ch1 qm-ii integer programming zafer

Integer programming

1

Integer Programming

Many real-life situations must be formulated as an integer programming problem (IP) in which some or all of the variables are required to be non-negative integers, such as the number of shirts or pants to be produced or the number of cars to be purchased or the number of workers to be hired etc. There are three types of integer programming problems;

1. An IP in which all variables are required to be integers is called a pure integer programming problem.

2. An IP in which only some of the variables are required to be integers is called a mixed integer programming problem.

3. An integer programming problem in which all the variables must equal 0 or 1 is called a 01 IP. Integer programming is a Linear programming problem with some additional constraints which states that some variables are integer or binomial (0 or 1), so the feasible region of IP is in feasible region of LP and objective function of LP is generally greater than objective function of IP.

Z linear programming Z integer programming

Rounding the solution set of an LP problem to make the variables integer will lead to answers which may not be in the feasible region. Such as; Max Z= 4X1+X2

s.t. 2X1+X2 5 2X1+3X2 = 5

X1,X2 0; X1, X2 integer

The optimal solution to the LP for this IP is Z = 10, X1=5

2 , X2=0. Rounding off this solution, we obtain either the

candidate X1=2, X2=0 or the candidate X1=3, X2=0. Neither candidate is a feasible solution to the IP, therefore to obtain the feasible solution, other methods (enumeration method or cutting plane algorithm or branch and bound algorithm) will be used.

1. Pure integer programming All variables have integer values.

Enumeration method

Listing all feasible solutions and selecting the one with the best objective function is called enumeration method. This method is suitable only for small problems. WORKED-OUT PROBLEM 1 (Harrison Electric Company) The Company produces two products popular with home renovators, old- fashioned chandeliers and ceiling fans. Both the chandeliers and fans require a two-step production process involving wiring and assembly. It takes about 2 hours to wire each chandelier and 3 hours to wire a ceiling fan. Final assembly of the chandeliers and fans requires 6 and 5 hours respectively. The production capability is such that only 12 hours of wiring time and 30 hours of assembly time are available. Each chandelier produced nets the firm $7 and each fan $6. Harrisons production mix decision can be formulated using LP as follows.

Maximize profit = $7X1 + $6X2 subject to 2X1 + 3X2 12 (wiring hours)

6X1 + 5X2 30 (assembly hours) X1 , X2 0 (nonnegative)

where X1 = number of chandeliers produced , X2 = number of ceiling fans produced

1

Integer programming

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The rounding solution of X1 = 4, X2 = 1 gives a profit of $34 but the optimal solution is X1 = 5, X2 = 0 which gives a

profit of $35.

Branch-and-Bound Method The basic concept underlying the branch-and-bound technique is to divide and conquer.Theprocess contains dividing (branching) original large problem into smaller subproblems and bounding the best solution in the subsets. The steps are;

Solve the problem without integer restrictions,

If the solution is integer , then this must be the solution to integer problem,

If these variables are not integer valued, the feasible region is divided by adding constraints restricting

the value of one of the variables that was not integer valued,

Bounds on the value of the objective function are found and used to help determine which sub-problems

can be eliminated and when the optimal solution has been found,

If a solution is not optimal, a new sub-problem is selected and branching continues.

CHANDELIERS (X

1)

0

1

2

3

4

5

0

1

2

3

4

0

1

2

3

0

1

0

CEILING FANS (X

2)

0

0

0

0

0

0

1

1

1

1

1

2

2

2

2

3

3

4

PROFIT

($7X1 + $6X

2)

$0

7

14

21

28

35

6

13

20

27

34

12

19

26

33

18

25

24

Optimal solution to integer

programming problem

Solution if rounding is used

Integer programming

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Recall that the Harrison Electric Companys integer programming formulation is

Maximize profit = $7X1 + $6X2 subject to 2X1 + 3X2 12 6X1 + 5X2 30 where X1 = number of chandeliers produced X2 = number of ceiling fans produced and the optimal non-integer solution is X1 = 3.75 chandeliers, X2 = 1.5 ceiling fans, profit = $35.25

Since X1 and X2 are not integers, this solution is not valid. The profit value of $35.25 will provide the initial upper

bound. We can round down to X1 = 3, X2 = 1, profit = $27, which provides a feasible lower bound.

