MIS 463: Decision Support Systems for Business Review of Linear Programming and Applications Aslı...
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Transcript of MIS 463: Decision Support Systems for Business Review of Linear Programming and Applications Aslı...
MIS 463: Decision Support Systems for Business
Review of Linear Programming and
Applications
Aslı Sencer
BIS 517-Aslı Sencer 2
Basic LP Models: Product Mix
Production System for tables and chairs.
Resource
Unit Requirements
Amount Available in
a PeriodTable Chair
Wood (ft) 30 20 300
Labor (hrs) 5 10 110
Unit profit $6 $8
BIS 517-Aslı Sencer 3
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
cX
cXMaximize 86XProfit t
)itynonnegativ(0, ct XX
cX
BIS 517-Aslı Sencer 4
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
BIS 517-Aslı Sencer 5
LP Formulation
vity)(nonnegati 0,X
(roughage) 150015.00.10X
(protein) 120010.00.12X
(fat) 48006.00.04X Subject to
10.00.18XCost Minimize
mixturein (lb)sunflower ofAmount :
mixturein (lb)buckweat ofAmount :
b
b
b
b
b
s
s
s
s
s
s
b
X
X
X
X
X
X
X
BIS 517-Aslı Sencer 6
Applications of LP:Transportation Models
Sporting goods company
Capacity Plants Warehouses Demand
Juarez
Seoul
Tel Aviv
Yokohama
Phoenx
NY
Frankfurt100
300
200
150
100
200
150
BIS 517-Aslı Sencer 7
LP:Transportation Models (cont’d.)
From Plant
Destination
Frankfurt NY Phoenix Yokohama
Juarez $19 $7 $3 $21
Seoul 15 21 18 6
Tel Aviv 11 14 15 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?
BIS 517-Aslı Sencer 8
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
+19X21+7X22+3X23+21X24
+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
BIS 517-Aslı Sencer 9
LP:Transportation Models (cont’d.)
subject 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+X41≥150 (Frankfurt)
X12+X22+X32+X42≥100 (NY)
X13+X23+X33+X43≥200 (Phoenix)
X14+X24+X34+X44 ≥150 (Yokohama)Nonnegativity
Xij ≥0 for all i,j.
BIS 517-Aslı Sencer 10
Capacity Plants Warehouses Demand
Juarez
Seoul
Tel Aviv
Yokohama
Phoenx
NY
Frankfurt100
300
200
150
100
200
150
LP:Transportation Models (cont’d.)Optimal Solution: Optimal cost=$6,250
100
50
100
150
100
100
BIS 517-Aslı Sencer 11
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
•No more than 5 ads in True and at least two ads in Playboy and Esquire
BIS 517-Aslı Sencer 12
LP: Marketing Applications (cont’d.)
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
BIS 517-Aslı Sencer 13
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
BIS 517-Aslı Sencer 14
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
BIS 517-Aslı Sencer 15
LP:Labor Planning Addresses 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.
BIS 517-Aslı Sencer 16
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
BIS 517-Aslı Sencer 17
LP:Labor Planning (cont’d.)
0,
)10...141210(5.04
12
10
15
17
18
16 5.0
14 5.0
12
10
)4(16$50$
.5...,2,1,slot timeat workstart who tellers timePart # :
dayper tellers time Full # :
5
54
543
5432
4321
321
21
1
i
i
i
i
PF
P
F
PF
PPF
PPPF
PPPPF
PPPPF
PPPF
PPF
PF
PCostPersonnelDailyMax
iiP
F
Alternative Optimal SolutionF=10, P2=2, P3=7, P4=5F=10, P1=6, P2=1, P3=2, P4=5at a cost of $724/day
BIS 517-Aslı Sencer 18
Solving Linear Programs with a Spreadsheet
Write out the formulation table Put the formulation table into a spreadsheet Use Excel’s Solver to obtain a solution
Step 1: The Formulation Table
Variables XE XL XR XS XM Sign RHS
Objective 58 43 25 17 28 = P(max)PC Board 25 15 10 5 1 < 50,000A Availability 28 24 18 12 5 < 10,000B Availability 52 48 40 60 75 < 25,000Assembly Time 1.50 1.25 1.00 0.75 1.50 < 2,000Regular Quantity 1 > 200Small Quantity 1 > 100Oversized Mixture 2 -1 < 0Miniature Mixture -0.50 -0.50 -0.50 -0.50 1.00 < 0
The formulation table arranges the problem in a tabular format, as shown below for the Microcircuit Production Plan.
