Graduate Program in Business Information Systems Integer and Goal Programming Aslı Sencer.

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Graduate Program in Business Information Systems Integer and Goal Programming Aslı Sencer

Transcript of Graduate Program in Business Information Systems Integer and Goal Programming Aslı Sencer.

Graduate Program in Business Information

Systems

Integer and Goal Programming

Aslı Sencer

BIS 517-Aslı Sencer 2

Integer Program

Structurally like the LP models where the solutions must be integers!

Ex: Production order quantities Number of items sent from plants to warehouses Binary decisions

produce or do not produce Assign staff i to job j or not Locate facility i in location j or not

BIS 517-Aslı Sencer 3

Ex:Redwood Furniture Model (revisited)

0

(labor)113105

(wood)3102030

86

CT

CT

CT

CT

,XX

XX

XXtosubject

XXMax

Optimal solution:

8.98

2.9

2.4

*

*

*

P

X

X

C

T

BIS 517-Aslı Sencer 4

The Branch and Bound Method

An effective search procedure that involves solving a succession of carefully formulated LP’s.

If the optimal solution of the parent program includes fractions, then two descendent LP’s are developed and solved to force the noninteger variable to be an integer.

The optimal objective function of the parent program constitues a bound on the optimal objective function value of the descendent LP’s.

BIS 517-Aslı Sencer 5

Graphical Solution

Problem1

Problem 2

11

15

10 22

Xt

Xc

4 5

Problem 3

BIS 517-Aslı Sencer 6

Branch and Bound Solution

4TX

9CX

5TX

Problem 1

Xt=4.2Xc=9.2P=98.8

Problem 2

Xt=4Xc=9.3P=98.4

Problem 3

Xt=5Xc=8P=94

Problem 4

Xt=4Xc=9P=96

Problem 5

Xt=2.6Xc=10P=95.6

OPTIMAL SOLUTION

10CX

BIS 517-Aslı Sencer 7

When do we round off the LP?

For problems having small number of variables, IP is solved in a short time.

Bigger problems require a lot of time.

If no integer solution has been found, a rounded LP may be used.

Which solution to be used is judgemental.

BIS 517-Aslı Sencer 8

Solving Integer Programs with a Spreadsheet

When using Excel Spreadsheet, it is sufficient to add a new constraint and define the decision variables as integers.

You can also use WinQSB or QuickQuant to solve LP and IP problems.

BIS 517-Aslı Sencer 9

Ex: Maui Miser Car Rentals Budget $500,000. Three types of cars: Economy vans, compact cars,

large cars Income/vehicle: $15,000, $7,600, $10,600 Costs/vehicle: $25,000, $15,000, $21,000 At least 25 vehicles At least 5 vans, at least 5 large cars, at least 12

passenger cars # large cars can not exceed the #compacts Average daily rental rate should be less than $39

where Daily rental prices/vehicle: $50, $30, $40

BIS 517-Aslı Sencer 10

Ex: Formulation

integers are ,,

0 911

0

12

5

5

25

500,000 000,21000,15000,25

600,10600,7000,15IncomeMax

purchase to cars large :#

purchase to carscompact :#

purchase to vans economy :#

LCV

LCV

LC

LC

L

V

LCV

LCV

LCV

L

C

V

XXX

XXX

XX

XX

X

X

XXX

XXXtosubject

XXX

X

X

X

BIS 517-Aslı Sencer 11

Optimal Solution

81.953,273$

5

46.11

92.8

Solution LP

P

X

X

X

L

C

V

800,271$

5

13

8

Solution IP

P

X

X

X

L

C

V

BIS 517-Aslı Sencer 12

Goal Programming

Maximize profit ORMinimize cost

Single Goal

Sometimes we have Multiple Goals.

Minimize Cost AND Maximize Customer SatisfactionMaximize Return AND Minimize Risk

What is the optimal balance in attaining the conflicting goals?

