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Chapter 6

Integer Linear Programming

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Integer Linear Programming All-Integer Linear Program

All variables must be integers Mixed-Integer Linear Program

Some, but not all variables must be integers 0-1 Integer Linear Program

Integer variables must be 0 or 1, also known as binary variables

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Integer Programming – All Integers

Northern Airlines is a small regional airline. Management is now considering expanding the company by buying additional aircraft. One of the main decisions is whether to buy large or small aircraft to use in the expansion. The table below gives data on the large and small aircraft that may be purchased.

As noted in the table, management does not want to buy more than 2 small aircraft, while the number of large aircraft to be purchased is not limited.

How many aircraft of each type should be purchased in order to maximize annual profit?

  Small Large Capital Available

Annual profit $1 million $5 million  

Purchase cost $5 million $50 million $100 million

Maximum purchase quantity 2 No maximum  

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Define Variables - Northern Airlines Let:

S = # of Small Aircraft

L = # of Large Aircraft

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General Form - Northern Airlines Max

1S + 5L

s.t.

5S + 50L <= 100

S <= 2

S, L >= 0 & Integer

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Northern Airlines – Graph Solution

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3

S

L Small AC

LP Relaxation(2, 1.8)

Budget

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Northern Airlines – Graph Solution

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3

S

L Small AC

RoundedSolution

(2, 1)

Budget

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Northern Airlines – Graph Solution

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3

S

L

Budget

Small AC

OptimalSolution

(0, 2)

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Integer Linear Programming All-Integer Linear Program

All variables must be integers Mixed-Integer Linear Program

Some, but not all variables must be integers 0-1 Integer Linear Program

Integer variables must be 0 or 1, also known as binary variables

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Integer Programming – Mixed Integer

Hart Manufacturing, a mixed integer production problem:

Hart Manufacturing makes three products. Each product goes through three manufacturing departments, A, B, and C. The required production data are given in the table below. (All data are for a monthly production schedule.)

Production Department Product 1 Product 2 Product 3 Hours available

A (hours/unit) 1.5 3 2 450

B (hours/unit) 2 1 2.5 350

C (hours/unit) 0.25 0.25 0.25 50

Profit Contributions per Unit $25 $28 $30  

Setup Costs per production run $400 $550 $600  

Max Production per production run (Units) 175 150 140  

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General Form – Hart Manu.

Let:

X1= units of product 1

X2= units of product 2

X3= units of product 3

Y1= 1 if production run, else = 0

Y2= 1 if production run, else = 0

Y3= 1 if production run, else = 0

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General Form – Hart Manu.

Max 25X1 + 28X2 + 30X3 – 400Y1 – 550Y2 – 600Y3

s.t. 1.5X1 + 3X2 + 2X3 <= 450 Dept. A 2X1 + X2 + 2.5X3 <= 350 Dept. B .25X + .25X + .25X <= 50 Dept. C X1 <= 175Y1

X2 <= 150Y2

X3 <= 140Y3

Xi >= 0 Yi = integer 0,1

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Integer Linear Programming All-Integer Linear Program

All variables must be integers Mixed-Integer Linear Program

Some, but not all variables must be integers 0-1 Integer Linear Program

Integer variables must be 0 or 1, also known as binary variables

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0-1 Integer Linear Program (Binary Integer Programming)

Assists in selection process 1 corresponding to undertaking 0 corresponding to not undertaking

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0-1 Integer Linear Program (Binary Integer Programming)

Allows for modeling flexibility through: Multiple choice constraints

k out of n alternatives constraint

Mutually exclusive constraints Conditional & co-requisite constraint

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Integer Programming - Binary Integer Programming

CAPEX Inc. is a high technology company that faces some important capital budgeting decisions over the next four years. The company must decide among four opportunities:

1. Funding of a major R&D project. 2. Acquisition of an existing company, R&D Inc. 3. Building a new plant, and 4. Launching a new product.

CAPEX does not have enough capital to fund all of these projects. The table below gives the net present value of each item together with the schedule of outlays for each over the next four years. All values are in millions of dollars.

 R&D Project

Acquisition of R&D Inc. New Plant

Launch New Product

Capital Available

Net Present Value (NPV) 100 50 30 50  

Year 1 10 30 5 10 40

Year 2 15 0 5 10 60

Year 3 15 0 5 10 80

Year 4 20 0 5 10 70

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General Form – CAPEX Inc.

Let:

X1= 1 if R&D Project funded, else = 0

X2= 1 if acquire company, else = 0

X3= 1 if build new plant, else = 0

X4= 1 if launch new project, else = 0

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General Form – CAPEX Inc Max

100X1 + 50X2 + 30X3 + 50X4

s.t.

10X1 + 30X2 + 5X3 + 10X4 <= 40 Yr 1

15X1 + 0X2 + 5X3 + 10X4 <= 60 Yr 2

15X1 + 0X2 + 5X3 + 10X4 <= 80 Yr 3

20X1 + 0X2 + 5X3 + 10X4 <= 70 Yr 4

Xi = 0,1

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Review Problems

Electrical Utility

Distribution Co.

