This paper aims to develop a data envelopment analysis (DEA) based model for allocating input resources and deciding output targets in organizations with a centralized decision-making environment, for example, banks, police stations, and supermarket chains. The central decision-maker has an interest in maximizing the total output production and at the same time minimizing the total input consumption. Traditionally, all decision-making units (DMUs) can be easily projected to the efficient frontier, which is a mathematical feasibility; however, it does not guarantee the managerial feasibility during the planning period. In this paper, we will take potential limitations of input-output changes into account by building a difficulty coefficient matrix of modifying their production in the current production possibility set so that the solution guarantees managerial feasibilities. Three objectives, namely, maximizing aggregated outputs, minimizing the consumption of input resources, and minimizing the total difficulty coefficient, are proposed and incorporated into the formation of resource allocation and target setting scheme. Building on this, we combine DEA and multiobjective programming to solve the resource allocation and target setting problem. In the end, we apply our proposed approach to a real-world problem of sixteen chain hotels to illustrate the efficacy and usefulness of the proposed approach.
In many real scenarios, there are many complex organizations typically consisting of a number of individual production units, such as an education authority that may correspond to different schools. In such organizational environment, all the decision-making units (DMUs) fall under the umbrella of a central unit which has power to oversee them. Management of these large companies and public service organizations needs coordinating mechanism to better manage its resources to fulfill its mission. How to allocate resource and set targets is an important thing for the central unit. In this research we deal with the problem of allocating input resources and setting output targets which happen on the organizational level. The organization will have to set targets for each unit which should be logically linked to the resource that organization allocates to each unit simultaneously.
Resource allocation and/or target setting is a typical problem in organizations, which has become one of hottest topics and classical applications in economic and managerial field. Data envelopment analysis (DEA) has proved to be a powerful tool for determining the relative efficiencies of the decision-making units, which consume multiple inputs to produce multiple outputs. Since DEA was first introduced by Charnes et al. [
Normally, the output target is set simultaneously by allocating some input resources. Amirteimoori and Kordrostami [
In many real managerial applications, there are situations where all DMUs are controlled by a centralized decision-maker. Under the circumstances, the centralized decision-maker would like to increase the overall output production and decrease the total input resource consumption. Lozano and Villa [
Traditionally, the DEA literature thinks that the unit is able to reach any point on the efficient frontier easily, which is a mathematical feasibility; however, it does not guarantee managerial feasibility during the planning period. Moreover, it projects on the efficiency frontier, which declares that the DMU has the potential to achieve the input reducing and output increasing, which provide a direction of each DMU’s improvement. If the center decision-maker does not consider the effect of the restricted input reducing and output increasing, he will make incorrect decisions. In this text we will consider the potential restrictions of the change in the unit’s production during the planning period when the center decision-maker faces the resource allocation and target setting problem. To this end, we build a difficulty coefficient matrix of modifying the current production so that the solution guarantees managerial feasibilities. Further, three objectives are used to generate the final resource allocation and target setting plan, namely, maximizing aggregated output, minimizing the consumption of input resources, and minimizing the total difficulty coefficient. The DEA methodology and multiobjective programming are combined to solve the problem.
The remainder of this paper is organized as follows: in Section
At the starting point, we assume there exist
In the above model,
Model (
Its dual problem is also useful in realistic applications, which is expressed as follows:
The optimal objective function of model (
Traditional resource allocation and target setting model based on DEA methodology considers one unit at a time in relation to the other units. We can discover that a great deal of DEA models take the organizational setting into account. Cook et al. [
In this side, our approach clearly deviates from earlier papers which use the DEA model to solve the resource allocation and target setting problem. We will take the potential limitations of changing the unit’s production into account to guarantee managerial feasibility by projecting onto efficiency frontier during the planning period. In general, there are a lot of influence factors which restrict the change and thus subtract a proportion of the potential output. There is no doubt that we could set a “price” for the change. But there are factors which cannot use the price to measure the restriction of change, such as management efforts. We will use the difficulty coefficient to weigh the limitations of reaching the efficiency frontier. During the planning period, the central decision-maker will set the difficulty coefficient of reducing centesimal input and increasing centesimal output based on the managerial experience, the limit of budget, and manager’s preference. The central decider could use the analytic hierarchy process (AHP) and group decision to set the difficulty coefficient matrix. With larger value of the difficulty coefficient, it is more difficult to change the current situation of the production in the next prior.
In this section, we will propose three objectives, namely, maximizing aggregated output, minimizing the consumption of input resources, and minimizing the total difficulty coefficient, and create a model with these objectives for producing a resource allocation and target setting scheme.
Before formulating the models, the required notation needs to be introduced. Here we summarize these mathematical notations as follows.