The problem is now divided into two sub-problems A and B.

Sub-problem A has branched into two new sub-problems, C and D.

Sub-problem C has no feasible solution because the all the constraints cannot be satisfied. We terminate this branch and do not consider this solution. Sub-problem Ds optimal solution is X1 = 4.17, X2 = 1, profit = $35.16 . This non-integer solution yields a new upper bound of $35.16 .

Sub-problem A

Maximize profit = $7X1 + $6X2 subject to 2X1 + 3X2 12 6X1 + 5X2 30 X1 4

Sub-problem C

Maximize profit = $7X1+$6X2 subject to 2X1 + 3X2 12 6X1 + 5X2 30 X1 4 X2 2

Sub-problem B

Maximize profit = $7X1 + $6X2 subject to 2X1 + 3X2 12 6X1 + 5X2 30 X1 3

Sub-problem Bs optimal solution: X1 = 3, X2 = 2, profit = $33.00

Sub-problem D

Maximize profit = $7X1+ $6X2 subject to 2X1 + 3X2 12 6X1 + 5X2 30 X1 4 X2 1

Integer programming

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Finally we derive sub-problems E and F.

No further branching is possible; the optimal solution is sub-problem Fs node.

Using Excels Solver to formulate Harrisons integer programming model

Sub-problem E

Maximize profit = $7X1+$6X2 subject to 2X1 + 3X2 12 6X1 + 5X2 30 X1 4 X1 4 X2 1 Optimal solution to E: X1 = 4, X2 = 1, profit = $34

Sub-problem F

Maximize profit = $7X1+$6X2 subject to 2X1 + 3X2 12 6X1 + 5X2 30 X1 4 X1 5 X2 1 Optimal solution to F: X1 = 5, X2 = 0, profit = $35

Integer programming

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Integer variables are specified with a drop-down menu in Solver.

WORKED-OUT PROBLEM 2 The Telfa Corporation manufactures tables and chairs. A table requires 1 hour of labor and 9 square board feet of wood, and a chair requires 1 hour of labor and 5 square board feet of wood. Currently, 6 hours of labor and 45 square board feet of wood are available. Each table contributes $8 to profit, and each chair contributes $5 to profit. Formulate and solve an IP to maximize Telfas profit.

Solution: Let X1= number of tables manufactured X2 = number of chairs manufactured

Because X1 and X2 must be integers, Telfa wants to solve the following IP: max Z = 8X1 + 5X2

s.t. X1 + X2 6 (Labor constraint)

s.t. 9X1 + 5X2 45 (Wood constraint)

X1, X2 0; X1, X2 integer

The branch-and-bound method begins by solving the LP.Unfortunately, the optimal solution to the LP is Z=

,

X1=

, X2 =

. we know that (optimal Z-value for IP) (optimal Z-value for LP). This implies that the optimal Z-value

for the IP cannot exceed

(upper bound). X1 and X2 are rounded down and found as X1=3 and X2=2 and Z= 34 is

lower bound, so optimal Z-value for the IP cannot be less than 34.

Among the candidate solutions, Z=40 is maximum value among them. So 5 tables (X1=5), and no chairs (X2 =0) will yield a profit of $40.

X1=3.75 X2 =2.25 Z=41.25

X1=3 X2 =3 Z=39

X1=4 X2 =1.8 Z= 41

X13

X14

X22

X21

X1=4.44 X2 =1

Z=40.5

X1=3.88 X2 =2

Z=41,1

X14 and it is found as X1=3.88 ,

so this set of solution is

unfeasible.

