Question

Infocomp Systems Lab is a research and development (R&D) company that develops computer systems and software primarily for the medical industry. The lab has proposals from its own researchers for eight new projects. Each of the proposed research projects requires limited resources, and it is not possible to undertake all of them. The following table shows the developmental budget, the number of researchers, and the expected annual sales from each project if successfully developed and implemented:

Project	Developmental Budget ($1,000,000s)	Number of Research Personnel	Expected Annual Sales ($1,000,0005)
1	$0.675	6	$0.82
2	1.050	5	1.75
3	0.725	7	1.60
4	0.430	8	1.90
5	1.240	10	0.93
6	0.890	6	1.70
7	1.620	7	1.30
8	1.200	6	1.80

The lab has developed the following set of prioritized goals for selecting which projects to initiate:

(1) The company would like to remain within a total developmental budget of $5,000,000.

(2) The number of available research personnel is 27, and Infocomp would like to avoid obtaining extra researchers.

(3) The company would like the expected future annual sales from the implemented projects to be at least $6,500,000.

(4) Projects 1, 3, 4, and 6 are considered offensive in that they represent new product initiatives, while projects 2, 5, 7, and 8 are existing product upgrades and thus defensive in nature. The lab would like to select at least two projects from each group.

(5) Projects 2, 3, 5, 6, and 7 are considered the most risky of the projects, and the company would prefer not to select any more than three of these projects.

(6) The lab’s owner has indicated that she would like to see projects 5 and 6 initiated if doing so would not interfere with the achievement of any of the more important goals determined by the lab’s top management.

a. Formulate a goal programming model to determine which projects Infocomp Systems Lab should select to best achieve its goals.

No need to solve in excel. Looking for Decision variables, objective function and constraints.

Answer 1

Answer: x2, x3, x4, x5, x6, x7, and x8 are defined
Objective function of LP without any goals is
Sales = 0.82x1 + 1.75x2 + 1.6x3 + 1.9x4 + 0.93x5 + 1.7x6 + 1.3x7 + 1.8x8 -----(Eq.1)
Availability of research personnel
6x1 + 5x2 + 7x3 + 8x4 + 10x5 + 6x6 + 7x7 + 6x8 <= 27 ---------(Eq.2)
Availability of funds
$0.675x1 + 1.050 x2 + 0.725 x3 + 0.430x4 + 1.240x5 + 0.890x6 + 1.620x7 + 1.200x8 <= 5 -------Eq(3)
No we look at goal program
1. The company wants to remain within a total development budget of $5,000,000
This is the hard constraints as development budget is not allowed to exceed. From eq(3)
$0.675x1 + 1.050 x2 + 0.725 x3 + 0.430x4 + 1.240x5 + 0.890x6 + 1.620x7 + 1.200x8 <= 5 Eq(3)

2. The company wants to avoid hiring additional research personnel beyond the current staff level of 27 researchers.
We treat this as the first priority goal. Let us define d1+ and d1- s the deviation variables to exceed and keeping less than the targeted staff level. The goal constraint is, from eq(2)
6x1 + 5x2 + 7x3 + 8x4 + 10x5 + 6x6 + 7x7 + 6x8 -d1+ + d1- = 27 Eq(4)

3. The company would like the expected future annual sales from the implemented projects to be at least $6,500,000.
This is a hard constraint as expected future sales is not allowed to fall below 6500000 mark. This constraints is given by (see eq(1))
0.82x1 + 1.75x2 + 1.6x3 + 1.9x4 + 0.93x5 + 1.7x6 + 1.3x7 + 1.8x >= 6.5 Eq(5)

4. Projects 2, 3, 5, 6, and 7 are considered the most dangerous of the tasks, and the organization would incline toward not to choose more than three from this gathering. The company would like to select at least two products from each group.
This again is hard constraints but with combination of binary variables. At least two projects from each of the two groups, 1,3,4,6 and 2,5,7,8, have to be selected. Selecting at least two from the first group is given by
x1 + x3 + x4 + x6 >=2 Eq(6).....(Note that the variables are binary)
Similarly for the other group we get
x2 + x5 + x7 + x8 >=2 Eq(7)

5. Projects 2, 3, 5, 6, and 7 are considered the riskiest of the projects, company would prefer not to select. The constraint for this is given by
x2 + x3 + x5 + x6 + x7 <= 3 Eq(8)

6. The president of Infocomp Systems has expressed an interest in the initiation of projects 5 and 6 unless these selections are in conflict with other goals.
No separate constraint is required for this goal as variable definition (binary variables) will take care of it.
Now our goal program for priority level 1 P1 is
Avoid the requirement of exceeding the current staff level. Hence, we minimize the d1+
(P1) Minimize: d1+
Subject to
6x1 + 5x2 + 7x3 + 8x4 + 10x5 + 6x6 + 7x7 + 6x8 -d1+ + d1- = 27 Eq(4)
$0.675x1 + 1.050 x2 + 0.725 x3 + 0.430x4 + 1.240x5 + 0.890x6 + 1.620x7 + 1.200x8 <= 5000000 Eq(3)
0.82x1 + 1.75x2 + 1.6x3 + 1.9x4 + 0.93x5 + 1.7x6 + 1.3x7 + 1.8x >= 6500000 Eq(5)
(Note that the sales target is also put as a hard constraint as it is said that sales cannot fall below 6500000. This can also be modeled through deviation variables and it will be then maximization of d+ deviation variable. It is same as modeling the constraint as in Eq(5))
x1 + x3 + x4 + x6 >=2 Eq(6)
x2 + x5 + x7 + x8 >=2 Eq(7)
x2 + x3 + x5 + x6 + x7 <= 3 Eq(8)
All the deviation variables are non-negative
x1, x2, x3, x4, x5, x6, x7, and x8 = {0 ,1}