INTEGER PROGRAMMING

INTEGER PROGRAMMING

In the LP models• The decision variables are allowed to have fractional values.• There is a unique objective function.• All mathematical expression (objective fn + constraints) have

to be linear.

Although fractional values (x=0.44, y=105.3) may be valid for decision variables in many problems, a large no. of business problems can be solved only if variables have integer values. For example , when an airline decides how many flights to operate on, it cannot decide to operate 2.38 flights; it must operate 2, 3 or some other integer number.

INTEGER PROGRAMMING CONTD.

Integer Variables • General integer variables (any non- negative

integer satisfying all constraints) Ex: 0, 1, 2, 3, …• Binary variables Ex: 0 or 1

IP• Pure IP problems (all decision variables are

integer) general, binary or combination of two.

• Mixed IP problem(some decision variables, not all are integers)

MODELS WITH GENERAL INTEGER VARIABLES

• Harrision Electric Company produces two expensive products that are popular with renovators of historic old homes; ornate lamps and old fashioned ceiling fans. Both lamps and ceiling fans require a two step production process involving wiring and assembly time. It takes 2 hours to wire each lamp and 3 hours to wire a ceiling fan. Finally assembly of each lamp and fan requires 6 & 5 hours respectively. The production capability is such that only 12 hours of wiring time and 30 hours of assembly time are available. Each lamp produced nets the firm $600 and each fan nets $700 in profit.

SOLUTION OF IP PROBLEMS

Specify the integer requirements• Use the add option to include integer constraint.Solving IP model• The time and computational effort required to

solve IP problems grows rapidly with problem size.How IP models solved• Optimal solution of IP problem need not be at a

corner point of the feasible region. The Branch and Bound (B & B) method is used to solve IP problem by most software packages including solver.

SOLUTION OF IP PROBLEMS CONTD.

Solver options• Max time options • The tolerance option: A tolerance value of

5% implies that we are willing to accept an IP solution that is within 5% of true optimal IP solution value. When solver finds a solution within the allowable tolerance, it stops and present as the final solution.

• Sensitivity reports are not available for IP models.

INTEGER PROGRAMMING (IP)

Where some or all decision variables are required to be whole numbers.

General Integer Variables (0,1,2,3,etc.)Values that count how many

Binary Integer Variables (0 or 1)Usually represent a Yes/No decision

GENERAL INTEGER EXAMPLE:HARRISON ELECTRIC CO.

Produce 2 products (lamps and ceiling fans) using 2 limited resources

Decision: How many of each product to make? (must be integers)

Objective: Maximize profit

DECISION VARIABLESL = number of lamps to makeF = number of ceiling fans to make

Lamps(per lamp)

Fans(per fan)

Hours Available

Profit Contribution $600 $700

Wiring Hours 2 hrs 3 hrs 12

Assembly Hours 6 hrs 5 hr 30

LP Model Summary

Max 600 L + 700 F ($ of profit)

Subject to the constraints:

2L + 3F < 12 (wiring hours)

6L + 5F < 30 (assembly hours)

L, F > 0

GRAPHICAL SOLUTION

PROPERTIES OF INTEGER SOLUTIONS Rounding off the LP solution might not

yield the optimal IP solution

The IP objective function value is usually worse than the LP value

IP solutions are usually not at corner points

BINARY INTEGER EXAMPLE:PORTFOLIO SELECTION

Choosing stocks to include in portfolio

Decision: Which of 7 stocks to include?

Objective: Maximize expected annual return (in $1000’s)

STOCK DATA

COMPANY NAME(LOCATION)

EXPECTED ANNUAL RETURN in $

COST FOR BLOCK OF SHARES (IN

THOUSANDS $)

Trans-Texas Oil (Texas) 50 480

British Petro (Foreign) 80 540

Dutch Shell (Foreign) 90 680

Houston Drilling (Texas) 120 1000

Lone Star Petro (Texas) 110 700

San Diego (California) 40 510

California Petro (California)

75 900

DECISION VARIABLES

Use the first letter of each stock’s name

Example for Trans-Texas Oil:

