Evaluating decision-making performance in a grid-computing environment using DEA

Evaluating decision-making performance in a grid-computing environment using DEA A. Fernández-Montes a, ⇑ , F. Velasco b , J.A. Ortega a a Departamento de Lenguajes y Sistemas Informáticos, Universidad de Sevilla, Spain b Departamento de Economía Aplicada I, Universidad de Sevilla, Spain article info Keywords: DEA Energy policies Efficiency Performance Return-to-scale abstract Energy saving involves two direct benefits: sustainability and cost reduction, both of which Information Technologies must be aware. In this context, clusters, grids and data centres represent the hungriest consumers of energy. Energy-saving policies for these infrastructures must be applied in order to maximize their resources. The aim of this paper is to compare how efficient these policies are in each location of a grid infrastructure. By identifying efficient policies in each location and the slack in inputs and outputs of the inefficient locations, Data Envelopment Analysis presents a very useful technique for comparing and improving efficiency level. This work enables managers to uncover any misuse of resources so that corrective action can be taken. Ó2012 Elsevier Ltd. All rights reserved. 1. Introduction Data Envelopment Analysis (DEA) is a nonparametric method to provide a relative efficiency assessment (called DEA efficient) for a group of decision-making units (DMU) or for productive efficiency (aka technical efficiency) with a multiple number of inputs and outputs. DEA was first proposed in Charnes, Cooper, and Rhodes (1978) and is commonly used in operations research and economics to empirically measure productive efficiency of DMUs. In order to determine whether a DMU is efficient is as easy as checking if the DMU is on the ‘‘frontier’’ of the production possibility set. In this way, DEA identifies a ‘‘frontier’’ on which the relative performance of all utilities in the sample can be compared. In recent years, a great variety of applications of DEA have appeared for the evaluation of the performances of many kinds of entities engaged in various contexts. DEA is especially useful when examining the nature of complex (often unknown) relations between multiple inputs and multiple outputs. DEA has been used both in private (Amirteimoori & Emrouznejad, 2012; Chiang & Hwang, 2010; Eilat, Golany, & Shtub, 2008; Emrouznejad, Parker, & Tavares, 2008) and in public contexts (Afonso, Schuknecht, & Tanzi, 2010; Gonzalez-Rodriguez, Velasco-Morente, & González-Abril, 2010). Regarding energy efficiency studies, DEA is commonly applied for the study and comparison of the performance and efficiency of energy industries, above all in the electricity industry, see (Pérez-Reyes & Tovar, 2009; Pombo & Taborda, 2006; Tovar, Javier Ramos-Real, & de Almeida, 2011; Vaninsky, 2006; Weyman-Jones, 1991). More recently, it has also been applied to IT companies in Serrano-cinca and Fuertes-calle (2005). Recently, it has also been popularized in environmental performance measurement due to its empirical applicability. In this work, DEA is used as a method to compare energy-consumption efficiency between each Grid’5000 location, where productive efficiency is measured as the energy consumed to run Grid’5000 jobs at each location. The rest of this paper is structured as follows: Section 2includes a brief introduction to DEA methodology used in this paper. Various on–off policies, designed to save energy are presented, and a comparison between current energy consumption and the results of each on–off policy are given in Section 3. The way in which jobs can be scheduled between resources is shown in Section 4. Software developed for testing and simulation is explained in Section 5and the dataset used for DEA is described and presented. Finally, in Sections 6 and 7, results are given and conclusions are drawn. 