See the attached file for complete solution. The text here may not be copied exactly as some of the symbols / tables may not print. Thanks

Note: This problem cannot be solved using the "Goal Programming" module in the QM software as the decision variables are 0/1.

See the attached files. I have used LINDO for solving the problem.

Goal programming using QM software

Hi!

I want you to show your work for me to study midterm and complete this study case...

the text book is "Quantitative Analysis for Management" Render/ Stair / Hanna(9th edition)

A small aerospace company is considering the following eight projects:

1. Develop an automated test facility

2.barcode all company inventory and machinery

3.introduce a CAD/CAM system

4. Buy a new lathe and deburring station

5.institute FMS (flexible manufacturing system)

6.install a LAN ( local area network)

7.develop AIS (artificial intelligence simulation)

8.set up a TQM (total quality management) initiative

Each project has been rated on the following five attributes:

...Return on investment (ROI)

...Cost

...Productivity improvement

...Worker requirements

...Degree of technological risk

The ratings for each of the projects are given in the table below:

Project ROI($) Cost($) Productivity Manpower Risk

1 2,070 900 3 18 3

2 456 240 2 18 2

3 670 335 2 27 4

4 350 700 0 36 1

5 495 410 1 42 1

6 380 190 0 6 0

7 1,500 500 3 48 2

8 480 160 2 24 3

The company has set the following five goals:

...Achieve a return on investment of at least $ 3,250

...Limit costs to $ 1,300

...Achieve a productivity improvement of at least 6

...Limit manpower to 108

...Limit technological risk to a total of 4

1. Formulate the problem as a zero-one integer program with 8 decision variables and 5 constraints.

...Use the QM software (mixed integer programming) to solve.

...Set the variable type for each of the projects to "0/1"

...Set all objective function values to zero
Decision variables:
xi=1 if project i is selected else 0 i= 1 to 8

Constraints:
1. Return on investment
2070x1+456x2+670x3+350x4+495x5+380x6+1500x7+480x8>=3250

2. Cost Limit
900x1+240x2+335x3+700x4+410x5+190x6+500x7+160x8<=1300

3. Productivity Improvement
3x1+2x2+2x3+0x4+1x5+0x6+3x7+2x8>=6

4. Manpower Limit
18x1+18x2+27x3+36x4+42x5+6x6+48x7+24x8<=108

5. Technological risk
3x1+2x2+4x3+x4+x5+0x6+2x7+3x8<=4

The objective function is
Maximum 0x1+0x2+0x3+0x4+0x5+0x6+0x7+0x8

When we solve the model, there is no integer feasible solution found. Hence, we cannot solve the problem using the integer programming.

Maximum 0x1+0x2+0x3+0x4+0x5+0x6+0x7+0x8;
subject to
2070x1+456x2+670x3+350x4+495x5+380x6+1500x7+480x8>=3250
900x1+240x2+335x3+700x4+410x5+190x6+500x7+160x8<=1300
3x1+2x2+2x3+0x4+1x5+0x6+3x7+2x8>=6
18x1+18x2+27x3+36x4+42x5+6x6+48x7+24x8<=108
3x1+2x2+4x3+x4+x5+0x6+2x7+3x8<=4
END
INTEGER 1
INTEGER 2
INTEGER 3
INTEGER 4
INTEGER 5
INTEGER 6
INTEGER 7
INTEGER 8

2. Formulate a goal program by modifying the constraints from (1) in the mixed integer-programming module.

...Set the variable type for each of the deviation variables to "Real"

...Use the following weights for undesirable deviations to arrive at your answer:

Goal1- 12.0 Goal2- 1.0 Goal3- 500.00

Goal4- 300.0 Goal5- 50.0

Use the QM software to solve the problem. only the results from (1) and (2)---File, Print, Original Problem with Answers. Do not print out the iterations to the problem!

Note: This problem cannot be solved using the "Goal Programming" module in the QM software as the decision variables are 0/1.
Hi, I have used LINDO to solve the problem.
Output
Model
Minimize 12d1minus+d2plus+500d3minus+300d4plus+50d5plus
subject to
2070x1+456x2+670x3+350x4+495x5+380x6+1500x7+480x8-d1plus+d1minus=3250
900x1+240x2+335x3+700x4+410x5+190x6+500x7+160x8-d2plus+d2minus=1300
3x1+2x2+2x3+0x4+1x5+0x6+3x7+2x8-d3plus+d3minus=6
18x1+18x2+27x3+36x4+42x5+6x6+48x7+24x8-d4plus+d4minus=108
3x1+2x2+4x3+x4+x5+0x6+2x7+3x8-d5plus+d5minus=4
END
INTEGER x1
INTEGER x2
INTEGER x3
INTEGER x4
INTEGER x5
INTEGER x6
INTEGER x7
INTEGER x8
LP OPTIMUM FOUND AT STEP 85
OBJECTIVE VALUE = 50.0000000

FIX ALL VARS.( 2) WITH RC > 90.0480
SET X1 TO >= 1 AT 1, BND= -98.20 TWIN= -3675. 122
SET X5 TO <= 0 AT 2, BND= -98.20 TWIN=-0.1000E+31 122
SET X6 TO <= 0 AT 3, BND= -98.20 TWIN=-0.1000E+31 122
SET X7 TO >= 1 AT 4, BND= -150.0 TWIN= -3128. 133

NEW INTEGER SOLUTION OF 150.000000 AT BRANCH 4 PIVOT 133
BOUND ON OPTIMUM: 150.0000
DELETE X7 AT LEVEL 4
DELETE X6 AT LEVEL 3
DELETE X5 AT LEVEL 2
DELETE X1 AT LEVEL 1
ENUMERATION COMPLETE. BRANCHES= 4 PIVOTS= 133

LAST INTEGER SOLUTION IS THE BEST FOUND
RE-INSTALLING BEST SOLUTION...

OBJECTIVE FUNCTION VALUE

1) 150.0000

VARIABLE VALUE REDUCED COST
X1 1.000000 1050.000000
X2 0.000000 340.000000
X3 0.000000 535.000000
X4 0.000000 750.000000
X5 0.000000 460.000000
X6 0.000000 190.000000
X7 1.000000 600.000000
X8 0.000000 310.000000
IMISE 0.000000 1.000000
D1MINUS 0.000000 12.000000
D2PLUS 100.000000 0.000000
D3MINUS 0.000000 500.000000
D4PLUS 0.000000 300.000000
D5PLUS 1.000000 0.000000
D1PLUS 320.000000 0.000000
D2MINUS 0.000000 1.000000
D3PLUS 0.000000 0.000000
D4MINUS 42.000000 0.000000
D5MINUS 0.000000 50.000000

ROW SLACK OR SURPLUS DUAL PRICES
2) 0.000000 0.000000
3) 0.000000 1.000000
4) 0.000000 0.000000
5) 0.000000 0.000000
6) 0.000000 50.000000

NO. ITERATIONS= 135
BRANCHES= 4 DETERM.= 1.000E 0

The solution is to choose project 1 and 7 and minimum penalty for the decision is 150.