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Goal programmingGoal programming may be used to solve may be used to solve linear programs with multiple objectives, linear programs with multiple objectives, with each objective viewed as a "goal". with each objective viewed as a "goal".
In goal programming, In goal programming, ddii++ and and ddii
-- , , deviation deviation variablesvariables, are the amounts a targeted goal , are the amounts a targeted goal ii is overachieved or underachieved, is overachieved or underachieved, respectively.respectively.
The goals themselves are added to the The goals themselves are added to the constraint set with constraint set with ddii
++ and and ddii-- acting as the acting as the
surplus and slack variables.surplus and slack variables.
One approach to goal programming is to satisfy One approach to goal programming is to satisfy goals in a goals in a priority sequencepriority sequence. Second-priority . Second-priority goals are pursued without reducing the first-goals are pursued without reducing the first-priority goals, etc.priority goals, etc.
For each priority level, the objective function is For each priority level, the objective function is to minimize the (weighted) sum of the goal to minimize the (weighted) sum of the goal deviations. deviations.
Previous "optimal" achievements of goals are Previous "optimal" achievements of goals are added to the constraint set so that they are not added to the constraint set so that they are not degraded while trying to achieve lesser priority degraded while trying to achieve lesser priority goals. goals.
Step 1: Decide the priority level of each goal.Step 1: Decide the priority level of each goal.
Step 2: Decide the weight on each goal.Step 2: Decide the weight on each goal.
If a priority level has more than one If a priority level has more than one goal, for goal, for each goal each goal ii decide the decide the weight, weight, wwi i , to be placed , to be placed on the on the deviation(s), deviation(s), ddii
++ and/or and/or ddii--, from the goal., from the goal.
Step 3: Set up the initial linear program.Step 3: Set up the initial linear program.
Min Min ww11dd11++ + + ww22dd22
--
s.t. Functional Constraints, s.t. Functional Constraints, and Goal Constraints and Goal Constraints
Step 4: Solve the current linear program.Step 4: Solve the current linear program.
If there is a lower priority level, go to If there is a lower priority level, go to step 5. step 5. Otherwise, a final solution Otherwise, a final solution has been reached.has been reached.
Step 5: Set up the new linear program.Step 5: Set up the new linear program.
Consider the next-lower priority level goals Consider the next-lower priority level goals and formulate a new objective function based on and formulate a new objective function based on these goals. Add a constraint requiring the these goals. Add a constraint requiring the achievement of the next-higher priority level achievement of the next-higher priority level goals to be maintained. goals to be maintained. The new linear The new linear program might be:program might be:
Min Min ww33dd33++ + + ww44dd44
--
s.t. Functional Constraints,s.t. Functional Constraints, Goal Constraints, andGoal Constraints, and
ww11dd11++ + + ww22dd22
-- = = kk
Go to step 4. (Repeat steps 4 and 5 until Go to step 4. (Repeat steps 4 and 5 until all priority levels have been examined.) all priority levels have been examined.)
Conceptual Products is a computer company Conceptual Products is a computer company thatthatproduces the CP400 and CP500 computers. Many ofproduces the CP400 and CP500 computers. Many ofthe components used in the twothe components used in the two
computer models are produced incomputer models are produced inabundant supply by the abundant supply by the company.company.However, the memory modules,However, the memory modules,external hard drives, and cases areexternal hard drives, and cases arebought from suppliers.bought from suppliers.
The CP400 model uses two memory modules The CP400 model uses two memory modules andandno external hard drive, whereas the CP500 uses oneno external hard drive, whereas the CP500 uses onememory module and one external hard drive. Bothmemory module and one external hard drive. Bothmodels use one case.models use one case.
To solve graphically, first graph the functional To solve graphically, first graph the functional constraints. Then graph the first goal: constraints. Then graph the first goal: xx11 = 200. = 200. Note on the next slide that there is a set of points Note on the next slide that there is a set of points that exceed that exceed xx11 = 200 (where = 200 (where dd11
Now add Goal 1 as Now add Goal 1 as xx11 >> 200 and graph Goal 2: 200 and graph Goal 2:
xx11 + + xx22 = 500. Note on the next slide that there is = 500. Note on the next slide that there is still a set of points satisfying the first goal that also still a set of points satisfying the first goal that also satisfies this second goal (where satisfies this second goal (where dd22
Now add Goal 2 as Now add Goal 2 as xx11 + + xx22 >> 500 and Goal 3: 500 and Goal 3:
.2.2xx11 + .5 + .5xx22 = 250. Note on the next slide that no = 250. Note on the next slide that no points satisfy the previous functional constraints and points satisfy the previous functional constraints and goals goals andand satisfy this constraint. satisfy this constraint.
Thus, to Min Thus, to Min dd33--, this minimum value is achieved , this minimum value is achieved
when we Max .2when we Max .2xx11 + .5 + .5xx22. Note that this occurs at . Note that this occurs at xx11 = 200 and = 200 and xx22 = 400, so that .2 = 400, so that .2xx11 + .5 + .5xx22 = 240 or = 240 or dd33
Scoring Model for Job SelectionScoring Model for Job Selection
Considering only the Considering only the long-run careerlong-run career
advancementadvancement criterion: criterion:
• The The financial analyst position infinancial analyst position in
ChicagoChicago is the best decision alternative. is the best decision alternative. Considering only the Considering only the locationlocation criterion: criterion:
• The The accountant position in Denveraccountant position in Denver
is the best decision alternative.is the best decision alternative. Considering only the Considering only the stylestyle criterion: criterion:
• The The auditor position in Houstonauditor position in Houston is is the the best alternative.best alternative.
Steps Required to Develop a Scoring Steps Required to Develop a Scoring ModelModel
Step 1:Step 1: List the decision-making criteria. List the decision-making criteria. Step 2:Step 2: Assign a weight to each criterion. Assign a weight to each criterion. Step 3:Step 3: Rate how well each decision alternative Rate how well each decision alternative
satisfies each criterion.satisfies each criterion. Step 4:Step 4: Compute the score for each decision Compute the score for each decision
alternative.alternative. Step 5:Step 5: Order the decision alternatives from Order the decision alternatives from
highest score to lowest score. highest score to lowest score. The The alternative with the highest alternative with the highest score is the score is the recommended recommended alternative.alternative.
Assigning a Weight to Each CriterionAssigning a Weight to Each Criterion
CriterionCriterion ImportanceImportance WeightWeight Career advancementCareer advancement Very importantVery important 55 LocationLocation Average importanceAverage importance 33 ManagementManagement Somewhat importantSomewhat important 44 SalarySalary Average importanceAverage importance 33 PrestigePrestige Somewhat unimportantSomewhat unimportant 22 Job securityJob security Somewhat importantSomewhat important 44 Enjoyable workEnjoyable work Very importantVery important 55
Order the decision alternatives from highestOrder the decision alternatives from highestscore to lowest score. The alternative with the highestscore to lowest score. The alternative with the highestscore is the recommended alternative.score is the recommended alternative.
• The The accountant position in Denveraccountant position in Denver has the highest has the highest score and is the score and is the recommended decision alternativerecommended decision alternative..
• Note that the analyst position in Chicago ranks first Note that the analyst position in Chicago ranks first in 4 of 7 criteria compared to only 2 of 7 for the in 4 of 7 criteria compared to only 2 of 7 for the accountant position in Denver.accountant position in Denver.
• But when the weights of the criteria are considered, But when the weights of the criteria are considered, the Denver position is superior to the Chicago job.the Denver position is superior to the Chicago job.