2. Data Envelopment Analysis DEA has been successfully applied to several sectors. The method establishes a best-practice production frontier (or envelop) based on the empirical input and output data on DMUs. It determines the level of production inefficiency of a DMU by projecting the unit onto the frontier. The original DEA model, introduced in Charnes et al. (1978), was set up with input orientation and assumes constant returns to scale (CRS). In an input-oriented model, the desired output level is achieved by minimizing the production inputs. The CRS 0957-4174/$ - see front matter Ó2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.eswa.2012.04.028 ⇑ Corresponding author. Tel.: +34 626215333; fax: +34 954557139. E-mail address: [email protected] (A. Fernández-Montes). Expert Systems with Applications 39 (2012) 12061–12070 Contents lists available at SciVerse ScienceDirect Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa assumption suggests that an increase in the amount of inputs utilized would lead to a proportional increase in the amount of outputs generated. The original model has been subsequently extended and numerous variations of DEA. For example, a DEA model can be set up to be output-oriented (Charnes, Cooper, & Rhodes, 1981), which attempts to maximize outputs with a set of available inputs. Another significant development of the DEA model by Banker, Charnes, and Cooper (BCC) (Banker, Charnes, & Cooper, 1984) allows for variable returns to scale (VRS). The VRS assumptionsuggests that an increase in the amount of inputs utilized can lead to a proportional or nonproportional change in the amount of outputs generated (Barkhi & Kao, 2010). 3. Energy policies at a glance Energy policies establish the managing of grid resources. While other research works try to reduce the make-span (Tseng, Chin, & Wang, 2009), the policies shown in this work try to describe and compute what to do with a resource once a job finishes its execution. Thus, each energy policy decides whether to leave a resource switched on or to switch it off depending on the purpose of the policy. The following subsections show energy policies implemented in Grid’5000 Toolbox. 3.1. Always On This is the simplest energy policy. It never switches resources off, under any condition, and hence resources stay idle, waiting for a new job to be run. Grid’5000 is currently running this way, and therefor these consumption results can be used for comparison with other energy policies in order to know how much energy would have been saved. The number of times resources are switched off or on are always zero, and therefor the stress upon the resource is minimal. 3.2. Always Off This policy always switches resources off, under any condition, and hence a resource starts shutting down immediately after any job finishes, and remains switched off. If a new job arrives, resources assigned have to be booted to run that job. This booting is carried out within reservation limits, and hence the user cannot make effective use of the resources until they are booted. This policy is usually the best regarding energy consumption results, but the number of times a resource is booted up and shut down is always maximum, and the stress produced on the hardware components is the highest, which is seldom desirable. 3.3. Switch off randomly This policy randomly switches resources off or leaves them idle by following a Bernoulli distribution whose parameter is equal to 0.5 when a job finishes. Hence, the number of times resources are switched off or left idle tends towards 50%, and results tend to be half-way between those of the Always Off and Always On policies (regarding the times resources are switched off and those of energy consumption). 3.4. Load Load can be defined as the percentage of resources that are On among the clusters of a location. This policy queries this information and leaves resources idle or switches resources off if the load when finishing a job is greater than a certain threshold or less than a threshold respectively. This threshold is a parameter selected from the GUI from 0 to 1. 3.5. Switch off T S T S is defined as the minimum time which ensures an energy saving if a resource is switched off between two jobs (Orgerie, Lefèvre, & Gelas, 2008). T S can be computed as follows: T S ¼E s P Off d tot þE On!Off þE Off!On P Idle P Off where P Off and P Idle refer to the power consumption in watts of a given resource when it is Off and Idle, respectively. E On?Off and E Off?On refers to the required energy in joules for a given resource to boot or switch it off respectively. E S is the energy saved during T S seconds. Finally, d tot =d On?Off +d Off?On , which is the total time a given resource needs for it to be switched off and switched on. This energy policy queries the agenda to check if the next submitted jobs are going to be run in the grid in less than T S . This policy computes the number of resources that are going to be Fig. 1. Configuration tab presenting setup parameters for a batch of simulations. 