American Institute of Mathematical Sciences

June  2020, 10(2): 127-142. doi: 10.3934/naco.2019043

Resource allocation and target setting based on virtual profit improvement

 1 Department of Mathematics, Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran 2 Faculty of Industrial Engineering and Management Sciences, Shahrood University of Technology, Shahrood, Iran 3 Operations and Information Management Department, Aston Business School, Aston University, Birmingham, B4 7ET, UK

Received  April 2018 Revised  July 2019 Published  September 2019

Fund Project: We thank four reviewers of this journal for their most constructive comments

One application of Data Envelopment Analysis (DEA) is the resource allocation and target setting among homogeneous Decision Making Units (DMUs). In this paper, we assume that all units are under the supervision and control of a central decision making unit, for instance chain stores, banks, schools, etc. The aim is to allocate available resources among units in a way that the so-called organisational overall "virtual profit" is maximized. Our method is highly flexible in decision making to achieve the goals of the Decision Maker (DM). The resulting production plans maintain the following characteristics: (1) the virtual profit of each unit is calculated with a common set of weights; (2) the selected weights for calculating the virtual profit prevent the virtual profit of the system from getting worse; (3) the virtual profits of less profitable units are improved as much as possible. The proposed method is illustrated with a simple numerical example and a real life application.

Citation: Jafar Sadeghi, Mojtaba Ghiyasi, Akram Dehnokhalaji. Resource allocation and target setting based on virtual profit improvement. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 127-142. doi: 10.3934/naco.2019043
References:

