linear programming and applications graduate program in business information systems aslı sencer

LİNEAR PROGRAMMİNG AND APPLİCATİONS

Graduate Program in Business Information

Systems

Aslı Sencer

OPERATIONS RESEARCH

BIS 517-Aslı Sencer

2

What is Operations Research? Collection of techniques used to allocate the scarce resources in the “best” –OPTIMAL – way!

Best of what? We need an “objective” to be minimized or

maximized! Profit, Cost, Utility, Delay, Distance, Flow, etc.

Some applications


3

Resource allocationProduction and inventory planningCapacity Planning Workers and machine schedulingInvestment planningFormulating marketing and military

strategies

Some news about OR:


4

A Wall Street Journal Article lists the use of LP as one of the greatest technological innovations of the past 1000 years.

1975 Nobel Prize for economics: T.C. Koopmans and L.V. Kantoprovich for their contributions in the field

1992 Nobel Prize: Harry Markowitz for his LP based research.

Basic Optimization Techniques


5

Linear ProgrammingInteger and Goal Programming

Transportation, Assignment Models Network Models

Nonlinear ProgrammingStochastic ProgrammingAlso Simulation

LINEAR PROGRAMMING


6

Most successful of all modern quantitative methods.

Program here is not a computer code! It is a plan that efficiently allocate limited resources to achive a goal.

Involves linear relationships, i.e. relations are in the form of lines, planes!

Basic LP Models: Product Mix

Redwood Furniture Co.

Resource

Unit Requirements

Amount Available in

a PeriodTable Chair

Wood (ft) 30 20 300

Labor (hrs) 5 10 110

Unit profit $6 $8


7

What is the optimal plan to max. Profit?


8

Option 1: Allocate all resources to the more profitable item.Total quantity, profit? Any resource left?

Option 2: Is it more profitable to produce less chairs and more tables?

Linear Programming

Formulating a Linear Problem

Define variables:

: number of tables produced in a period

: number of chairs produced in a period

Define constraints:

Define Objective Function

labor)(110105

wood)(3002030

ct

ct

XX

XX

tX


9

cX

cXMaximize 86XProfit t

)itynonnegativ(0, ct XX

tX

cX

How is an LP solved?


10

Graphical Method: Applicable to a maximum of two decision variables.

Simplex Method: Applicable to all LP.Takes long to implement manually.

Use softwares based on simplex and other techniques.

Graphical Solution


11

Constraint 1

Constraint 2

11

15

10 22

Optimal Solution: Xt=4 tables, Xc=9 ChairsProfit*=$96

(4,9)

Xt

Xc

Basic LP Models: Feed Mix

Two types of seeds are mixed to formulate the wheat of wild birdseed.

Nutritional Item

Proportional Content

Total RequirementBuckwheat

Sunflower wheat

Fat .04 .06 ≥480 lb

Protein .12 .10 ≥1200 lb

Roughage .10 .15 ≤1500

Cost per lb $.18 $.10


12

LP Formulation


13

vity)(nonnegati 0, X

(roughage) 150015.00.10X

(protein) 120010.00.12X

(fat) 48006.00.04X toSubject

10.00.18XCost Minimize

mixture in (lb)sunflower of Amount :

mixture in (lb)buckweat of Amount :

b

b

b

b

b

s

s

s

s

s

s

b

X

X

X

X

X

X

X

Graphical Solution to Feed Mix Problem


14

Fat RoughageProtein

(15000,0)(10000,0)

(3750,7500)

Xb

Xs

Optimal Solution:Xb*=3750 lb, Xs*=7500 lbCost*=$1425

Types of Feasible Regions


15

Bounded Feasible Region

Unbounded FeasibleRegion

Cases in an LP: Infeasible Solution


16

No feasible region

If an LP has no feasible region, then the solution is INFEASIBLE!

Do all LP has an optimal Solution?

Cases in an LP: Multiple Optima

0, X

4553X

7248X

6065X toSubject

12X01P Maximize

21

21

21

21

21

X

X

X

X

X

7

24

7

66

)1(

7

36

7

24

*2

*1 X

X

10


17

Optimal Solutions:Point(1): X1*=4 2/7, X2*=6 3/7Point(2): X1*=6 6/7, X2*=4 2/7P*=$120

(1)

(2)

Infinite number of optimal solutions exist in the form

Cases in an LP: Unbounded Optimal Solution


18

Optimal Solution:X1*=, X2*=P*=

Solving Linear Programs with a Spreadsheet


19

Write out the formulation tablePut the formulation table into a spreadsheetUse Excel’s Solver to obtain a solution

Solution in the Excel Solver


20

Applications of LP:Transportation Models


21

Sporting goods company

Capacity Plants Warehouses Demand

Juarez

Seoul

Tel Aviv

Yokohama

Phoenx

NY

Frankfurt100

300

200

150

100

200

150

LP:Transportation Models (cont’d.)

