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Many times, a supply chain manager requires an objective performance measurement tool for evaluating the relative efficiency of decision-making units (DMUs) in supply chain. Here, a decision-making unit is a distinct unit within supply chain that has flexibility with respect to some of the decisions it makes, but not necessarily the total freedom with respect to these decisions. Here, we illustrate an objective performance measurement tool with an example. |
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| Simple Example: |
Consider a number of bank branches. Each branch has a single output measure (number of personal transactions done) and a single input measure (number of staff). The data is as follows: |
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| Branch |
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Personel Transactions |
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No. of Staff |
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| A |
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125,000 |
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18 |
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| B |
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44,000 |
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16 |
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| C |
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80,000 |
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17 |
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| D |
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23,000 |
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11 |
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To compare the efficiencies of various branches, we can use the ratio of number of Personal Transactions per Staff. The values of this ratio for the branches A, B, C and D are 6944, 2750, 4706 and 2091 respectively. This analysis implies that branch ‘A’ is the most efficient branch and can be used to set the target for other branches. |
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| More Complex Example: |
In the previous example, we had one input and one output. But if there are multiple inputs and multiple outputs, we cannot use simple ratio. Let us consider the following data wherein there are two inputs and one output. |
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| Branch |
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Personel Transactions |
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Bussiness Transactions |
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No. of Staff |
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| A |
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125,000 |
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50,000 |
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18 |
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| B |
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44,000 |
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20,000 |
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16 |
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| C |
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80,000 |
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55,000 |
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17 |
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| D |
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23,000 |
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12,000 |
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11 |
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Let us we find the ratio of number of business transactions per staff. The values of this ratio for the branches A, B, C and D are 2778, 1250, 3235 and 1091 respectively. Based on the ratio of number of personal transactions per staff, branch ‘A’ is the most efficient. Based on the ratio of number of business transactions per staff, branch ‘C’ is the most efficient. This renders our decision-making process difficult. |
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In this situation, we can use the optimization technique to solve this problem. The optimization model is given below: |
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| Subject to |
| 16 WS = 1 |
| 125000 WP + 50000 WB <= 18 WS |
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44000 WP + 20000 WB <= 16 WS |
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80000 WP + 55000 WB <= 17 WS |
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23000 WP + 12000 WB <= 11 WS |
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WP,WB and WS = 0 |
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Where WP – weight attached to personal transactions |
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WB – weight attached to business transactions |
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WS – weight attached to number of staff |
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In the above model for the calculation of the efficiency of a particular DMU, the weights are chosen so as to maximize its efficiency, thereby presenting the DMU in the best possible light. The above-model is solved using the CPLEX solver. The optimal values of WP, WB and WS are 0.00394, 0.0149 and 0.0625 respectively. The efficiency of each branch that is calculated as the ratio of weighted output to weighted input is given below: |
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| Branch |
Efficiency |
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| A |
1.00 |
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| B |
0.43 |
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| C |
1.00 |
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| D |
0.36 |
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From the above table, the branches ‘A’ and ‘C’ are equally efficient. Branch ‘D’ is the least efficient. Thus, the optimization problem helps us find the relative efficiency of various DMUs when there are multiple inputs and outputs. |
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