According to the efficiency objective, it seeks to achieve “more-for-less”; in other words, if the resources are allocated, the amount of the total outputs of the DMUs should be maximized, or the target plan of output produced is given; the resources of all the DMUs consumed will be minimized. Also we will emphasize that it is very important to take potential limitations of the change during the planning time into account.
To illustrate resource allocation and target setting, we will consider the situations based on the following general constraints. The targeted outputs shall not decrease for any of the DMUs while the unit is able to reach any point on the efficient frontier. This idea is modelled as The allocated inputs shall not increase for any of the DMUs while the unit is able to reach any point on the efficient frontier. Similarly, we have
The multiobjectives we impose are as follows. Maximizing the total output of the all DMUs: Minimizing the total input of the all DMUs: Minimizing the total difficulty coefficient of the all DMUs:
Thus, the first objective function is to maximize the average percentage increase in the total outputs when the resource variable of each DMU is given. The second objective function is to minimize the average percentage decrease in the total inputs when the target of each unit is set. The third objective function is to minimize the total difficulty coefficient; in other words, it is to minimize the cost of change of the current production in the next period. Also it is to engage the managerial feasibility and coincidence of the company’s actual production. Traditionally, we maximize the total outputs and minimize the total inputs as the objective function. However, when the inputs or the outputs are more than one, the problem is a multiple-criteria problem. Now we choose the proportion form which has an interesting feature of the three objective functions which is invariance to the unit measurement. If any input variable or output variable changes its units of assessment, the objective function values do not change for any feasible solution. Because of the dimensionless quality of the ratios that are summed in each of the expressions, the optimal solutions of the proposed models will not change with changes in the units of assessment.
Based on the given some objective functions and some general constraints, we form a multiple objective model for allocating resource and setting target for each DMU. Let
Model (
As to the multiple objective linear programming of model (
When we take the total difficulty coefficient of changing the production situation which the company can accept into account, we will first fix the level of DMU’s difficulty coefficient; then we will adjust the difficulty coefficient which the company can accept to observe the change of the total output increased or total input saved. The priorities of the total maximum output and the total minimal input can be done by the user. In this paper we will consider two alternations. In the first alternative, the priority is that
Model (
Let
Model (
In addition to the proposed approach being the feature of unit invariant, obviously, an additional important feature is its flexibility. Thus, we can give a different relative weight which can be reflected in decision-maker’s subjective preference for different outputs, inputs, and difficulty coefficients. Assuming a weight
In this section, the advanced characteristic and practical values in our proposed model are demonstrated. To this end, we analyzed resource allocation and target setting simultaneously based on 2012 annual report of 16 sets of chain hotels located in Anhui, China. These hotels belong to the same chain with single central decision-making team to supervise all their branches. The company consists of 16 branches (DMUs), each of which uses two inputs and two outputs. The two output variables are room occupancy rate and turnover. Room occupancy rate is an index of hotel popularity defined by the ratio of number of occupied rooms to total number of guest rooms for the hotel business reference analysis. Turnover profits from room revenue, food and beverage income, and other income. On the other hand, two input variables are fixed assets and operating costs. Fixed assets are from facilities and equipment and other supplies that are used for hotel operations, while rent, staff salaries, office expenses, material consumption, and other expenses form operating costs. The raw data of the example studied in this paper is shown in Table
Data for the illustration example.
DMU | Inputs | Outputs | CCR efficiency | ||
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Fixed asset (104 Yuan) | Operating cost (104 Yuan) | Occupancy rate (%) | Turnover (104 Yuan) | ||
1 | 245.00 | 284.00 | 31.27 | 204.00 | 0.7874 |
2 | 384.00 | 304.00 | 38.22 | 624.00 | 1.0000 |
3 | 229.00 | 310.00 | 27.58 | 324.00 | 0.8134 |
4 | 272.00 | 311.00 | 44.09 | 410.00 | 1.0000 |
5 | 314.00 | 334.00 | 40.89 | 630.00 | 1.0000 |
6 | 266.00 | 393.00 | 37.51 | 494.00 | 1.0000 |
7 | 379.00 | 493.00 | 42.85 | 794.00 | 1.0000 |
8 | 565.00 | 504.00 | 56.54 | 569.00 | 0.8023 |
9 | 625.00 | 567.00 | 57.14 | 582.00 | 0.7223 |
10 | 624.00 | 627.00 | 86.30 | 884.00 | 0.9875 |
11 | 623.00 | 640.00 | 51.77 | 574.00 | 0.5899 |
12 | 599.00 | 647.00 | 91.50 | 769.00 | 0.9976 |
13 | 583.00 | 698.00 | 81.60 | 600.00 | 0.8635 |
14 | 668.00 | 722.00 | 83.20 | 869.00 | 0.8322 |
15 | 754.00 | 773.00 | 54.63 | 838.00 | 0.5733 |
16 | 671.00 | 776.00 | 84.40 | 756.00 | 0.7760 |
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First of all, we can use model (
Then, we solve model (
Resource allocation and target setting for alternative 1.