12062 A. Fernández-Montes et al. /Expert Systems with Applications 39 (2012) 12061–12070 needed within a time period less than T S , and leaves idle or shuts resources down of the job which has just finished, accordingly. In this way, the simulator attempts to minimize the cycles of booting up and shutting down when these cycles are not going to save energy. 3.6. Exponential The Exponential distribution, denoted by Exp(k), describes the time between events in a Poisson process, i.e. a process in which events occur continuously and independently at a constant average rate (1/k). Under the hypothesis that the arrival of new jobs follows an Exponential distribution, this energy policy attempts to predict the arrival of new jobs. Thus, to compute the kparameter, every time a job finishes, then the mean time between the last jobs is computed, denoted by l . Hence, k=1/ l according to the of method of maximum likelihood. The probability of the arrival of a new job is then computed by means of the Exponential cumulative density function (cdf) as cdf ðT s Þ¼1e T s = l . Therefore, given a threshold value: if cdfðT s ÞPthreshold then leave resources Idle if cdfðT s Þ<threshold then switch resources Off  3.7. Gamma The Gamma distribution, denoted by C (h, j ), is frequently used as a probability model for waiting times, and is a more general model than that given by the Exponential. Under the hypothesis that the arrival of new jobs follows a Gamma distribution, this energy policy attempts to predict the arrival of new jobs. The parameters computed every time a job finishes are: Number of resources available, as resourcesAvailable. These are the resources that are Idle and ready to accept new jobs. Mean resources used by the last jobs, as meanResources. The total number of resources used by the last jobs is computed and divided by the number of jobs. The number of last jobs number is a selected window size. Mean duration of these last jobs, as meanDuration. The sum of the duration of the last jobs is computed and divided by the number of the last jobs. The floor of resourcesAvailable/meanResources,asz. The parameters of the Gamma distribution are then estimated as: h=1/meanDuration and j =z+ 1. The probability of the arrival of a new job is then computed by means of the cumulative density function (cdf) with Fig. 2. Statistics tab presenting results for a batch of simulations. A. Fernández-Montes et al. / Expert Systems with Applications 39 (2012) 12061–12070 12063 cdfðT s Þ¼ c ð j ;T s =hÞ C ð j Þ Hence, given a threshold value: if cdfðT s ÞPthreshold then leave resources Idle if cdfðT s Þ<threshold then switch resources Off  4. Arranging policies at a glance Arranging policies establish the arrangement of jobs for their execution. A job can be moved from a set of resources to another, or a planned job execution can even be moved in time in order to take advantages of resources that are already switched on. Do Nothing (DN): Neither does this policy move jobs in time nor from one resource to another; jobs are executed as defined in the agenda. This is the current behaviour in Grid’5000. The combination of this arranging policy with the energy policy Always On in a simulation offers the current Grid’5000 behaviour, and includes results of energy consumption. Simple Aggregation of Jobs (SA): This policy attempts to find resources available (Idle) for new jobs. In this way, if a job is assigned to a set of resources which are Off and some resources are already switched on and available, we can save the time and Table 1 Summary of inputs and outputs. Location Outputs Inputs Saved energy (kW h) # Jobs deployed # Resources # Bootings Always Off Bordeaux 128,697 345,218 650 4,036,514 Lille 238,159 62,451 618 327,408 Lyon 57,715 134,719 322 927,472 Nancy 94,932 73,934 574 1,668,946 Orsay 132,518 89,048 684 2,111,974 Rennes 152,832 57,987 714 2,328,890 Sophia 48,848 57,533 568 2,337,336 Toulouse 86,531 165,995 434 1,754,930 Random Bordeaux 115,539 345,218 650 2,225,174 Lille 220,282 62,451 618 168,398 Lyon 51,771 134,719 322 494,442 Nancy 64,407 73,934 574 904,920 Orsay 105,075 89,048 684 1,141,004 Rennes 141,222 57,987 714 1,205,530 Sophia 39,918 57,533 568 1,198,338 Toulouse 71,738 165,995 434 922,932 Load Bordeaux 127,089 345,218 650 3,675,094 Lille 238,159 62,451 618 327,408 Lyon 57,708 134,719 322 926,028 Nancy 74,616 73,934 574 1,176,234 Orsay 125,703 89,048 684 1,922,154 Rennes 152,832 57,987 714 2,328,890 Sophia 41,063 57,533 568 1,475,640 Toulouse 86,057 165,995 