Programming
Note: Goal See problem 13.13 for the problem statement. We assume that part-time (fractional) workers are allowed. Example 1: Preemptive Goal Programming The problem is currently stated as a preemptive goal program. In a preemptive GP, we have one LP/ILP/MILP for each priority level. If you are working on priority level k, the LP/ILP/MILP looks like: min sum of weighted deviations for level k goals subject to ALL functional constraints ALL goal constraints ALL nonfunctional constraints OF1 = v1 OF2 = v2 . . .
OFk-1 = vk-1 where OFj is the objective function for the j-th priority level and vj is the optimal value of the objective function for the j-th priority level (1 j k - 1).
Let x1 = number employees assigned to phones x2 = number employees assigned to door-to-door The functional constraints (i.e. the constraints that MUST be satisfied) are x1 + x2 35 there are 35 employees total
The goals may be written as constraints as follows: 1. Level 1 Achieve (at least) $20K in expected weekly sales 20(0.06)(400)x1 + 20(0.20)(150)x2 20000 1
Spend no more than $10K in weekly salaries 240x1 + 300x2 10000 2. Level 2 Reach (at least) 6000 potential customers per week 400x1 + 150x2 6000 3. Level 3 Assign at least 10 employees to phones x1 10 Assign at least 10 employees to door-to-door x2 10 Based on the above, we can see that the detrimental deviations are Goal Deviation Reason 1 U1 implies that sales were under $20,000 2 E2 implies that salaries exceeded $10,000 3 U3 implies that the potential customers reached was under 6,000 4 U4 implies that the number employees assigned to phones was under 10 5 U5 implies that the number employees assigned to door-to-door was under 10 Priority Level 1 Program There are two goals in priority level one. These goals are equally important so they can have the same weight. min U1 + E2 subject to x1 + x2 480x1 + 600x2 + U1 - E1 240x1 + 300x2 + U2 - E2 400x1 + 150x2 + U3 - E3 x1 + U4 - E4 x2 + U5 - E5 x, U, E 2 = = = = = 35 20000 10000 6000 10 10 0
Priority Level 2 Program There is one goal in priority level two. The priority one objective is now included as a constraint, highlighted below. min U3 subject to x1 + x2 480x1 + 600x2 + U1 - E1 240x1 + 300x2 + U2 - E2 400x1 + 150x2 + U3 - E3 x1 + U4 - E4 x2 + U5 - E5 U1 + E1 x, U, E = = = = = = 35 20000 10000 6000 10 10 v1 0
Priority Level 3 Program There are two goals in priority level three. These goals are equally important they so can have the same weight. The priority one and two objectives are now included as constraints, highlighted below. min U4 + U5 subject to x1 + x2 480x1 + 600x2 + U1 - E1 240x1 + 300x2 + U2 - E2 400x1 + 150x2 + U3 - E3 x1 + U4 - E4 x2 + U5 - E5 U1 + E1 U3 x, U, E = = = = = = = 35 20000 10000 6000 10 10 v1 v2 0
The Preemptive GP Solution If you are using the template, you can just enter in the functional constraints and the 5 goals, specifying their priority levels then run solver. But, if you are not using the templates, you'll need to solve the 3 LPs IN THE ORDER they are listed above. When we do this, we obtain the following:
3
Level x1 x2 Alternate Optima OF Value 1 3.57 30.48 Yes 0 2 3.57 30.48 Yes 0 3 8.33 26.67 No 1.67 Based upon the solution to the level 3 program, we can state the following: Goal Achieved Amt Under 1 Yes 0 2 Yes 0 Yes 0 3 4 No 1.67 Yes 0 5 Amt Over 0 0 1333.33 0 16.67
Example 2: Non-Preemptive Goal Programming A non-preemptive goal program looks like min sum of weighted deviations for ALL goals subject to ALL functional constraints ALL goal constraints ALL nonfunctional constraints Suppose we weren't given the priority levels for the KarKleen problem. Suppose instead we were given the following information: The sales goal and the salary goal are equally important. The employee assignment (phone or door-to-door) goals are equally important. The sales goal is twice as important as the potential customers goal. The...