DMU | Fixed asset | Operating cost | Occupancy rate | Turnover | Efficiency |
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1 | 245.00 | 260.61 | 31.90 | 491.56 | 1.0000 |
2 | 384.00 | 304.00 | 38.22 | 624.00 | 1.0000 |
3 | 229.00 | 243.59 | 29.82 | 459.46 | 1.0000 |
4 | 272.00 | 311.00 | 44.09 | 410.00 | 1.0000 |
5 | 314.00 | 334.00 | 40.89 | 630.00 | 1.0000 |
6 | 266.00 | 393.00 | 37.51 | 494.00 | 1.0000 |
7 | 379.00 | 493.00 | 42.85 | 794.00 | 1.0000 |
8 | 565.00 | 504.00 | 62.63 | 997.63 | 1.0000 |
9 | 625.00 | 567.00 | 70.35 | 1116.86 | 1.0000 |
10 | 624.00 | 627.00 | 86.30 | 945.07 | 1.0000 |
11 | 623.00 | 640.00 | 78.57 | 1218.17 | 1.0000 |
12 | 599.00 | 647.00 | 91.50 | 863.22 | 1.0000 |
13 | 583.00 | 698.00 | 81.60 | 1084.03 | 1.0000 |
14 | 668.00 | 710.55 | 86.99 | 1340.25 | 1.0000 |
15 | 754.00 | 773.00 | 94.91 | 1472.11 | 1.0000 |
16 | 671.00 | 713.74 | 87.38 | 1346.27 | 1.0000 |
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1.0000 |
When we will pursue the total minimum input which has precedence over the total maximum outputs. We will use alternative 2 and solve model (
Resource allocation and target setting for alternative 2.
DMU | Fixed asset | Operating cost | Occupancy rate | Turnover | Efficiency |
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1 | 192.91 | 220.57 | 31.27 | 204.00 | 1.0000 |
2 | 384.00 | 304.00 | 38.22 | 624.00 | 1.0000 |
3 | 186.84 | 206.86 | 27.58 | 324.00 | 1.0000 |
4 | 272.00 | 311.00 | 44.09 | 410.00 | 1.0000 |
5 | 314.00 | 334.00 | 40.89 | 630.00 | 1.0000 |
6 | 266.00 | 393.00 | 37.51 | 494.00 | 1.0000 |
7 | 379.00 | 493.00 | 42.85 | 794.00 | 1.0000 |
8 | 359.49 | 406.71 | 56.54 | 569.00 | 1.0000 |
9 | 365.03 | 412.29 | 57.14 | 582.00 | 1.0000 |
10 | 552.54 | 623.61 | 86.30 | 884.00 | 1.0000 |
11 | 342.27 | 382.07 | 51.77 | 574.00 | 1.0000 |
12 | 564.48 | 645.42 | 91.50 | 769.00 | 1.0000 |
13 | 503.41 | 575.59 | 81.60 | 600.00 | 1.0000 |
14 | 536.84 | 604.26 | 83.20 | 869.00 | 1.0000 |
15 | 418.60 | 445.56 | 54.63 | 838.00 | 1.0000 |
16 | 520.68 | 595.34 | 84.40 | 756.00 | 1.0000 |
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1.0000 |
Then we will consider potential limitations of changing the current production. For simplicity and without loss of generality, we will analyze alternative 1. After allocating resource and setting output target, we will find all DMU projecting the efficient frontier. This is impossible to achieve for the next term’s production planning. If the decision-makers do not pay attention to the potential limitations and their abilities of changing the current production condition, they will make some incorrect decisions. So we want to take the problems of restrictions on projecting onto the efficiency frontier which might be very expensive into account. The decision-makers define the difficulty coefficient based on the company’s strategy. If the company can accept the total difficulty coefficient, the company will reach the efficient frontier easily in the next term. Also, the company can save the total input and gain more total output. Now the company identifies the difficulty coefficient matrix which is from the current production to next production condition.
DEA has been traditionally used for measuring the performance of individual units. Currently, DEA is also a useful approach for solving the problem of resource allocation and target setting popularly, since Lozano and Villa [
In this paper we use the CCR model; we also can extend the approach to other DEA models. For instance, we can obtain the scheme of resource allocation and target setting under the condition of variable returns to scale (VRS). In addition, the inputs and outputs are only measured by exact values in our work. Future research can extend the present approach to consider the cases where the inputs and outputs of DMUs are bounded data, ordinal data, and ratio-bounded data. Also, one can take the interaction among DMUs into account. By saying interaction, we mean that DMUs can cooperate, coordinate, and even negotiate towards the completion of the problem of resource allocation and target setting. That is a direction for further research where there are interdependencies between DMUs.
No potential conflicts of interest were reported by the authors.