434 1,667,222 T s Bordeaux 127,018 345,218 650 2,238,318 Lille 236,793 62,451 618 299,846 Lyon 57,299 134,719 322 538,154 Nancy 90,771 73,934 574 1,297,252 Orsay 130,825 89,048 684 1,384,922 Rennes 152,226 57,987 714 1,392,750 Sophia 46,332 57,533 568 1,271,836 Toulouse 85,250 165,995 434 876,026 Exponential Bordeaux 119,779 345,218 650 1,574,410 Lille 237,688 62,451 618 122,680 Lyon 56,349 134,719 322 612,766 Nancy 92,168 73,934 574 1,168,646 Orsay 127,303 89,048 684 1,387,566 Rennes 152,141 57,987 714 1,770,858 Sophia 48,360 57,533 568 1,847,484 Toulouse 86,203 165,995 434 671,122 Gamma Bordeaux 67,374 345,218 650 1,141,048 Lille 159,213 62,451 618 884 Lyon 31,532 134,719 322 131,106 Nancy 18,833 73,934 574 156,116 Orsay 61,581 89,048 684 623,515 Rennes 116,158 57,987 714 644,109 Sophia 20,017 57,533 568 510,400 Toulouse 39,395 165,995 434 153,326 12064 A. Fernández-Montes et al. /Expert Systems with Applications 39 (2012) 12061–12070 the energy needed for them to be switched on. Notice that this policy does not change start or stop times, and hence is transparent to users. 5. Methodology In order to compare energy efficiency between the locations of the Grid’5000, a software simulator has been developed. Grid’5000 Toolbox 1 replays the progress of the real grid regarding the operation of jobs and resources. Grid’5000 Toolbox is able to compute energy consumption of Grid’5000, and enables the user to establish several parameters including: (a) simulation start-time, (b) simulation stop-time, (c) location, (d) energy policy, and (e) arranging policy. These parameters can be set up through the Configuration tab as shown in Fig. 1. The simulator operation is based on an agenda where jobs are registered, and on a list of resources representing the real resources at the sites. The simulator queries the agenda from simulation start-time to simulation stop-time. Each query is related to current simulation time (the moment in past-time the software is replaying), and hence the agenda seeks jobs and events that occur at given current time. Once the agenda returns new events, the simulator processes them and changes the states of the resources as would be needed for execution in the real world, whilst taking into account the policies selected in order to manage resources and jobs. The energy consumed is computed step by step by means of the information on energy consumption of each resource and on the resource states detailed in the resource list. The results of simulation executions are stored on a spreadsheet where researchers can find details about consumption, the number of times the resources are shut down and booted up, the comparison between minimal energy consumable and current energy consumed, etc. Results are also shown in the Statistics tab in a more visual way (see Fig. 2). A battery of tests has been performed in order to compute energy-saving results based on: One period of 12 months. From 1st January to 31st December 2008. Two arranging policies, Do Nothing and Simple Aggregation of Jobs. The seven energy policies listed in Section 3. Various values of several parameters as follows: 1. Load policy. Load threshold parameter from 0.0 to 1 in steps of 0.3. A total of four scenarios. 2. Exponential and Gamma. Threshold probability parameter from 0.0 to 1 in steps of 0.3, and window size from 2 0 to 2 8 . Hence there are 36 different scenarios for each policy. From the 162 setups run, the best energy savers have been selected of each policy. From computed results, we select the following inputs and outputs to measure relative efficiency between locations: Inputs: 1. The number of resources at the location. This parameter remains unchanged between simulations. Resources are the entities that run jobs. 2. The number of times resources have been switched off and booted during the simulation. Each energy policy shows different behaviour when a job finishes, and therefore this input changes between each energy policy simulated. Outputs: 1. The energy saved, in kW h, using a given energy policy. This is the amount of energy that the location would save if a given energy policy were applied. 2. The number of jobs deployed at each location. The following table shows the summary of inputs and outputs for each energy policy for which the DEA methodology is computed using, Coelli software (Coelli, 1996) due to its simplicity usage. Results are compared with those produced by other tools, such as Benchmarking library in R language (Bogetoft & Otto, 2010). 