X

X14

X15

X1=4 X2 =1 Z=37

X1=5 X2 =0 Z=40

Candidate solution

Candidate solution

Candidate solution

Integer programming

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PROBLEMS (Pure integer programming) Use branch-and-bound to solve the following IPs.

1. max Z = 5x1 + 2x2

s.t. 3x1 + x2 12

s.t . x1 + x2 5

s.t. 3x1 x1, x2 0; x1, x2 integer

2. max Z = 4x1 + 3x2

s.t. 4x1 + 9x2 26

8x1 + 5x2 17


3. max Z = 4x1 + 5x2

s.t. 3x1 + 2x2 10

x1 + 4x2 11

3x1 + 3x2 13


4. max Z = 2x1 + 3x2

s.t. x1 + 2x2 10

s.t .x1 + 4x2 25


5. max Z = 4x1 + 5x2

s.t. x1 + 4x2 5

3x1 + 2x2 7


6. max Z = 7x1 + 3x2

s.t. 2x1 + x2 9

3x1 + 2x2 13


2. Mixed-Integer Programming There are many situations in which some of the variables are restricted to be integers and some are not. To solve a mixed IP by the branch-and-bound method, branching is applied only on variables that are required to be integers. Consider the following case;

max Z = 2x1 + x2

s.t. 5x1+2x2 8

x1+x2 3

x1, x2 0; x1 integer

As before, we begin by solving the LP. The optimal solution of the LP relaxation is Z=

, x1=

, x2=

Because x2 is allowed to be fractional, we do not branch on x2.

Z=

, x1= 1 , x2=

is the optimum solution to the mixed IP.

Candidate solution

Z= 11

3

x1= 2

3

x2= 7

3

Z= 3 x1= 0 x2= 3

Z=7

2

x1= 1

x2= 3

2

X10

X11

Candidate solution

Answers: 1. x1=4,x2=0, Z=20 2. x1=2,x2=0, Z=8 3. x1=2,x2=2, Z=18 4. x1=10, x2=0, Z=20 5. x1=1, x2=1, Z= 9 6. x1=4, x2=0, Z=28

Integer programming

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WORKED-OUT PROBLEM 3 Maximize profit Z= $85X + $1.50Y subject to 30X + 0.5Y 2,000 18X + 0.4Y 800 2X + 0.1Y 200

X, Y 0 and X integer Solution with Excel:

PROBLEMS (Mixed-Integer Programming)

Use Excel to find the solution of the given LPs and use branch and-bound to find the solutions for IPs:

1. max Z = 3x1 + x2

s.t. 5x1 + 2x2 10

4x1 + x2 7


2. min Z = 3x1 + x2

s.t. x1 + 5x2 8

x1 + 2x2 4


3. min Z = 4x1 +3x2 + x3

s.t. 3x1 + 2x2 + x3 7

2x1 + x2 + 2x3 11

x1, x2 ,x3 0; x2 ,x3 integer

4. (12.1-6./Hillier)Vincent Cardoza is the owner and manager of a machine shop that does custom order work. This Wednesday afternoon, he has received calls from two customers who would like to place rush orders. One is a trailer hitch company which would like some custom-made heavy-duty tow bars. The other is a mini-car-carrier company which needs some customized stabilizer bars. Both customers would like as many as possible by the end of the week (two working days). Since both products would require the use of the same two machines, Vincent needs to decide and inform the customers this afternoon about how many of each product he will agree to make over the next two days. Each tow bar requires 3.2 hours on machine 1 and 2 hours on machine 2. Each stabilizer bar requires 2.4 hours on machine 1 and 3 hours on machine 2. Machine 1 will be available for 16 hours

