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


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


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


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


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


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?



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



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.


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


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


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


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


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


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.


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


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


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.


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


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).


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.


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.


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.


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.


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.


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


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:

mis 463: decision support systems for business review of linear programming and applications aslı...

Documents

applicationsasl sencerbis

basic lp models

applications of lp

lp formulationbis

transportation models

transportat models contd

optimal cost

shipping costs