The formulation table arranges the problem in a tabular format, as shown below for the Microcircuit Production Plan.
Step 2: The Excel Spreadsheet
123
456789101112
A B C D E F G H
Variables XE XL XR XS XM Sign RHS
Objective 58 43 25 17 28 = P(max)PC Board 25 15 10 5 1 < 50000A Availability 28 24 18 12 5 < 10000B Availability 52 48 40 60 75 < 25000Assembly Time 1.50 1.25 1.00 0.75 1.50 < 2000Regular Quantity 1 > 200Small Quantity 1 > 100Oversized Mixture 2 -1 < 0Miniature Mixture -0.50 -0.50 -0.50 -0.50 1.00 < 0
Microcircuit Production Plan
The numbers in the Excel spreadsheet come from the formulation table.
The numbers in the Excel spreadsheet come from the formulation table.
BIS 517-Aslı Sencer 21
Step 3: Expanded Spreadsheet
123456789
10111213141516171819202122
A B C D E F G H I J
Variables XE XL XR XS XM Sign RHSObjective 58 43 25 17 28 = P(max) Profit 8420PC Board 25 15 10 5 1 < 50000 PC Board 3060A Availability 28 24 18 12 5 < 10000 A Availability 5610B Availability 52 48 40 60 75 < 25000 B Availability 16230Assembly Time 1.50 1.25 1.00 0.75 1.50 < 2000 Assembly Time 330Regular Quantity 1 > 200 Regular Quantity 200Small Quantity 1 > 100 Small Quantity 100Oversized Mixture 2 -1 < 0 Oversized Mixture 0Miniature Mixture -0.50 -0.50 -0.50 -0.50 1.00 < 0 Miniature Mixture -155
XE XL XR XS XM
10.00 20.00 200.00 100.00 10.00
Microcircuit Production Plan
Solution
456789
101112
J=SUMPRODUCT(B4:F4,$B$15:$F$15)=SUMPRODUCT(B5:F5,$B$15:$F$15)=SUMPRODUCT(B6:F6,$B$15:$F$15)=SUMPRODUCT(B7:F7,$B$15:$F$15)=SUMPRODUCT(B8:F8,$B$15:$F$15)=SUMPRODUCT(B9:F9,$B$15:$F$15)=SUMPRODUCT(B10:F10,$B$15:$F$15)=SUMPRODUCT(B11:F11,$B$15:$F$15)=SUMPRODUCT(B12:F12,$B$15:$F$15)
The expanded spreadsheet contains the formulas necessary to use Solver. Put =SUMPRODUCT(B4:F4,$B$15:$F$15) in cell J4 and copy it down to cell J12. Cell J4 gives the value of the objective function.
The expanded spreadsheet contains the formulas necessary to use Solver. Put =SUMPRODUCT(B4:F4,$B$15:$F$15) in cell J4 and copy it down to cell J12. Cell J4 gives the value of the objective function.
The solution is found here (the values of the decision variables).
The solution is found here (the values of the decision variables).
BIS 517-Aslı Sencer 22
Using Excel’s Solver to Solve Linear Programs
Click on Tools on the menu bar, select the Solver option, and the Solver Parameters dialog box shown next appears.
BIS 517-Aslı Sencer 23
Solver Parameters Dialog Box
1. Enter the value of the objective function, J4, in the Target Cell line, either with or without the $ sign.
1. Enter the value of the objective function, J4, in the Target Cell line, either with or without the $ sign.
2. The Target Cell is to be maximized so click on Max in the Equal To line.
2. The Target Cell is to be maximized so click on Max in the Equal To line.
3. Enter the decision variables in the By Changing Cells line, B15:F15.
3. Enter the decision variables in the By Changing Cells line, B15:F15.
4. The constraints are entered in the Subject to Constraints box by using the Add Constraints dialog box shown next (obtained by clicking on the Add button). If a constraint needs to be changed, click on the Change button. The Change and Add Constraint dialog box function in the same manner.
4. The constraints are entered in the Subject to Constraints box by using the Add Constraints dialog box shown next (obtained by clicking on the Add button). If a constraint needs to be changed, click on the Change button. The Change and Add Constraint dialog box function in the same manner.
NOTE: Normally all these entries appear in the Solver Parameter dialog box so you only need to click on the Solve button. However, you should always check to make sure the entries are correct for the problem you are solving.
NOTE: Normally all these entries appear in the Solver Parameter dialog box so you only need to click on the Solve button. However, you should always check to make sure the entries are correct for the problem you are solving.
