11.IntegerProgramming

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Integer Programming

Introduction to Integer Programming (IP)

Difficulties of LP relaxation

IP Formulations

Branch and Bound Algorithms

Reference: Chapter 9 in W. L. Winston’s book.



1

Integer Programming Model

An Integer Programming model is a linear programmingproblem where some or all of the variables are required to

be non-negative integers.

These models are in general substantially harder thansolving linear programming models.

Network models are special cases of integer programmingmodels and are very efficiently solvable.

We will discuss several applications of integer programmingmodels.

We will study the branch and bound technique, one of themost popular algorithm to solve integer programmingmodels.



2

Classifications of IP Models

Pure IP Model: Where all variables must take integer values.

Maximize z = 3x1 + 2x2 subject to x1 + x2 6

x1, x2 0, x1 and x2 integer

Mixed IP Model: Where some variables must be integer while

others can take real values.

Maximize z = 3x1 + 2x2 subject to x1 + x2 6

x1, x2 0, x1 integer

0-1 IP Model: Where all variables must take values 0 or 1 .

Maximize z = x1 - x2 subject to x1 + 2x2 2

2x1 - x2 1, x1, x2 = 0 or 1



3

Classifications of IP Models (contd.)

LP Relaxation: The LP obtained by omitting all integer or 0-1constraints on variables is called the LP relaxation of IP.

IP:Maximize z = 21x1

+ 11x2 subject to

7x1 + 4x2 13x1, x2 0, x1 and x2 integer

LP Relaxation:Maximize z = 21x1

+ 11x2

subject to7x1 + 4x2 13x1, x2 0

Result:

Optimal objective function value of IP



4

IP and LP Relaxation

x x xx

x1

x2

1 2 3

1

3

2

7x1 + 4x2= 13



5

Simple Approaches for Solving IP

Approach 1:

Enumerate all possible solutions Determine their objective function values Select the solution with the maximum (or, minimum) value.

Any potential difficulty with this approach?

Approach 2:

Solve the LP relaxation Round-off the solution to the nearest feasible integer

solution

Any potential difficulty with this approach?



6

Capital Budgeting Problem

Stockco Co. is considering four investments

It has $14,000 available for investment

Formulate an IP model to maximize the NPV obtained fromthe investments

IP:

Maximize z = 16x1 + 22x2 + 12x3 + 8x4 subject to

5x1 + 7x2 + 4x3 + 3x4 14x1, x2,,x3, x4 0, 1

Investmentchoice

1 2 3 4

Cashoutflow

$5000 $7000 $4000 $3000

NPV $16000 $22000 $12000 $8000



7

Fixed Charge Problem

Gandhi cloth company manufactures three types ofclothing: shirts, shorts, and pants

Machinery must be rented on a weekly basis to make eachtype of clothing. Rental Cost:

$200 per week for shirt machinery

$150 per week for shorts machinery $100 per week for pants machinery

There are 150 hours of labor available per week and 160square yards of cloth

Find a solution to maximize the weekly profit

Labor hr Cloth yd Var. cost PriceShirts 3 4 $12 $6Shorts 2 3 $8 $4Pants 6 4 $15 $8



8

Fixed Charge Problem (contd.)

Decision Variables:

x1 = number of shirts produced each weekx2 = number of shorts produced each weekx3 = number of pants produced each week

y1 = 1 if shirts are produced and 0 otherwise

y2 = 1 if shorts are produced and 0 otherwisey3 = 1 if pants are produced and 0 otherwise

Formulation:

Max. z = 6x1

+ 4x2

+ 7x3

- 200y1

- 150 y2

- 100y3subject to

3x1 + 2x2 + 6x3 1504x1 + 3x2 + 4x3 160

x1 M y1, x2 M y2, x3 M y3

x1, x2,,x3 0, and integer; y1, y2,,y3 0 or 1



9

Either-Or Constraints

Dorian Auto is considering manufacturing three types ofauto: compact, midsize, large.

Resources required and profits obtained from these carsare given below.

We have 6,000 tons of steel and 60,000 hours of labor

available.

If any car is produced, we must produce at least 1,000 unitsof that car.

Find a production plan to maximize the profit.

