a company is trying to allocate its limited resources to a number of different projects. It has three resources and three projects to which they can be allocated. For each unit of each resource allocated to a project the project duration is reduces the project duration for P3 by 1.4 hours.
Project P1 P2 P3 R1 1.0 1.9 0.4
Resource R2 0.7 1.0 1.4 R3 2.8 3.5 3.8
It is believed that the company has available 1200 units of resource R1, 1500 units of resource R2 and 900 units of resouce R3.
Technological restrictions mean that for every 3 units of resource R3 allocated to project P3 at most 2 units of resource R2 can be allocated to project P3.
The company would like to use goal programming to allocate resources to projects and ideally would allocate enough resource to each project so as to reduce the duration of each individual project by 1200 minutes. However they accept it may be necessary to deviate from this ideal, but are currently uncertain as to the weights they would use for deviations in any goal program. However they are clear that project P1 is the most important project, followed by project P2, with project P3 being the least important project. An additional condition that needs to be considered is that a company goal is to reduce the total project duration time (over all three projects) by 3600 minutes.
(a) Formulate this problem as a weighted goal program with linear constraints. Note here that you should not attempt to solve this problem numerically or attempt to simplify any linear equations that you give.
(b) How does your formulation change if instead of using goal programming you formulate the problem the problem as describe above but using linear programming(so do not include goal deviation variables)?What might be the advantages and disadvantages of doing this? Note here that you should not attempt to solve this problem numerically or attempt to simplify any linear equations that you give.