BIS 517-Aslı Sencer 24
The Add Constraint Dialog BoxTo represent the constraints in rows 5 - 8:
1. Enter J5:J8 (or $J$5:$J$8) in the Cell Reference line. This is the total amount of these resources used.
To represent the constraints in rows 5 - 8:
1. Enter J5:J8 (or $J$5:$J$8) in the Cell Reference line. This is the total amount of these resources used.
3. Enter the amounts of the resources available H5:H8 in the Constraint line (or =$H$5:$H$8).
3. Enter the amounts of the resources available H5:H8 in the Constraint line (or =$H$5:$H$8).
4. Click Add and repeat Steps 1 - 3 if another constraint is to be added. If this is the last constraint, click OK.
4. Click Add and repeat Steps 1 - 3 if another constraint is to be added. If this is the last constraint, click OK.
Normally, all these entries already appear. You will need to use this dialog box only if you need to add a constraint.
Normally, all these entries already appear. You will need to use this dialog box only if you need to add a constraint.
If you need to change a constraint, the Change Constraint dialog box functions just like this one.
If you need to change a constraint, the Change Constraint dialog box functions just like this one.
2. Enter <= as the sign because the resources used must be equal to or less than the amounts available, given next in Step 3. If another sign is needed, see the next slide.
2. Enter <= as the sign because the resources used must be equal to or less than the amounts available, given next in Step 3. If another sign is needed, see the next slide.
BIS 517-Aslı Sencer 25
The Solver Options Dialog Box
Click on the Options button in the Solver Parameters dialog box to check the Solver Options dialog box to ensure that the Assume Linear Model and Assume Non-Negative boxes are checked.
Click on the Options button in the Solver Parameters dialog box to check the Solver Options dialog box to ensure that the Assume Linear Model and Assume Non-Negative boxes are checked.
BIS 517-Aslı Sencer 26
Solver Results Dialog Box(Figure 9-9)
Be sure to check the message in the Solver Results dialog box. In this case it indicates that a solution has been found. What happens when Solver does not find a solution will be discussed latter. Click OK and the spreadsheet with the solution, shown next, is obtained.
Be sure to check the message in the Solver Results dialog box. In this case it indicates that a solution has been found. What happens when Solver does not find a solution will be discussed latter. Click OK and the spreadsheet with the solution, shown next, is obtained.
BIS 517-Aslı Sencer 27
Spreadsheet with Optimal Solution
123456789
101112131415
A B C D E F G H I J
Variables XE XL XR XS XM Sign RHSObjective 58 43 25 17 28 = P(max) Profit 16800.65PC Board 25 15 10 5 1 < 50000 PC Board 6228.10A Availability 28 24 18 12 5 < 10000 A Availability 10000.00B Availability 52 48 40 60 75 < 25000 B Availability 25000.00Assembly Time 1.50 1.25 1.00 0.75 1.50 < 2000 Assembly Time 565.24Regular Quantity 1 > 200 Regular Quantity 200.00Small Quantity 1 > 100 Small Quantity 100.00Oversized Mixture 2 -1 < 0 Oversized Mixture 0.00Miniature Mixture -0.50 -0.50 -0.50 -0.50 1.00 < 0 Miniature Mixture -237.92
XE XL XR XS XM
67.54 135.08 200.00 100.00 13.39
Microcircuit Production Plan
Solution
2. Enter the data: the coefficients of the objective function in cells B4:F4, the right-hand sides in cells H5:H12, and the exchange coefficients in cells B5:F12.
2. Enter the data: the coefficients of the objective function in cells B4:F4, the right-hand sides in cells H5:H12, and the exchange coefficients in cells B5:F12.
3. To find the solution, click on Tools and Solver to obtain the Solver Parameters dialog box and then click the Solve button.
3. To find the solution, click on Tools and Solver to obtain the Solver Parameters dialog box and then click the Solve button.
4. For bigger problems insert additional rows or columns. Insert them in the middle of the table and not at the beginning or the end. Copy the formulas in column J to any new cells created by inserting rows. Check to make sure the ranges of the formulas and signs in the Solver Parameters dialog
box are correct.
4. For bigger problems insert additional rows or columns. Insert them in the middle of the table and not at the beginning or the end. Copy the formulas in column J to any new cells created by inserting rows. Check to make sure the ranges of the formulas and signs in the Solver Parameters dialog
box are correct.
1. To solve other problems:
1. To solve other problems: