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ISE 170

Homework 4

Due Thursday, April 7

Question 1) (20 points)

a) Write the dual of the following primal LP (use the Table that has the rules in your slides about the

relationship between signs, and variables of the primal and dual problem)

Min 2x1+x2+x35x1+2x2+7x3 = 420

3x1+2x2+5x3 >= 280

x1,x2 >= 0

 b) solve the primal problem with a computer software, write down optimal solution and optimal objective

function value. Also write down the shadow prices (dual prices) of the primal constraints.

c) solve the dual problem and write down optimal solution of the dual problem and optimal objective function

value. Also write down the shadow prices (dual prices) of the primal constraints. (if a variable is unrestricted

in sign, you can use @FREE(variable_name) command)

d) Compare the solutions of the primal and dual problems. What do you observe?

e) Check whether the complementary slackness equations are held.

Question 2) (20 points)Consider the following problem:

Minimize Z= – 5x1 – 15x2

s.t.

2x1 – 4x2 =0

!"  #$%&' )*+, -. /+)* 01!2*+3!% 4')*$56

7"  8$9,)1:3) )*' 5:!% 21$7%'46

3"  #$%&' )*' 5:!% 21$7%'4 /+)* 01!2*+3!% 4')*$56 ;*!) 5$

Question 3) (20 points)A student must complete at least two math courses, at least two OR courses, and at least two computer course

to graduate with a major in operations research. Some courses can be used to fulfill more than one

requirement: Calculus can fulfill the math requirement; operations research can fulfill math and OR

requirements; data structures can fulfill computer and math requirements; business statistics can fulfill math

and OR requirements; computer simulation can fulfill OR and computer requirements; introduction to

computer programming can fulfill computer requirement; and forecasting can fulfill OR and math

requirements.

Some courses are prerequisites for others: Calculus is a prerequisite for business statistics; introduction to

computer programming is a prerequisite for computer simulation and for data structures; and business statistic

is a prerequisite for forecasting. Formulate an IP that minimizes the number of courses needed to satisfy the

major requirements.

Question 4) (20 points)

A computer company produces two types of computers: Pear computers and Banana computers. To be able to

 produce these computers 3000 chips and 1200 hours of labor are available. Following data is available.

computer Labor Chips Equipment costs ($) Selling price ($)

pear 1 hr 2 5000 400

banana 2 hr 5 7000 900

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Question 5) (20 points) 

Gotham City has been divided into eight districts. The time (in minutes) it takes an ambulance to travel from

one district to another is shown below. Suppose Gotham City must select cities to locate ambulances, and each

city should be within 4 minutes of an ambulance location.

DistrictDistrict

1 2 3 4 5 6 7 8

1 0 3 4 6 8 9 8 102 3 0 5 4 8 6 12 9

3 4 5 0 2 2 3 5 7

4 6 4 2 0 3 2 5 4

5 8 8 2 3 0 2 2 4

6 9 6 3 2 2 0 3 2

7 8 12 5 5 2 3 0 2

8 10 9 7 4 4 2 2 0

a)Formulate an integer program to minimize the number of ambulance locations

 b) Solve it using either Excel solver or Lingo