Compact Midsize Large

Steel Req. 1.5 tons 3 tons 5 tons

Labor Req. 30 hours 25 hours 40 hours

Profit $2000 $3000 $4000



Either-Or Constraints (contd.)

Decision Variables:

x1, x2, x3 = number of compact, midsize and large cars producedy1, y2, y3 = 1 if compact , midsize and large cars are produced or

not

Formulation:

Maximize z = 2x1 + 3x2 + 4x3 subject tox1 My1; x2 My2; x3 My3 1000 - x1 M(1-y1)

1000 - x2

M(1-y2

)1000 - x3 M(1-y3)

1.5 x1 + 3x2 + 5x3 600030 x1 + 25x2 + 40 x3 60000

x1, x2, x3 0 and integer; y1, y2, y3 = 0 or 1



Set Covering Problems

Western Airlines has decided to have hubs in USA.

Western runs flights between the following cities: Atlanta,Boston, Chicago, Denver, Houston, Los Angeles, NewOrleans, New York, Pittsburgh, Salt Lake City, SanFrancisco, and Seattle.

Western needs to have a hub within 1000 miles of each ofthese cities.

Determine the minimum number of hubs

Cities within 1000 milesAtlanta (AT) AT, CH, HO, NO, NY, PI

Boston (BO) BO, NY, PI

Chicago (CH) AT, CH, NY, NO, PI

Denver (DE) DE, SL

Houston (HO) AT, HO, NO

Los Angeles (LA) LA, SL, SF

New Orleans (NO) AT, CH, HO, NO

New York (NY) AT, BO, CH, NY, PIPittsburgh (PI) AT, BO, CH, NY, PI

Salt Lake City (SL) DE, LA, SL, SF, SE

San Francisco (SF) LA, SL, SF, SE

Seattle (SE) SL, SF, SE



Formulation of Set Covering Problems

Decision Variables:

xi = 1 if a hub is located in city ixi = 0 if a hub is not located in city i

Minimize xAT + xBO + xCH + xDE + xHO + xLA + xNO + xNY + xPI + xSL

+ xSF + xSE

subject toAT BO CH DE HO LA NO NY PI SL SF SE Required

AT 1 0 1 0 1 0 1 1 1 0 0 0 xAT >= 1

BO 0 1 0 0 0 0 0 1 1 0 0 0 xBO >= 1

CH 1 0 1 0 0 0 1 1 1 0 0 0 xCH >= 1

DE 0 0 0 1 0 0 0 0 0 1 0 0 xDE >= 1

HO 1 0 0 0 1 0 1 0 0 0 0 0 xHO >= 1

LA 0 0 0 0 0 1 0 0 0 1 1 0 xLA >= 1NO 1 0 1 0 1 0 1 0 0 0 0 0 xNO >= 1

NY 1 1 1 0 0 0 0 1 1 0 0 0 xNY >= 1

PI 1 1 1 0 0 0 0 1 1 0 0 0 xPI >= 1

SL 0 0 0 1 0 1 0 0 0 1 1 1 xSL >= 1

SF 0 0 0 0 0 1 0 0 0 1 1 1 xSF >= 1

SE 0 0 0 0 0 0 0 0 0 1 1 1 xSE >= 1



Additional Applications

Location of fire stations needed to cover all cities

Location of fire stations to cover all regions

Truck despatching problem

Political redistricting

Capital investments



Branch and Bound Algorithm

Branch and bound algorithms are the most popular methodsfor solving integer programming problems

They enumerate the entire solution space but only implicity;hence they are called implicit enumeration algorithms.

A general-purpose solution technique which must bespecialized for individual IP's.

Running time grows exponentially with the problem size, butsmall to moderate size problems can be solved inreasonable time.



An Example

Telfa Corporation makes tables and chairs

A table requires one hour of labor and 9 square board feetof wood

A chair requires one hour of labor and 5 square board feetof wood

Each table contributes $8 to profit, and eachchair contributes $5 to profit.