6. Input-orientated DEA results The results computed are input orientated since firms are able to modify their inputs, and hence our study is focused on reducing inputs while maintaining the level of outputs (see Table 1). Table 2 Summary of DEA results for CRS, VRS, and scale efficiency. 1 This software can be downloaded and executed from the web of the Idinfor research group (Idinfor, 2011). A. Fernández-Montes et al. / Expert Systems with Applications 39 (2012) 12061–12070 12065 Table 2 shows the results generated by the DEA tool (Coelli, 1996) for an input-orientated DEA with 2 inputs, 2 outputs and 8 firms (locations 2 ), and these are grouped by energy policy. CRSTE (constant returns-to-scale technical efficiency), VRSTE (variable returns-to-scale technical efficiency) and Scale (scale efficiency) results are shown. Mean and standard deviation are computed for each energy policy and each location. Results in Table 2 and Fig. 3 show that the most efficient energy policies are those of Exponential and Gamma (Sections 3.6 and 3.7) in terms of VRSTE ð x¼0:817Þ, followed by the Load and Always Off energy policies ð x¼0:815Þ. On the other hand, the overall results of Random policy show this to be the least efficient ð x¼0:754Þ. In terms of dispersion, the least dispersion is reached using the Load policy ( r = 0.172), which indicates that this policy works homogeneously for any of the policies. Fig. 3 shows a graphical comparison of scale efficiency per energy policy. In the analysis of locations, it can be observed that Bordeaux, Lille and Lyon are the most efficient locations (VRSTE equals 1.000 for these policies), followed by Toulouse, and that the least efficient locations are Sophia and Orsay, followed by Nancy and Rennes. In terms of dispersion, Bordeaux, Lille and Lyon have the most homogeneous behaviour between policies, followed by Sophia, with Toulouse being the location whose performance is the most dispersed between policies, followed by Sophia, Rennes and Nancy. Fig. 4 shows this graphical comparison of VRSTE per locations. As a consequence of these analyses, corrections on inputs and outputs can be carried out. Table 3 shows peers per location, including weights and corrections proposed per location/policy. Notice that the type of correction (increase or decrease) remains the same within each location, which constitutes further confirmation of the validity of these corrections. For example, the proposed corrective actions for Nancy are: increase the number of jobs deployed, decrease the number of resources (as they are underused) and decreasing the number of power cycles (since the policies are not working as efficiently as those in other locations). By taking into account that certain locations are underused, the system manager could better balance the workload through the relocation of jobs from efficient locations to underused Bordeaux 100% Lille 100% Lyon 100% Nancy 65% Orsay 63% Rennes 68% Sophia 57% Toulouse 95% 0% 20% 40% 60% 80% 100% 120% Scale efficiency VR Scale Efficiency Comparison by Locations Fig. 4. Comparison of locations VR scale technical efficiency. Always Off 81.20% Random 79.40% Load 81.50% Ts 81.00% Exponential 81.70% Gamma 81.70% 78.00% 78.50% 79.00% 79.50% 80.00% 80.50% 81.00% 81.50% 82.00% Scale Efficiency VR Scale Efficiency Comparison by Energy policy Fig. 3. Comparison of energy policies for VR scale technical efficiency. 2 B, Li, Ly, N, O, R, S, and T stand for Bordeaux, Lille, Lyon, Nancy, Orsay, Rennes, Sophia, and Toulouse, respectively. 12066 A. Fernández-Montes et al. /Expert Systems with Applications 39 (2012) 12061–12070 locations. The system manager could also unplug a number of resources at underused locations, in the search for a threshold which guarantees both satisfaction of users and energy saving objectives. 6.1. Detailed analysis of Always Off energy policy technical efficiency Sophia is selected to illustrate this energy policy. Sophia is the least efficient location in general, and also the least efficient Table 3 Peers per location and per energy policy and correction proposals. Policy Peers Corrections Jobs Resources Bootings Bordeaux Alwz. Off B (1.000) MMM Random B (1.000) MMM Load B (1.000) MMM T s B (1.000) MMM Exp. B (1.000) MMM Gamma B (1.000) MMM Summary Bordeaux MMM Lille Alwz. Off Li (1.000) MMM Random Li (1.000) MMM Load Li (1.000) MMM T s Li (1.000) MMM Exp. Li (1.000) MMM Gamma Li (1.000) MMM Summary Lille MMM Lyon Alwz. