A firm will decide on the percentage order-quantity...

A firm will decide on the percentage order-quantity allocation among three vendors, A, B or C. Each vendor is willing and has the capacity to satisfy any portion of the buyer's need at the quoted price. The criteria that are considered are quality, price, lead time, and service. Criteria are independent. The preference matrix is given as: $$ \begin{array}{l|l|l|l|l} \hline & \text { Price } & \text { Quality } & \text { Lead time } & \text { Service } \\ \hline \text { Price } & 1 & 1 / 7 & 3 & 5 \\ \hline \text { Quality } & 7 & 1 & 8 & 9 \\ \hline \text { Lead time } & 1 / 3 & 1 / 8 & 1 & 3 \\ \hline \text { Service } & 1 / 5 & 1 / 9 & 1 / 3 & 1 \\ \hline \end{array} $$ From the pairwise comparisons the weights for each of the vendor with respect to criteria can be presented as: $$ \begin{array}{l|l|l|l} \hline & \text { Vendor A } & \text { Vendor B } & \text { Vendor C } \\ \hline \text { Price } & 0.195 & 0.717 & 0.088 \\ \hline \text { Quality } & 0.731 & 0.081 & 0.188 \\ \hline \text { Lead time } & 0.058 & 0.663 & 0.278 \\ \hline \text { Service } & 0.731 & 0.188 & 0.081 \\ \hline \end{array} $$ (a) What is the desired solution for the firm? Check the preference matrix for consistency. (b) When comparing the vendors under the price criterion, how do you phrase the question to the decision maker? Give an example.

A firm will decide on the percentage order-quantity allocation among three vendors, A, B or C. Each vendor is willing and has the capacity to satisfy any portion of the buyer's need at the quoted price. The criteria that are considered are quality, price, lead time, and service. Criteria are independent. The preference matrix is given as:
$$
\begin{array}{l|l|l|l|l}
\hline & \text { Price } & \text { Quality } & \text { Lead time } & \text { Service } \\
\hline \text { Price } & 1 & 1 / 7 & 3 & 5 \\
\hline \text { Quality } & 7 & 1 & 8 & 9 \\
\hline \text { Lead time } & 1 / 3 & 1 / 8 & 1 & 3 \\
\hline \text { Service } & 1 / 5 & 1 / 9 & 1 / 3 & 1 \\
\hline
\end{array}
$$
From the pairwise comparisons the weights for each of the vendor with respect to criteria can be presented as:
$$
\begin{array}{l|l|l|l}
\hline & \text { Vendor A } & \text { Vendor B } & \text { Vendor C } \\
\hline \text { Price } & 0.195 & 0.717 & 0.088 \\
\hline \text { Quality } & 0.731 & 0.081 & 0.188 \\
\hline \text { Lead time } & 0.058 & 0.663 & 0.278 \\
\hline \text { Service } & 0.731 & 0.188 & 0.081 \\
\hline
\end{array}
$$
(a) What is the desired solution for the firm? Check the preference matrix for consistency.
(b) When comparing the vendors under the price criterion, how do you phrase the question to the decision maker? Give an example.

Key Concepts

Preference Elicitation

Preference elicitation refers to the process of asking decision makers to compare criteria or alternatives in a structured way. It involves framing questions that enable the expression of relative importance, typically by asking which of two elements is more significant and by how much, thereby capturing the decision maker's judgments systematically.

Weight Derivation

Weight derivation involves calculating numerical values that represent the relative importance of each criterion or alternative based on pairwise comparisons. These derived weights are then used to aggregate scores in a manner that reflects the decision maker’s preferences.

Consistency Ratio

The consistency ratio is a measure used to determine whether the judgments in a pairwise comparison matrix are consistent. It indicates if the comparisons made by the decision maker are reliable; a low consistency ratio suggests that the comparisons are logical and coherent, while a high ratio can signify the need for revisiting and revising the evaluations.

Analytic Hierarchy Process (AHP)

AHP is a structured decision-making methodology used to organize and analyze complex decisions based on mathematics and psychology. It decomposes a decision problem into a hierarchy of more easily comprehended sub-problems, each of which can be analyzed independently, and then synthesizes the results to determine the best overall decision.

Pairwise Comparison Matrix

This is a tool used in AHP to compare the relative importance or performance of elements two at a time. Decision makers evaluate one element against another using a standardized scale, and the resulting matrix is used to derive weights or priority values for the criteria or alternatives.

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