T = 1 if Trans-Texas Oil is included T = 0 if not included

RESTRICTIONS Invest up to $3 million Include at least 2 Texas companies Include no more than 1 foreign

company Include exactly 1 California company If British Petro is included, then

Trans-Texas Oil must also be included

OBJECTIVE FUNCTION (IN $1000’S RETURN)

Max 50T + 80B + 90D + 120H + 110L + 40S + 75C

Subject to the constraints:

Invest up to $3 Million480T + 540B + 680D + 1000H + 700L + 510S + 900C <

3000

Include At Least 2 Texas CompaniesT + H + L > 2

Include No More Than 1 Foreign Company

B + D < 1

Include Exactly 1 California CompanyS + C = 1

If British Petro is included (B=1), then Trans-Texas Oil must also be included

(T=1)T=0 T=1

B=0 ok ok

B=1 not ok ok

B < T allows the 3 acceptable combinations and prevents the unacceptable one

Combinationsof B and T

Model Summary

Max 50T+80B+90D+120H+110L+40S+75C ($ of return)

Subject to the constraints:

480T+540B+680D+1000H+700L+510S+900C <= 3000 (investment limit)

T +H +L >= 2 (Texas Cos)

B + D <= 1 (Foreign Cos)

S +C = 1 (California Cos)

-T + B <= 0 (T-T & BP)

All variables are 0 or 1

SET COVERING PROBLEM

A set covering problem typically deal with trying to identify the optimal set of locations to cover or to serve a specified set of customers. Consider a case which needs to build health care clinics to serve seven sectors (named A to G) in a region. Each clinic can serve sectors within a maximum radius of 30 minutes driving time, and a sector may be served by more than one clinic. Table in the next slide shows the time it takes to travel between the seven sectors

SET COVERING PROBLEM (CONTD.)

WHAT IS THE MINIMUM NUMBER OF CLINICS THAT WOULD BE NEEDED, AND IN WHICH SECTORS SHOULD BE LOCATED

ToFrom A B C D E F G

A 0 15 20 35 35 45 40

B 15 0 35 20 35 40 40

C 20 35 0 15 50 45 30

D 35 20 15 0 35 20 20

E 35 35 50 35 0 15 40

F 45 40 45 20 15 0 35

G 40 40 30 20 40 35 0

SET COVERING PROBLEM (CONTD.)SECTOR COVERAGE

Sector Sectors within 30 minutes of drive

A A, B,C

B A, B,D

C A, C, D,G

D B, C,D, F, G

E E, F

F D, E, F

G C, D, G

IN ACTION

Selling seats at American Airlines using Integer Programming

American Airlines (AA) describes yield management as “selling the right seats to the right customers at the right prices.” The role of yield management is to determine how much of each product to put on the shelf (i.e., make available for sale) at a given point in time. American’s storefront is the computerized reservations system called SABRE.

SELLING SEATS AT AMERICAN AIRLINES USING INTEGER PROGRAMMING CONTD.

The AA yield-management problem is a mixed-integer program that requires data such as passenger demand, cancellations, and other estimates of passenger behavior that are subject to frequent changes. To solve the systemwide yield-management problem would require approximately 250 million decision variables.

To bring this problem down to a manageable size, AA’s IP model creates three smaller and easier sub problems.

SELLING SEATS AT AMERICAN AIRLINES USING INTEGER PROGRAMMING CONTD.

The airline looks at Overbooking, which is the practice of intentionally

selling more reservations for a flight than there are actual seats on the aircraft

Discount allocation, which is the process of determining the number of discount fares to offer on a flight

Traffic management, which is process of controlling reservations by passenger origin and destination to provide the mix of markets that maximizes revenue

 

SELLING SEATS AT AMERICAN AIRLINES USING INTEGER PROGRAMMING CONTD.

Yield management, much disliked by airline passengers, who view it as a way of squeezing the most money out of travelers as possible, has been a big winner for AA and other airlines. Each year, AA estimates that profits increase by several million dollars to the use of this approach.

MIXED INTEGER MODELS

• So far we have discussed with situation in which the total cost is directly proportional to the magnitude of the decision variable.

Ex: $10x• In many situation there are fixed cost in

addition to the per unit variable costs. These cost may include the costs to set up machines for the production run, construction cost to build a new facility, or design cost to develop a new product.