6 hours of labor and 45 square board feet isavailable

Find a product mix to maximize the profit

Maximize z = 8x1 + 5x2 subject to x1 + x2 6; 9x1 + 5x2 45; x1, x2 0; x1, x2 integer



Feasible Region for Telfa’s Problem

Subproblem 1 : The LP relaxation oforiginal

Optimal LP Solution: x1 = 3.75 andx2 = 2.25 and z = 41.25

Subproblem 2: Subproblem 1 +Constraint x1 4




Feasible Region for Subproblems

Branching : The process ofdecomposing a subproblem into two

or more subproblems is calledbranching.

Optimal solution of Subproblem 2:

z = 41, x1 = 4, x2 = 9/5 = 1.8

Subproblem 4: Subproblem 2 +Constraint x

2

2




Feasible Region for Subproblems 4 & 5



The Branch and Bound Tree

Subproblem 1z = 41.25x1 = 3.75x2 = 2.25

Optimal solution of Subproblem 5:

z = 40.05, x1 = 4.44, x2 = 1

Subproblem 6: Subproblem 5 + Constraint x1 5

Subproblem 5: Subproblem 5 + Constraint x1 4

3

Subproblem 2z = 41x1 = 4

x2 = 1.8

Subproblem 3

Subproblem 4Infeasible

Subproblem 5

x1 4 x1 3

x2 2 x2 1

1

2

4



Feasible Region for Subproblems 6 & 7

Optimal solution ofSubproblem 7:

z = 37, x1 = 4, x2 = 1

Optimal solution ofSubproblem 6:

z = 40, x1 = 5, x2 = 0




Subproblem 1z = 41.25x1 = 3.75x2 = 2.25


x2 = 1.8


x2 = 1, LB = 39

Subproblem 4Infeasible

Subproblem 5z = 40.55x1 = 4.44

x2 = 1

x1 4 x1 3

x2 2 x2 1

Subproblem 6z = 40x

1= 5

x2 = 0, LB = 37


x2 = 1

1

2

3 4

7

6 5



Solving Knapsack Problems

Max z = 16x1+ 22x2 + 12x3 + 8x4 subject to

5x1+ 7x2 + 4x3 + 3x4 14xi = 0 or 1 for all i = 1, 2, 3, 4

LP Relaxation:

Max z = 16x1+ 22x2 + 12x3 + 8x4 subject to

5x1+ 7x2 + 4x3 + 3x4 140 xi 1 for all i = 1, 2, 3, 4

Soving the LP Relaxation: Order xi’s in the decreasing order of ci /ai where ci are thecost coefficients and ai’s are the coefficients in theconstraint

Select items in this order until the constraint is satisfied withequality




x3 = 0x3 = 1

x2 = 0x4 = 1

Subproblem 1z = 44

x1 = x2 = 1x

3

=.51

72

8 9

Subproblem 5z = 43.6

x1 =.6, x2=x3=1x4 = 0, LB = 36

Subproblem 4z = 36

x1 = x3=1x2 = 0, x4 =1

Subproblem 3z = 43.7

x1 =x3= 1,x2 = .7, x4=0

Subproblem 6z = 42

x1 =0, x2=x3=1x4 = 1, LB = 36

5

Subproblem 7LB = 42

Infeasible6

x2 = 1

x1 = 0 x1 = 1

Subproblem 2z = 43.3, LB=42

x1 = x2=1x3 = 0, x4 =.67

Subproblem 8z = 38, LB=42

x1 = x2=1x3 = x4 = 0

Subproblem 9z= 42.85, LB=42

x1 = x4 =1x3 = 0, x2 = .85

x4 = 0

3 4



Strategies of Branch and Bound

The branch and bound algorithm is a divide and conqueralgorithm, where a problem is divided into smaller and smaller

subproblems. Each subproblem is solved separately, and thebest solution is taken.

Lower Bound (LB): Objective function value of the best solution

found so far.

Branching Strategy : The process of decomposing a subprobleminto two or more subproblems is called branching.



Strategies of Branch and Bound (contd.)

Upper Bounding Strategy: The process of obtaining an upperbound (UB) for each subproblem is called an upper bounding

strategy.

Pruning Strategy: If for a subproblem, UB LB, then thesubproblem need not be explored further.

Searching Strategy: The order in which subproblems areexamined. Popular search strategies: LIFO and FIFO.

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