Alpha Airlines

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Integer Programming - Review Electrical Utility, a mixed integer set-up problem:

A problem faced by an electrical utility each day is that of deciding which generators to start up in order to minimize total cost. The utility in question has three generators with the characteristics shown in the table below. There are two periods in a day, and the number of megawatts needed in the first period is 2900. The second period requires 3900 megawatts. A generator started in the first period may be used in the second period without incurring an additional startup cost. All major generators (e.g. A, B, and C) are turned off at the end of the day. (Assume all startups occur in time period 1.)

Generator Fixed Startup CostCost Per Period Per

Megawatt Used

Maximum Capacity In Each Period

(MW)

A $3,000 $5 2,100

B $2,000 $4 1,800

C $1,000 $7 3,000

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General Form – Electrical Utility

Let: XA1 = Power from Gen A in Period 1 XB1 = Power from Gen B in Period 1 XC1 = Power from Gen C in Period 1 XA2 = Power from Gen A in Period 2 XB2 = Power from Gen B in Period 2 XC2 = Power from Gen C in Period 2 YA = 1 if Generator A started; else = 0 YB = 1 if Generator A started; else = 0 YC = 1 if Generator A started; else = 0

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General Form – Electrical Utility

Min 5(XA1+XA2) + 4(XB1+XB2) + 7(XC1+XC2) + 3000YA + 2000YB + 1000YC

s.t. XA1 + XB1 + XC1 >= 2900 XA2 + XB2 + XC2 >= 3900 XA1 <= 2100YA

XA2 <= 2100YA

XB1 <= 1800YB

XB2 <= 1800YB

XC1 <= 3000YC

XC2 <= 3000YC

Xij >= 0 Yi = 0, 1

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Integer Programming - Review Distribution Company, a integer transportation problem:

A distribution company wants to minimize the cost of transporting goods from its warehouses A, B, and C to the retail outlets 1, 2, and 3. The costs (in $’s) for transporting one unit from warehouse to retailer are given in the following table.

The fixed cost of operating a warehouse is $500 for A, $750 for B, and $600 for C, and at least two of them have to be open. The warehouses can be assumed to have adequate storage capacity to store all units demanded, ie., assume each warehouse can store 525 units.

  Retailer    

Warehouse 1 2 3

A $15 $32 $21

B $9 $7 $6

C $11 $18 $5

Demand 200 150 175

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General Form – Distribution Co.

Let:

Xij = units shipped from i to j

YA = 1 if warehouse A opens, else = 0

YB = 1 if warehouse B opens, else = 0

YC = 1 if warehouse C opens, else = 0

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General Form – Distribution Co.

Min 500YA + 750YB + 600YC + 15XA1 + 32XA2 + 21XA3 + 9XB1 + 7XB2 + 6XB3 + 11XC1 +

18XC2 + 5XC3

s.t. XA1 + XB1 + XC1 = 200 XA2 + XB2 + XC2 = 150 XA3 + XB3 + XC3 = 175 XA1 + XB1 + XC1 <= 525YA

XA2 + XB2 + XC2 <= 525YB

XA3 + XB3 + XC3 <= 525YC

YA + YB + YC >= 2

Xij >= 0

Yi = 0, 1

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Integer Programming - Review Alpha Airlines, a integer scheduling problem:

Alpha Airlines wishes to schedule no more than one flight out of Chicago to each of the following cities: Columbus, Denver, Los Angeles, and New York. The available departure slots are 8 A.M., 10 A.M., and 12 noon. Alpha leases the airplanes at the cost of $5000 before and including 10 A.M. and $3000 after 10 A.M., and is able to lease at most two per departure slot. Also, if a flight leaves for New York in a time slot, there must be a flight leaving for Los Angeles in the same time slot. The expected profit contribution before rental costs per flight is shown below (in K$)

  Time Slot    

Cities8:00 AM 10:00 AM 12:00 Noon

Columbus 10 6 6

Denver 9 10 9

Los Angeles 14 11 10

New York 18 15 10

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General Form – Alpha Airlines

Let:

Xij= 1 if flight to i occurs in time slot j, else = 0

Yj = number of planes leased for time slot j

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General Form – Alpha AirlinesMax 10XC1 + 6XC2 + 6XC3 + 9XD1 + 10XD2 + 9XD3 + 14XL1+ 11XL2 + 10XL3 + 18XN1 + 15XN2 + 10XN3 – 5Y1

– 5Y2 – 3Y3

s.t.

XC1 + XC2 + XC3 <= 1 XD1 + XD2 + XD3 <= 1 XL1 + XL2 + XL3 <= 1 XN1 + XN2 + XN3 <= 1 XC1 + XD1 + XL1 + XN1 = Y1

XC2 + XD2 + XL2 + XN2 = Y2

XC3 + XD3 + XL3 + XN3 = Y3

Y1 <= 2 Y2 <= 2 Y3 <= 2 XN1 <= XL1

XN2 <= XL2

XN3 <= XL3

Xij= 0,1 Yj = INTEGER

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