(a) A site manager has a gang of carpenters, a gang of steel fixers and a gang o
ID: 1714296 • Letter: #
Question
(a) A site manager has a gang of carpenters, a gang of steel fixers and a gang of concreters working under him. There are three jobs on the site. The first job requires the carpenter gang and then the concreter gang. The second job requires the steel fixer gang and then the concreter gang. The third job requires the carpenter gang, then the steel fixer gang and then the concreter gang. One gang can only work on one job at one time. The time required by the gangs on each job is given by Carpenter gang Steel fixer gang Concreter gang ½day 1 day 1% days 2 days Job 1 ob 2 ob 3 3 days 4 days 5 days Help the site manager to minimize the completion time of these three jobs by formulating the problem as a mixed integer programming model. (You do not have to solve the model formulated) (18 marks) Explain the consistency and inconsistency of a reciprocal matrix in the Analytic Hierarchy Process (AHP). How is the matrix's largest eigenalues) related to the consistency? (No mathematical proof is required) (b) (9 marks) In the application of AHP, what type of reciprocal matrix (consistent or inconsistent) is usually used? Why? (6 marks)Explanation / Answer
Answer b:
Analytic Hierarchy Process (AHP) is one of Multi Criteria decision making method that was originally developed by Prof. Thomas L. Saaty. In this method we find paired comparison based on derived ratio scales. The input can be obtained from actual measurement such as price, weight etc., or from subjective opinion such as satisfaction feelings and preference. In this method small inconsistency is allowed as the solution is not always consistent.
The consistency and inconsistency is explained with the help of example. Suppose we need to buy a bicycle and we have a choice of two bicycles that out of two we need to purchase one. Based on AHP we have to follow following steps.
1. To develop a model for the decision based on hierarchy of goals and criterias.
2. Derive priortities for the criteria developed. The importance of criteria are compared pairwise with respect to the desired goal to derive their weights. In short to check our judgements are consistent.
3. Derive priorities and alternatives with respect to each criteria. That is to check and adjust the consistency as required.
4. Derive overall priorities. That is the alternative with highest overall priorities constitues the best choice.
5. Perform sensitivity analysis. A study of how changes in the weights of the criteria could affect the result.
6. Make final decision.
So as an example our goal is to purchase a bicycle, the criterias are based on cost, easiness in riding {whether the bicycle consists of gear or non gear and alternative whether out of two bicycles which one to purchase.
The idea of consistency and inconsistency in AHP is best illustrated by examples as follows. If we prefer an
apple twice as much than a pear and a pear twice as much than an orange; how much would you prefer an apple with respect to an orange? The mathematically consistent answer is 4. Similarly, in a comparison matrix criteria, if we provide a value of 2 to the first criterion over the second and a assign a value of 3 to the second criterion with respect to the third one, the value of preference of the first criterion with respect to the third one should be 2 x 3 = 6. However, if the decision-maker has assigned a value such as 4, 5, or 7, there would be a certain level of inconsistency in the matrix of judgments. Some inconsistency is expected and allowed in AHP analysis. So how much inconsistency is allowed for that AHP analysis calculates the terms like consistency ratio [CR] which is ratio of consistency index CI of matrix in question to CI of random or reciprocal matrix RI.
So consistency ratio CR = CI / RI
As per Satty consistency ratio of 0.10 or less is acceptable to continue the AHP analysis. If the consistency ratio is greater than 0.10, it is necessary to revise the judgments to locate the cause of the inconsistency and correct it.
So from above equation it can be said that in the application of AHP consistent reciprocal matrix is used as it is based on consistency ratio CR.
The largest eigen value [lmax] related to consistency can be obtained by following equation:
CI = [lmax - n] / [n-1]
where n is number of compared elements [in our case of bicycle example it is alternative 2].
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.