20 lb of zinc, and each yields a profit of $90.
Each model B gate requires 100 lb of steel and 30 lb of zinc and can be
sold for a profit of $70. Find by graphical LP the best production mix of yard
gates.:
3.
Each coffee table produced by Kevin Watson Designers nets the firm a
profit of $9. Each bookcase yields a $12 profit. Watson’s firm is small and
its resources limited. During any given production period (of 1 week), 10
gallons of varnish and 12 lengths of high-quality redwood are available.
Each coffee table requires approximately 1 gallon of varnish and 1 length
of redwood. Each bookcase takes 1 gallon of varnish and 2 lengths of
wood. Formulate Watson’s production-mix decision as a linear
programming problem, and solve. How many tables and bookcases
should be produced each week? What will the maximum profit be?
4.
Walter Wallace is trying to determine how many units each of two
commercial multiline telephones to produce each day. One of these is
the standard model; the other one is the deluxe model. The profit per unit
on the standard model is $40, on the deluxe model $60. Each unit requires
30 minutes assembly time. The standard model requires 10 minutes of
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In class Exercise/T3
inspection time, the deluxe model 15 minutes. The company must fill an
order for six standard phones. There are 450 minutes of assembly time and
180 minutes of inspection time available each day. How many units of
each product should be manufactured to maximize profits?
5.
Solve the following LP problem graphically:
Minimize cost = 24X + 15Y
Subject to: 7X + 11Y ≥ 77
16X + 4Y ≥ 80
X, Y ≥ 0.
6.
Solve the following LP problem graphically:
Minimize cost = 4X1 + 5X2
Subject to: X1 + 2X2 ≥ 80
3X1 + X2 ≥ 75
X1, X2 ≥ 0.
7.
This is the slack time of year at JES, Inc. The firm would actually like to shut
down the plant, but if it laid off its core employees, they would probably
go to work for a competitor. JES could keep its core (full-time, yearround) employees busy by making 10,000 round tables per month, or by
making 20,000 square tables per month (or some ratio thereof). JES does,
however, have a contract with a supplier to buy a minimum of 5,000
square tables per month. Handling and storage costs per round table
will be $10; these costs would be $8 per square table.
Draw a graph, algebraically describe the constraint inequalities and the
objective function, identify the points bounding the feasible solution
area, and find the cost at each point and the optimum solution. Let X1
equal the thousands of round tables per month and X2 equal the
thousands of square tables per month.
8.
A tailor sews dresses and suits. For a suit the tailor uses 3 m2 of wool and
1 m2 of cotton. For a dress the tailor uses 2 m2 of wool and 2 m2 of
cotton. The tailor has 80 m2 of cotton and 120 m2 of wool in storage.
How many dresses and suits will the tailor be able to sew to maximize his
profit from the cloth if he is able to sell each dress and suit for €300?
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