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References:
Farell's frontier before resource allocation
Farell's frontier after resource allocation in cases I and II
Data set and results of numerical example
 Case I Case II $DMU$ $I_1$ $I_2$ $O$ $I_1$ $I_2$ $O$ $I_1$ $I_2$ $O$ A 8 24 4 6.4 24 4.1 6.4 21.32 4.2 B 27 27 9 32.4 27 8.55 32.4 32.4 9 C 56 8 8 54.3 8 7.6 67.2 9.6 8 D 8 14 2 6.4 14 2.1 6.4 11.2 2.1 E 42 24 6 33.6 24 6.3 33.6 20.28 6.3 F 32 8 4 25.6 8 4.2 30.18 9.6 4.2 G 30 3 3 24 3 3.15 26.82 3.6 3.15 Central 203 108 36 182.7 108 36 203 108 36.95
 Case I Case II $DMU$ $I_1$ $I_2$ $O$ $I_1$ $I_2$ $O$ $I_1$ $I_2$ $O$ A 8 24 4 6.4 24 4.1 6.4 21.32 4.2 B 27 27 9 32.4 27 8.55 32.4 32.4 9 C 56 8 8 54.3 8 7.6 67.2 9.6 8 D 8 14 2 6.4 14 2.1 6.4 11.2 2.1 E 42 24 6 33.6 24 6.3 33.6 20.28 6.3 F 32 8 4 25.6 8 4.2 30.18 9.6 4.2 G 30 3 3 24 3 3.15 26.82 3.6 3.15 Central 203 108 36 182.7 108 36 203 108 36.95
Results of numerical example
 Case I Case II $DMU$ $p_j$ $p_j^\prime$ $\theta_j$ $\theta_j^\prime$ $p_j$ $p_j^\prime$ $\theta_j$ $\theta_j^\prime$ A -0.0457 -0.0239 1 1 -0.0457 -0.0091 1 1 B 0 0 1 1 0 0 1 1 C -0.0047 0 1 1 -0.0047 0 1 1 D -0.042 -0.0287 0.63 0.7 -0.042 -0.0151 0.59 0.73 E -0.0869 -0.0459 0.59 0.77 -0.0869 -0.0151 0.59 0.91 F -0.0307 -0.0037 0.75 0.97 -0.0307 -0.0099 0.69 0.9 G -0.0239 -0.0028 0.85 1 -0.0239 0 0.85 1 Central -0.234 -0.1049 0.77 0.9 -0.234 -0.0491 0.77 0.95
 Case I Case II $DMU$ $p_j$ $p_j^\prime$ $\theta_j$ $\theta_j^\prime$ $p_j$ $p_j^\prime$ $\theta_j$ $\theta_j^\prime$ A -0.0457 -0.0239 1 1 -0.0457 -0.0091 1 1 B 0 0 1 1 0 0 1 1 C -0.0047 0 1 1 -0.0047 0 1 1 D -0.042 -0.0287 0.63 0.7 -0.042 -0.0151 0.59 0.73 E -0.0869 -0.0459 0.59 0.77 -0.0869 -0.0151 0.59 0.91 F -0.0307 -0.0037 0.75 0.97 -0.0307 -0.0099 0.69 0.9 G -0.0239 -0.0028 0.85 1 -0.0239 0 0.85 1 Central -0.234 -0.1049 0.77 0.9 -0.234 -0.0491 0.77 0.95
Data set and results of numerical example
 Case I Case II $DMU$ $I_1$ $I_2$ $O_1$ $O_2$ $I_1$ $I_2$ $O_1$ $O_2$ $I_1$ $I_2$ $O_1$ $O_2$ 1 79.1 4.99 115.3 1.71 87.01 4.99 121.06 1.8 71.19 4.49 121.06 1.8 2 60.1 3.3 75.2 1.81 66.11 3.3 78.96 1.9 54.09 2.97 78.96 1.9 3 126.7 8.12 225.5 10.39 139.37 8.12 214.22 9.87 139.37 8.93 225.5 10.91 4 153.9 6.7 185.6 10.42 169.29 6.7 194.88 10.94 138.51 7.37 194.88 10.94 5 65.7 4.74 84.5 2.36 72.27 4.74 88.73 2.48 59.13 4.27 88.73 2.48 6 76.8 4.08 103.3 4.35 84.48 4.08 108.46 4.57 69.12 4.24 108.46 4.57 7 50.2 2.53 78.8 0.16 55.22 2.53 82.74 0.17 55.22 2.78 82.74 0.17 8 44.8 2.47 59.3 1.3 49.28 2.47 62.27 1.37 40.32 2.72 62.27 1.37 9 48.1 2.32 65.7 1.49 52.91 2.32 68.99 1.56 43.29 2.55 68.99 1.56 10 89.7 4.91 163.2 6.26 98.67 4.91 155.04 5.95 98.67 5.4 163.2 6.26 11 56.9 2.24 70.7 2.8 62.59 2.24 74.23 2.94 51.21 2.46 74.24 2.94 12 112.6 5.42 142.6 2.75 123.86 5.42 149.73 2.89 101.34 4.88 149.73 2.89 13 106.9 6.28 127.8 2.7 117.59 6.28 134.19 2.84 96.21 5.65 134.19 2.84 