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Basic Idea

It is impossible to meet all goals simultaneously.

The focus aims at achieving certain targets for each goal.

The overall objective is to find the solution that collectively minimizes deviations from these targets.

BIS 517-Aslı Sencer 14

Ex:Goal Programming in Redwood Furniture Co.

Objectives Maximize Profit Maximize Revenue to maintain marketshare growth Maximize training time to increase work-force productivity

Table Chair

Profit/unit $6 $8

Revenue/unit $50 $25

Train. time/unit 1hr. 3hr.

Each LP has a different optimal solution. An idea is to set targets for the goals.

Target

$90

$450

30hr.

BIS 517-Aslı Sencer 15

LP with Goal Targets

30)target (Training 303 :3 Goal

450)target (Revenue 4502550 :2 Goal

90)target(Profit 9086 :1 Goal

CT

CT

CT

XX

XX

XX

Warning: When these constraints are added to the Redwood Product Mix Formulation there may be no feasible solution satisfying the minimum requirements of all goals!

What should we do then? Any comments?Answer: Try to lower the targets

Idea 1: We can treat targets as the lower bounds on the goals

BIS 517-Aslı Sencer 16

Goal Deviations

Idea 2: We can allow some goals to be partially met!

Compared to LP, goal programming is flexible. Optimal balance between goals can be achieved in several ways.

Goal Deviation: Amount of deviation from the goal target is a variable to be minimized.

BIS 517-Aslı Sencer 17

Defining Goal Deviations

Yp: The amount by which profit deviates from the target level

90)86( then

,90)86(Let

)(9086:1

PCT

CTP

CT

YXX

XXY

profitXXGoal

When Yp is positive,

When Yp is negative

0 , PPP YYY

0 , PPP YYY

0, , , PPPPP YYYYYHere

BIS 517-Aslı Sencer 18

Goal Deviation Constraints

30)(3

450)(2550

90)(86

TTCT

RRCT

PPCT

YYXX

YYXX

YYXX

110105

3002030

CT

CT

XX

XX

Regular Constrains:

BIS 517-Aslı Sencer 19

Goal Programming Objective

It costs $1 to fall a dollar below the profit target. However exceeding the profit target is desirable.

Falling below revenue target lowers market share, lost earnings is $2 for a dollar below revenue target.

Training time upto 30hrs enhances future efficiency. It costs $0.50 per hour obtained. Beyond 30hrs. training effect is negligable.

TTRRPPCT YYYYYYXXCMinimize 5.00201000

190)(8689

PPPCT YYYXX

BIS 517-Aslı Sencer 20

Ex: Optimal Solution

9

6Y 0Y 6 Y 6

0Y 0 Y 0 Y 6

TRP

TRP

C

X

X

C

T

Profit and Training levels are $6 and 6hrs. below their targets.Revenue target is met.

BIS 517-Aslı Sencer 21

Ex: Maui Miser Case (revisited)

Achieve annual rental income of at least $300,000 from fleet addition

Spend as close to as possible to $500,000 Do not exceed average daily rental of $39 for

the vehicles in the fleet addition.

Penalty Costs:

$1 for each dollar violation of goals 1 and 2.

$10,000 for each dollar violation of goal 3.

BIS 517-Aslı Sencer 22

Ex: Maui Miser Case GP Formulation

positive are variables all

0

12

5

5

25

0 911

500,000000,21000,15000,25

000,300600,10600,7000,15

Y000,10YYYC Min

33

22

11

3221

LC

LC

L

V

LCV

LCV

LCV

LCV

XX

XX

X

X

XXX

YYXXX

YYXXX

YYXXX

BIS 517-Aslı Sencer 23

Ex: Maui Miser Case GP Optimal Solution

154.046,26$

154.046,26

5

462.11

923.8

Solution GP

1

C

Y

X

X

X

L

C

V

200,28$

24

200,28

5

13

8

Solution IGP

3

1

C

Y

Y

X

X

X

L

C

V