Integer programming

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over the next two days and machine 2 will be available for 15 hours. The profit for each tow bar produced would be $130 and the profit for each stabilizer bar produced would be $150. Vincent now wants to determine the mix of these production quantities that will maximize the total profit. (a) Formulate an IP model for this problem. (b) Use a graphical approach to solve this model. (c) Use the computer to solve the model. 5. (12.1-7. /Hillier) Pawtucket University is planning to buy new copier machines for its library. Three members of its Operations Research Department are analyzing what to buy. They are considering two different models: Model A, a high-speed copier, and Model B, a lower-speed but less expensive copier. Model A can handle 20,000 copies a day, and costs $6,000. Model B can handle 10,000 copies a day, but costs only $4,000. They would like to have at least six copiers so that they can spread them throughout the library. They also would like to have at least one high-speed copier. Finally, the copiers need to be able to handle a capacity of at least 75,000 copies per day. The objective is to determine the mix of these two copiers which will handle all these requirements at minimum cost. (a) Formulate an IP model for this problem. (b) Use a graphical approach to solve this model. (c) Use the computer to solve the model.

Answers: 1. x1=1.2, x2=2, Z=5.6 2. x1=0, x2=2, C=2 3. x1=0.3, x2=3, x3=0, C=10.33 4. x1=0, x2=5, P=750 5. x1=2, x2=4, C=28,000

Integer programming

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3. Modeling With 0-1 (Binary) Variables

A 0-1 variable is assigned a value of 0 if a certain condition is not met and a 1 if the condition is met. This is also called a binary variable.

WORKED-OUT PROBLEM 4 A common capital budgeting problem is selecting from a set of possible projects when budget limitations make it impossible to select them all.A 0-1 variable is defined for each project. Quemo Chemical Company is considering three possible improvement projects for its plant.

A new catalytic converter.

A new software program for controlling operations.

Expanding the storage warehouse. It cannot do them all.They want to maximize net present value of projects undertaken.

The basic model is Maximize net present value of projects undertaken subject to

Total funds used in year 1 $20,000 Total funds used in year 2 $16,000

The mathematical statement of the integer programming problem becomes Maximize NPV = 25,000X1 + 18,000X2 + 32,000X3 subject to 8,000X1 + 6,000X2 + 12,000X3 20,000 7,000X1 + 4,000X2 + 8,000X3 16,000 X1, X2, X3 = 0 or 1 Solution: Excel solution of problem is X1 = 1, X2 = 0, and X3 = 1 with an objective function value of 57,000. One common use of 0-1 variables involves limiting the number of projects or items that are selected from a group;

Suppose Quemo Chemical is required to select no more than two of the three projects regardless of the funds available. This would require adding a constraint

X1 + X2 + X3 2 (Limited selection)

If they had to fund exactly two projects the constraint would be X1 + X2 + X3 = 2 (Limited selection)

PROJECT NET

PRESENT VALUE

YEAR 1 YEAR 2

Catalytic Converter $25,000 $8,000 $7,000

Software $18,000 $6,000 $4,000

Warehouse expansion $32,000 $12,000 $8,000

Available funds $20,000 $16,000

Integer programming

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At times the selection of one project depends on the selection of another project;

Suppose Quemos catalytic converter could only be purchased if the software was purchased The following constraint would force this to occur

X1 X2 or X1 X2 0 (Dependent Selections) (X1 will be selected if X2 is selected. If X2 is zero than X1 is zero, if X2 is 1 than X1 can be zero or 1)

If we wished for the catalytic converter and software projects to either both be selected or both not be selected, the constraint would be

X1 = X2 or X1 X2 = 0 (Dependent Selections)

WORKED-OUT PROBLEM 5 Sitka Manufacturing is planning to build at least one new plant and three cities are being considered in

Baytown, Texas

Lake Charles, Louisiana

Mobile, Alabama

Once the plant or plants are built, the company wants to have capacity to produce at least 38,000 units each year.