14 54.9 3.14 62.4 1.42 60.39 3.14 65.52 1.49 60.39 3.45 65.52 1.49 15 48.8 4.43 55.2 1.38 53.68 4.43 57.96 1.45 53.68 3.99 57.96 1.45 16 59.2 3.98 95.9 0.74 65.12 3.98 100.7 0.78 65.12 4.38 100.7 0.78 17 74.5 5.32 121.6 3.06 81.95 5.32 127.68 3.21 67.05 5.85 127.68 3.21 18 94.6 3.69 107 2.98 104.06 3.69 112.35 3.13 102.17 3.32 112.35 3.13 19 47 3 65.4 0.62 51.7 3 68.67 0.65 42.3 2.7 68.67 0.65 20 54.6 3.87 71 0.01 60.06 3.87 74.55 0.01 57.72 3.48 74.55 0.01 21 90.1 3.31 81.2 5.12 99.11 3.31 85.26 5.38 99.11 2.98 85.26 5.38 22 95.2 4.25 128.3 3.89 104.72 4.25 134.72 4.08 104.72 3.83 134.72 4.08 23 80.1 3.79 135 4.73 88.11 3.79 135.39 4.97 87.47 4.17 141.75 4.97 24 68.7 2.99 98.9 1.86 75.57 2.99 103.85 1.95 75.57 2.69 103.85 1.95 25 62.3 3.1 66.7 7.41 68.53 3.1 63.37 7.04 68.53 3.41 66.7 7.41 Central 1901.5 102.97 2586.1 81.72 2091.65 102.97 2663.52 83.42 1901.5 102.96 2692.66 85.14
 Case I Case II $DMU$ $I_1$ $I_2$ $O_1$ $O_2$ $I_1$ $I_2$ $O_1$ $O_2$ $I_1$ $I_2$ $O_1$ $O_2$ 1 79.1 4.99 115.3 1.71 87.01 4.99 121.06 1.8 71.19 4.49 121.06 1.8 2 60.1 3.3 75.2 1.81 66.11 3.3 78.96 1.9 54.09 2.97 78.96 1.9 3 126.7 8.12 225.5 10.39 139.37 8.12 214.22 9.87 139.37 8.93 225.5 10.91 4 153.9 6.7 185.6 10.42 169.29 6.7 194.88 10.94 138.51 7.37 194.88 10.94 5 65.7 4.74 84.5 2.36 72.27 4.74 88.73 2.48 59.13 4.27 88.73 2.48 6 76.8 4.08 103.3 4.35 84.48 4.08 108.46 4.57 69.12 4.24 108.46 4.57 7 50.2 2.53 78.8 0.16 55.22 2.53 82.74 0.17 55.22 2.78 82.74 0.17 8 44.8 2.47 59.3 1.3 49.28 2.47 62.27 1.37 40.32 2.72 62.27 1.37 9 48.1 2.32 65.7 1.49 52.91 2.32 68.99 1.56 43.29 2.55 68.99 1.56 10 89.7 4.91 163.2 6.26 98.67 4.91 155.04 5.95 98.67 5.4 163.2 6.26 11 56.9 2.24 70.7 2.8 62.59 2.24 74.23 2.94 51.21 2.46 74.24 2.94 12 112.6 5.42 142.6 2.75 123.86 5.42 149.73 2.89 101.34 4.88 149.73 2.89 13 106.9 6.28 127.8 2.7 117.59 6.28 134.19 2.84 96.21 5.65 134.19 2.84 14 54.9 3.14 62.4 1.42 60.39 3.14 65.52 1.49 60.39 3.45 65.52 1.49 15 48.8 4.43 55.2 1.38 53.68 4.43 57.96 1.45 53.68 3.99 57.96 1.45 16 59.2 3.98 95.9 0.74 65.12 3.98 100.7 0.78 65.12 4.38 100.7 0.78 17 74.5 5.32 121.6 3.06 81.95 5.32 127.68 3.21 67.05 5.85 127.68 3.21 18 94.6 3.69 107 2.98 104.06 3.69 112.35 3.13 102.17 3.32 112.35 3.13 19 47 3 65.4 0.62 51.7 3 68.67 0.65 42.3 2.7 68.67 0.65 20 54.6 3.87 71 0.01 60.06 3.87 74.55 0.01 57.72 3.48 74.55 0.01 21 90.1 3.31 81.2 5.12 99.11 3.31 85.26 5.38 99.11 2.98 85.26 5.38 22 95.2 4.25 128.3 3.89 104.72 4.25 134.72 4.08 104.72 3.83 134.72 4.08 23 80.1 3.79 135 4.73 88.11 3.79 135.39 4.97 87.47 4.17 141.75 4.97 24 68.7 2.99 98.9 1.86 75.57 2.99 103.85 1.95 75.57 2.69 103.85 1.95 25 62.3 3.1 66.7 7.41 68.53 3.1 63.37 7.04 68.53 3.41 66.7 7.41 Central 1901.5 102.97 2586.1 81.72 2091.65 102.97 2663.52 83.42 1901.5 102.96 2692.66 85.14
Results of numerical example
