Goal achievement

Infocomp Systems is a research and development laboratory firm that develops computer systems and software primarily for the medical industry. The laboratory 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 reflects the development budget, the number of researchers, and the expected annual sales from each project if successfully developed and implemented.

Project Developmental Budget
($ millions) #of Research Personnel Expected Annual Sales
Personnel ($ millions)
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 firm has developed the following set of prioritized goals for selecting projects to initiate:

? The company wants to remain within a total development budget of $5,000,000.
? The company wants to avoid hiring additional research personnel beyond the current staff level of 27 researchers.
? The company would like the expected future annual sales from the implemented projects to be at least $6,500,000.
? Projects 1, 3, 4, and 6 are considered strategically offensive in that they represent new product initiatives, while 2, 5, 7, and 8 are existing product upgrades and therefore defensive in nature. The company would like to select at least two products from each group.
? Projects 2, 3, 5, 6, and 7 are considered the most risky of the projects, and the company would prefer not to select more than three from this group.
? 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.

a. Formulate a linear goal programming model to determine the projects to select that will best achieve the company's goals.

Answer: First, LP formulation is provided. LP formulation helps in giving clear direction for which of the constraints are goals.
The company wants to decide on which projects to choose. A project if chosen it has to be chosen completely. The decision variables are binary/Boolean/bivalent decision variables.
x1 be the indicator variable for the choice of project 1. The variable will be 1 if the project is chosen, 0 otherwise. Similarly 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 1, 3, 4, and 6 are considered strategically offensive in that they represent new product initiatives, while 2, 5, 7, and 8 are existing product upgrades and therefore defensive in nature. 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 most risky of the projects, and the company would prefer not to select more than three from this group.
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)
(Note that even the budgetary constraint is the top priority goal we do not include it in the objective function. The reason is that it is modeled as a hard constraint, so in any case this requirement is met)
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}

b. Determine the solution that will best achieve the company's goals in project selection, including the projects selected and the levels of goal achievement.

Answer: Solve the above program using any package and you will get the solution as required. Observe the value of positive deviation variable.