SITE ANNUAL

FIXED COST VARIABLE COST PER

UNIT ANNUAL CAPACITY

Baytown, TX $340,000 $32 21,000

Lake Charles, LA $270,000 $33 20,000

Mobile, AL $290,000 $30 19,000

Solution: We can define the decision variables as

The integer programming formulation becomes

Minimize cost = 340,000X1 + 270,000X2 + 290,000X3 + 32X4 + 33X5 + 30X6 subject to X4 + X5 + X6 38,000 X4 21,000X1 X5 20,000X2 X6 19,000X3 X1, X2, X3 = 0 or 1; X4, X5, X6 0 and integer

The optimal solution is X1 = 0, X2 = 1, X3 = 1, X4 = 0, X5 = 19,000, X6 = 19,000, Objective function value = $1,757,000

Integer programming

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WORKED-OUT PROBLEM 6 One client has the following specifications

At least two Texas firms must be in the portfolio,

No more than one investment can be made in a foreign oil company,

One of the two California oil stocks must be purchased. The client has $3 million to invest and wants to buy large blocks of shares

Solution:

Maximize return = 50X1 + 80X2 + 90X3 + 120X4 + 110X5 + 40X6 + 75X7 subject to X1 + X4 + X5 2 (Texas constraint) X2+ X3 1 (Foreign oil constraint) X6 + X7 =1 (California constraint) 480X1 + 540X2 + 680X3 + 1,000X4 + 700X5 + 510X6 + 900X7 3,000 ($3 million limit) All variables must be 0 or 1

STOCK COMPANY

NAME EXPECTED ANNUAL RETURN ($1,000s)

COST FOR BLOCK OF SHARES ($1,000s)

1 Trans-Texas Oil 50 480

2 British Petroleum 80 540

3 Dutch Shell 90 680

4 Houston Drilling 120 1,000

5 Texas Petroleum 110 700

6 San Diego Oil 40 510

7 California Petro 75 900

Integer programming

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PROBLEMS (Binary Integer Programming)

1. (12.1-3./Hillier) A real estate development firm, Peterson and Johnson, is considering five possible development projects. The following table shows the estimated long-run profit (net present value) that each project would generate, as well as the amount of investment required to undertake the project, in units of millions of dollars.

Development Project

1 2 3 4 5

Estimated profit 1 1.8 1.6 0.8 1.4

Capital required 6 12 10 4 8

The owners of the firm, Dave Peterson and Ron Johnson, have raised $20 million of investment capital for these projects. Dave and Ron now want to select the combination of projects that will maximize their total estimated long-run profit (net present value) without investing more than $20 million. (a) Formulate a BIP model for this problem. (b) Display this model on an Excel spreadsheet. (c) Use the computer to solve this model.

2. (12.1-4./Hillier) The board of directors of General Wheels Co. is considering seven large capital investments. Each investment can be made only once. These investments differ in the estimated long-run profit (net present value) that they will generate as well as in the amount of capital required, as shown by the following table (in units of millions of dollars):

Investment Opportunity

1 2 3 4 5 6 7

Estimated profit 17 10 15 19 7 13 9

Capital required 43 28 34 48 17 32 23

The total amount of capital available for these investments is $100 million. Investment opportunities 1 and 2 are mutually exclusive, and so are 3 and 4. Furthermore, neither 3 nor 4 can be undertaken unless one of the first two opportunities is undertaken. There are no such restrictions on investment opportunities 5, 6, and 7. The objective is to select the combination of capital investments that will maximize the total estimated long-run profit (net present value). (a) Formulate a BIP model for this problem. (b) Use the computer to solve this model.

3. (12.1-1./Hillier) The CALIFORNIA MANUFACTURING COMPANY is considering expansion by building a new factory in either Los Angeles or San Francisco, or perhaps even in both cities. It also is considering building at most one new warehouse, but the choice of location is restricted to a city where a new factory is being built. The net present value (total profitability considering the time value of money) of each of these alternatives is shown in the fourth column of the table. The rightmost column gives the capital required (already included in the net present value) for the respective investments, where the total capital available is $10 million.