 Case I Case II $DMU$ $p_j$ $p_j^\prime$ $\theta_j$ $\theta_j^\prime$ $p_j$ $p_j^\prime$ $\theta_j$ $\theta_j^\prime$ 1 -0.0237 -0.0205 0.359 0.418 -0.0246 -0.0136 0.343 0.575 2 -0.0176 -0.0154 0.431 0.494 -0.0178 -0.0098 0.423 0.62 3 0 0 1 1 -0.0015 0 1 1 4 -0.0196 -0.0109 0.849 0.923 -0.0181 -0.0006 0.849 0.992 5 -0.0213 -0.0188 0.5 0.566 -0.023 -0.0138 0.436 0.618 6 -0.0127 -0.0085 0.712 0.809 -0.0128 -0.0036 0.709 0.915 7 -0.0146 -0.0128 0.078 0.095 -0.0143 -0.0154 0.078 0.085 8 -0.0125 -0.0107 0.431 0.502 -0.0126 -0.0092 0.427 0.566 9 -0.0113 -0.0093 0.516 0.57 -0.011 -0.0068 0.516 0.659 10 0 0 1 1 0 0 1 1 11 -0.0097 -0.0069 0.805 0.86 -0.0088 -0.0027 0.805 0.902 12 -0.0321 -0.0281 0.401 0.451 -0.0315 -0.0169 0.401 0.59 13 -0.0362 -0.0328 0.367 0.418 -0.0371 -0.0237 0.346 0.502 14 -0.019 -0.0174 0.366 0.414 -0.0194 -0.0214 0.346 0.37 15 -0.0225 -0.0211 0.393 0.444 -0.0248 -0.0236 0.306 0.353 16 -0.018 -0.0156 0.243 0.3 -0.019 -0.0207 0.232 0.247 17 -0.0159 -0.0118 0.628 0.757 -0.0175 -0.011 0.583 1 18 -0.0247 -0.0214 0.562 0.617 -0.0233 -0.0196 0.562 0.739 19 -0.0164 -0.0149 0.229 0.267 -0.017 -0.011 0.215 0.332 20 -0.0243 -0.0232 0.003 0.004 -0.0255 -0.0237 0.003 0.004 21 -0.02 -0.0164 0.729 0.799 -0.0185 -0.0153 0.729 1 22 -0.0182 -0.0137 0.677 0.735 -0.0173 -0.0125 0.677 0.914 23 -0.0038 0 1 1 -0.0031 0 1 1 24 -0.0146 -0.0117 0.573 0.617 -0.0138 -0.0107 0.573 1 25 0 0 1 1 0 0 1 1 central -0.4088 -0.3419 0.591 0.658 -0.4123 -0.2856 0.588 0.714
 Case I Case II $DMU$ $p_j$ $p_j^\prime$ $\theta_j$ $\theta_j^\prime$ $p_j$ $p_j^\prime$ $\theta_j$ $\theta_j^\prime$ 1 -0.0237 -0.0205 0.359 0.418 -0.0246 -0.0136 0.343 0.575 2 -0.0176 -0.0154 0.431 0.494 -0.0178 -0.0098 0.423 0.62 3 0 0 1 1 -0.0015 0 1 1 4 -0.0196 -0.0109 0.849 0.923 -0.0181 -0.0006 0.849 0.992 5 -0.0213 -0.0188 0.5 0.566 -0.023 -0.0138 0.436 0.618 6 -0.0127 -0.0085 0.712 0.809 -0.0128 -0.0036 0.709 0.915 7 -0.0146 -0.0128 0.078 0.095 -0.0143 -0.0154 0.078 0.085 8 -0.0125 -0.0107 0.431 0.502 -0.0126 -0.0092 0.427 0.566 9 -0.0113 -0.0093 0.516 0.57 -0.011 -0.0068 0.516 0.659 10 0 0 1 1 0 0 1 1 11 -0.0097 -0.0069 0.805 0.86 -0.0088 -0.0027 0.805 0.902 12 -0.0321 -0.0281 0.401 0.451 -0.0315 -0.0169 0.401 0.59 13 -0.0362 -0.0328 0.367 0.418 -0.0371 -0.0237 0.346 0.502 14 -0.019 -0.0174 0.366 0.414 -0.0194 -0.0214 0.346 0.37 15 -0.0225 -0.0211 0.393 0.444 -0.0248 -0.0236 0.306 0.353 16 -0.018 -0.0156 0.243 0.3 -0.019 -0.0207 0.232 0.247 17 -0.0159 -0.0118 0.628 0.757 -0.0175 -0.011 0.583 1 18 -0.0247 -0.0214 0.562 0.617 -0.0233 -0.0196 0.562 0.739 19 -0.0164 -0.0149 0.229 0.267 -0.017 -0.011 0.215 0.332 20 -0.0243 -0.0232 0.003 0.004 -0.0255 -0.0237 0.003 0.004 21 -0.02 -0.0164 0.729 0.799 -0.0185 -0.0153 0.729 1 22 -0.0182 -0.0137 0.677 0.735 -0.0173 -0.0125 0.677 0.914 23 -0.0038 0 1 1 -0.0031 0 1 1 24 -0.0146 -0.0117 0.573 0.617 -0.0138 -0.0107 0.573 1 25 0 0 1 1 0 0 1 1 central -0.4088 -0.3419 0.591 0.658 -0.4123 -0.2856 0.588 0.714

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