From Plant

Destination

Frankfurt NY Phoenix Yokohama

Juarez $19 $7 $3 $21Seoul 15 21 18 6

Tel Aviv 11 14 15 22


22

Shipping Costs per pair of skis

What are the optimal shipping quantities from the plants to the warehouses, if the demand has to be met by limited capacities while the shipping cost is minimized?


Xij: Number of units shipped from plant i to warehouse j. i=1,2,3 and j=1,2,3,4.

Minimize shipping costs=19X11+7X12+3X13+21X14

+15X21+21X22+18X23+6X24

+11X31+14X32+15X33+22X34

From Plant

Destination

CapacityFrankfurt NY Phoenix Yokohama

Juarez X11 X12 X13 X14 100

Seoul X21 X22 X23 X24 300

Tel Aviv X31 X32 X33 X34 200

Demand 150 100 200 150 600


23



24subject to#shipped from a plant can not exceed the capacity:

X11+X12+X13+X14≤100 (Juarez Plant)

X21+X22+X23+X24≤300 (Seoul Plant)

X31+X32+X33+X34≤200 (Tel Aviv Plant)

#shipped to a warehouse can not be less than the demand:

X11+X21+X31≥150 (Frankfurt)

X12+X22+X32≥100 (NY)

X13+X23+X33≥200 (Phoenix)

X14+X24+X34≥150 (Yokohama)Nonnegativity

Xij ≥0 for all i,j.



25

Capacity Plants Warehouses Demand

Juarez

Seoul

Tel Aviv

Yokohama

Phoenx

NY

Frankfurt100

300

200

150

100

200

150

Optimal Solution: Optimal cost=$6,250

100

50

100

150

100

100

LP: Marketing Applications

How to allocate advertising budget between mediums such as TV, radio, billboard or magazines?

Ex: Real Reels Co. Allocated ad. Budget=$100,000

Playboy True Esquire

Readers 10 million 6 million 4 million

Significant Buyers

10% 15% 7%

Cost per ad $10,000 $5,000 $6,000

Exposures per ad

1,000,000 900,000 280,000


26

•No more than 5 ads in True and at least two ads in Playboy and Esquire

LP: Marketing Applications (cont’d.)


27

million 11.36Exposure Optimal

ads. 2 Xads, 5 Xads, 3.6 X:Solution Optimal

0,,

2

2

5

)(000,100000,6000,5000,10

28.09.0Exposure TotalMax

Esquire in adds#:

True in adds#:

Playboy in adds#:

ETp

EPT

E

P

T

ETp

ETp

E

T

p

XXX

X

X

X

budgetXXX

XXX

X

X

X

Not integer?

LP: Assignment Models

Assignment of a set of workers to a set of jobs

Individual

Time required to complete one job

Drilling Grinding Lathe

Ann 5min 10min 10min

Bud 10 5 15

Chuck 15 15 10


28

LP: Assignment Models (cont’d.)

333231

232221

131211

101515

15510

10105Jobs of Time TotalMin

3,2,1,

j job toassignednot is i worker if,0

j job toassigned is i worker if,1

XXX

XXX

XXX

ji

X ij

.min20Time Total

1 X,1 X,1X

:Solution Optimal

332211

0

)(1

)(1

)(1

worker single a by performed is job Each

)(1

)(1

)(1

job single a to assigned be canworker A

332313

322212

312111

333231

232221

131211

ijX

LatheXXX

GrindingXXX

DrillingXXX

ChuckXXX

BuddXXX

AnnXXX


29

LP: Diet Problem


30

How much to use of each ingredient so that the nutritional requirements are met in the cheapest way?

Ex: Feed Mix problem given at the beginning of the lecture

LP:Labor Planning


31Addresses staffing needs over a specific time

period.

Hong Kong Bank of Commerce: 12 Full time workers available, but may fire some. Use part time workers who has to work for 4

consequtive hours in a day. Luch time is one hour between 11a.m. and 1p.m.

shared by full time workers. Total part time hours is less than 50% of the day’s

total requirement. Part-timers earn $4/hr (=$16/day) and full timers

earn $50/day.

LP:Labor Planning (Cont’d.)

Time Period Minimum labor required

9a.m.-10a.m. 10

10a.m.-11a.m. 12

11a.m.-noon 14

Noon-1p.m. 16

1p.m.-2p.m. 18

2p.m.-3p.m. 17

3p.m.-4p.m. 15

4p.m.-5p.m. 10


32

LP:Labor Planning (cont’d.)


33

0,

)10...141210(5.04

12

10

15

17

18

16 5.0

14 5.0

12

10

16$50$

.5...,2,1,slot at time start work who tellersPart time # :

dayper tellers timeFull # :

5

54

543

5432

4321

321

21

1

i

i

i

i

PF

P

F

PF

PPF

PPPF

PPPPF

PPPPF

PPPF

PPF

PF

PFCostPersonnelDailyMin

iiP

F

linear programming and applications graduate program in business information systems aslı sencer

Documents

linear programming slide

linear programming bis

applications bis

simulation slide

lp formulation bis

graphical solution bis

infeasible solution

operations research