Data for the California Manufacturing Co. example

Decision Number

Yes-or-No Question

Decision Variable

Net Present Value

Capital Required

1 Build factory in Los

Angeles?

x1 $9 million

$6 million

2 Build factory in San

Francisco?

x2 $5 million

$3 million

3 Build warehouse

in Los Angeles?

x3 $6 million

$5 million

4 Build warehouse

in San Francisco?

x4 $4 million

$2 million

Capital available: $10 million

(a) Formulate a BIP model for this problem. (b) Display this model on an Excel spreadsheet. (c) Use the computer to solve this model.

solution

Answers: 1. x1=x3=x4=1, x2=x5=0, Z=3.4 2. x1=x3=x7=1, x2=x4=x5=x6=0 , Z=41 3. x1=x3=0, x2=x4=1, NPV=9

Integer programming

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4. (11-12/Render) Seok Tang is the owner and general manager of Princess Brides, which provides a wedding planning service in Southwest Louisiana. She uses radio advertising to market her business. Two types of ads are available- those during prime hours and those at other times. Each prime time ad costs $390 and reaches 8,200 people, while the off peak ads each cost $240 and reach 5,100 people. Bailey has budgeted $1,800 per week for advertising. Based on comments from her customers, Tang wants to have at least 2 prime time ads and no more than 6 off peak ads. a) Formulate this as a linear program and solve using

a computer.

b) Find a good or optimal integer solution for part a)

by rounding off or making an educated guess at the

answer.

c) Solve this as an integer programming problem using

the branch and bound method.

5. (11-13/Render) A group of college students is planning a camping trip during the upcoming break. The group must hike several miles through the woods to get to the campsite, and anything that is needed on this trip must be packed in a knapsack and carried to the campsite. One particular student, Hoi Nei, has identified eight items that she would like to take on the trip, but the combined weight is too great to take all of them. She has decided to rate the utility of each item on a scale of 1 to 100, with 100 being the most beneficial. The item weights in pounds and their utility values are given below.

ITEM 1 2 3 4 5 6 7 8

WEIGHT 8 1 7 6 3 12 5 14

UTILITY 80 20 50 55 50 75 30 70

Recognizing that the height to the campsite is a long

one, a limit of 35 pounds has been sat as the maximum

total weight of the items to be carried.

a) Formulate this as 0-1 programming problem to

maximize the total utility of the items carried.

Solve this knapsack problem using a computer.

b) Suppose item number 3 is an extra battery pack,

which may be used with several of the other items.

Hoi has decided that she will only take item number

5, a CD player, if she also takes item number 3. On

the other hand, if she takes item number 3, she

may or may not take item number 5. Modify this

problem to reflect this and solve the new problem.

6. (11-14 /Render) Roneo Enterprises sells two sizes of wall posters, a large 3- by 4- foot poster and a smaller 2- by 3- foot poster. The profit earned from the sale of each large poster is $3; each smaller poster earns $2. The firm, although profitable, is not large; it consists of one art student, Iumiya Ro, at the Pepperdine University. Because of her classroom schedule, Iumiya has the following weekly constraints: 1) up to three large posters can be sold; 2) up to five smaller posters can be sold; 3) up to 10 hours can be spent on posters during the week, with each large poster requiring 2 hours of work and each small one taking 1 hour. With the semester almost over, Iumiya plans on taking a three-month summer vacation to England and doesnt want to leave any unfinished posters behind. Find the integer solution that will maximize her profit. 7. (11-15/Render) An airline owns an aging fleet of Boeing 737 jet airplanes. It is considering a major purchase of up to 17 new Boeing model 757 and 767 jets. The decision must take into account numerous cost and capability factors, including the following: 1) the airline can finance up to $1.6 billion in purchases; 2) each Boeing 757 will cost $80 million, and each Boeing 767 will cost $110 million; 3) at least one-third of the planes purchased should be the longer-range 757; 4) the annual maintenance budget is to be no more than $8 million; 5) the annual maintenance cost per 757 is estimated to be $800,000, and it is $500,000 for each 767 purchased; and 6) each 757 can carry 125,000 passengers per year, whereas each 767 can fly 81,000 passenger annually. Formulate this as an integer programming problem to maximize the annual passenger-carrying capability. What category of integer programming problem is this? Solve this problem.

