linear programming and applications graduate program in business information systems aslı sencer
TRANSCRIPT
LİNEAR PROGRAMMİNG AND APPLİCATİONS
Graduate Program in Business Information
Systems
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OPERATIONS RESEARCH
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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
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Resource allocationProduction and inventory planningCapacity Planning Workers and machine schedulingInvestment planningFormulating marketing and military
strategies
Some news about OR:
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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
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Linear ProgrammingInteger and Goal Programming
Transportation, Assignment Models Network Models
Nonlinear ProgrammingStochastic ProgrammingAlso Simulation
LINEAR PROGRAMMING
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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
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What is the optimal plan to max. Profit?
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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
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cX
cXMaximize 86XProfit t
)itynonnegativ(0, ct XX
tX
cX
How is an LP solved?
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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
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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
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LP Formulation
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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
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Fat RoughageProtein
(15000,0)(10000,0)
(3750,7500)
Xb
Xs
Optimal Solution:Xb*=3750 lb, Xs*=7500 lbCost*=$1425
Types of Feasible Regions
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Bounded Feasible Region
Unbounded FeasibleRegion
Cases in an LP: Infeasible Solution
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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
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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
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Optimal Solution:X1*=, X2*=P*=
Solving Linear Programs with a Spreadsheet
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Write out the formulation tablePut the formulation table into a spreadsheetUse Excel’s Solver to obtain a solution
Solution in the Excel Solver
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Applications of LP:Transportation Models
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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
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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?
LP:Transportation Models (cont’d.)
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
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LP:Transportation Models (cont’d.)
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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.
LP:Transportation Models (cont’d.)
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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
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•No more than 5 ads in True and at least two ads in Playboy and Esquire
LP: Marketing Applications (cont’d.)
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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
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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
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LP: Diet Problem
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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
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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
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LP:Labor Planning (cont’d.)
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)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
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