Problem Set – 2
(Some More Formulations and Sensitivity Analysis)
1. Consider the following linear programming problem.
Subject to
Answer each of the following questions using the graphical method, do NOT use Excel solver
to answer these questions. You may use Excel solver to verify your answers.
a. Plot the feasible region for the above problem. Identify all the feasible corner points
for this problem.
b. Determine the optimal solution to this linear programming problem graphically.
c. Determine the shadow prices associated with each of the constraints at the optimal
solution. Also, determine the range of RHS values for which these shadow prices
remain valid.
d. Determine the set of binding constraints at optimality when the RHS of the third
constraint is reduced by 5 units, i.e. it becomes 25. Does the optimal solution
correspond to a degenerate solution? If so, what are its implications?
e. Suppose the objective function coefficient of x 2 is changed to 8, what are the
implications of this change on optimal solution? What happens to the optimal
objective function value?
2. An auto company manufactures cars and trucks. Each vehicle must be processed in the
paint shop and body assembly shop. If the paint shop were only painting trucks, then 40 per
day could be painted. If the paint shop were only painting cars, then 60 per day could be
painted. If the body assembly shop were only producing cars, then it could process 50 per
day. If the body assembly shop were only producing trucks, then it could process 50 per
day. Each truck contributes $300 to profit and each car contributes $200 to profit.
Determine a daily production plan that would maximize the company’s profit, assuming that
both the shops could process any feasible combination of cars and trucks.
3. A post office requires different number of full-time employees on different days of the
week. The number of full-time employees required on each day is given in the table below.
Union rules state that each full-time employee must work five consecutive days and then
receive two days off. For example, an employee who works Monday to Friday must get an
off on Saturday and Sunday. The post office wants to meet its daily requirements using only
full-time employees. Formulate an LP that the post office can use to minimize the number
of full-time employees who must be hired.
Day
Monday Tuesday
Number
of 17
13
Full-time
Employees
required
Wednesday Thursday Friday
15
19
14
Saturday Sunday
16
11
a. Note that the solution obtained by solving the linear programming problem
formulated above is not a fair schedule i.e. while some employees get their days off
on the weekend, there are others who work on weekends. How can the solution
obtained above be implemented in a fair way?
4. Rylon Corporation manufactures Brute and Chanelle perfumes. The raw material needed to
manufacture each type of perfume can be purchased for $3 per pound. Processing 1 lb of
raw material requires 1 hour of laboratory time. Each pound of processed raw material
yields 3 oz of Regular Brute Perfume and 4 oz of Regular Chanelle Perfume. Regular Brute
can be sold for $7/oz and Regular Chanelle for $6/oz. Rylon also has the option of further
processing Regular Brute and Regular Chanelle to produce Luxury Brute, sold at $18/oz and
Luxury Chanelle, sold at $14/oz. Each ounce of Regular Brute processed further requires an
additional 3 hours of laboratory time and $4 processing cost and yields 1 oz of Luxury Brute.
Each ounce of Regular Chanelle processed further requires an additional 2 hours of
laboratory time and $4 processing cost and yields 1 oz of Luxury Chanelle. Each year, Rylon
has 6000 hours of laboratory time available and can purchase up to 4000 lb of raw material.
Formulate an LP that can be used to determine how Rylon can maximize profits. Assume
that the cost of laboratory hours is a fixed cost.
5. Salico Corporation must determine how many sailboats should be produced during each of
the next four quarters. The demand during each of the next four quarters is as follows:
Quarter
Demand
1
40
2
60
3
75
4
25
At the beginning of the first quarter, Salico has an inventory of 10 sailboats. At the
beginning of each quarter, salico must decide how many sailboats should be produced
during that quarter. For simplicity, we assume that sailboats manufactured during a quarter
can be used to meet the demand for that quarter as well. During each quarter, Salico can
produce up to 40 sailboats with regular time labor at a total cost of $400 per sailboat. By
having employees work overtime during a quarter, Salico can produce additional sailboats
with overtime labor at a total cost of $450 per sailboat.
At the end of each quarter (after production has occurred and the current quarter’s
demand has been satisfied), a carrying or holding cost of $20 per sailboat is incurred for the
left over sailboats. Use linear programming to determine a production plan to minimize the
sum of production and inventory costs during the next four quarters.
6. The Leon Burnit Advertising Agency is trying to determine a TV advertising schedule for
Priceler Auto Company. Priceler has three goals.
Goal 1: Its ads should be seen by at least 40 million high-income men (HIM).
Goal 2: Its ads should be seen by at least 60 million low-income people (LIP).
Goal 3: Its ads should be seen by at least 35 million high-income women (HIW).
Leon Burnit can purchase two types of ads: those shown during football games and those
shown during soap operas. At most, $600,000 can be spent on ads. The advertising costs
and potential audiences of a one-minute ad of each type are shown in the table below. Leon
Burnit must determine how many football ads and soap opera ads to purchase for Priceler.
Using Linear Programming determine an advertising plan for Leon Brunit.
Ad
Football
Soap Opera
HIM
7
3
LIP
10
5
Millions of Viewers
HIW
5
4
Cost ($)
100,000
60,000
a. If each million exposures by which Priceler falls short of the HIM goal costs Priceler a
$200,000 penalty because of lost sales, each million exposures by which Priceler falls
short of the LIP goal costs Priceler a $100,000 penalty because of lost sales, and each
million exposures by which Priceler falls short of the HIW goal costs Priceler a
$50,000 penalty because of lost sales, determine an advertising plan for Leon Brunit
that minimizes the total penalty.