8. (11-16/Render) Diya Investments is a venture capital firm that is currently evaluating six different investment opportunities. There is not sufficient capital to invest in all of these, but more than one will be selected. A 0-1 integer programming model is planned to help determine which of the six opportunities to choose. Variables x1, x2, x3, x4, x5, and x6 represent the six choices. For each of the following situations write a constraint (or several constraints), that would be used. a) At least 3 of these choices are to be selected.

b) Either investment 1 or investment 4 must be

undertaken, but not both.

c) If investment 4 is selected, then investment 6 must

also be selected. However, if investment 4 is not

selected, it is still possible to select number 6.

d) Investment 5 cannot be selected unless both

investments 2 and 3 are also selected.

Integer programming

14

e) Investment 5 must be selected if both investments

2 and 3 are also selected.

9. (11-18/Render) Malik Construction Company specializes in building moderately priced homes in Cincinnati, Ohio. Anu Malik has identified eight potential locations to construct new single-family dwelling, but he cannot put up homes on all of the sites because he has only $300,000 to invest in all projects. The accompanying table shows the cost of constructing homes in each area and the expected profit to be made from the sale of each home. Note that the home-building costs differ considerably due to low costs, site preparation and differences in the models to be build. Note also that a fraction of a home cannot be build.

LOCATION COST OF

BUILDING

AT THIS SITE

($)

EXPECTED

PROFIT

($)

Clifton 60,000 5,000

Mt. Auburn 50,000 6,000

Mt. Adams 82,000 10,000

Amberly 103,000 12,000

Norwood 50,000 8,000

Covington 41,000 3,000

Roselawn 80,000 9,000

Eden Park 69,000 10,000

a) Formulate Maliks problem using 0-1 integer

programming.

b) Solve with QM for Windows or Excel.

10. (11-19/Render) A real estate developer is considering three possible projects a small apartment complex, a small shopping center, and a mini-warehouse. Each of these requires different funding over the next two years, and the net present value of the investments also varies. The following table provides the required investment amounts ( in $1,000s) and the net present value (NPV) of each (also expressed $1,000s):

INVESTMENT

NPV YEAR 1 YEAR 2

Apartment 18 40 30

Shopping center 15 30 20

Mini warehouse 14 20 20

The company has $80,000 to invest in year 1 and

$50,000 to invest in year 2.

a) Develop an integer programming model to

maximize the NPV in this situation.

b) Solve the problem in part-a using computer

software. Which of the three projects would be

undertaken if NPV is maximized? How much money

would be used each year?

11. (11-21/Render) Pashwa Utilities provides electricity for three cities. The company has four electric generators that are used to provide electricity. The main generator operates 24 hours per day, with an occasional shutdown for routine maintenance. Three other generators (1, 2, and 3) are available to provide additional power when needed. A startup cost is incurred each time one of these generators is started. The startup costs are $6,000 for 1, $5,000 for 2, and $4,000 for 3. These generators are used in the following ways: A generator may be started at 6:00 A.M. and run for either 8 hours or 16 hours, or it may be started at 2:00 P.M. and run for 8 hours (until 10:00 P.M.). All generators except the main generator are shut down at 10:00 P.M. Forecasts indicate the need for 3,200 megawatts more than provided by the main generator before 2:00 P.M., and this need goes up to 5,700 megawatts between 2:00 and 10:00 P.M. Generator 1 may provide up to 2,400 megawatts, generator 2 may provide up to 2,100 megawatts, and generator 3 may provide up to 3,300 megawatts. The cost per megawatt used per eight hour period is $8 for 1, $9 for 2, and $7 for 3. a) Formulate this problem as an integer programming

problem to determine the least-cost way to meet

the needs of the areas.

b) Solve using computer software.

ch1 qm-ii integer programming zafer

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