Graduate Program in Business Information Systems Integer and Goal Programming Aslı Sencer.
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Transcript of 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 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?
BIS 517-Aslı Sencer 13
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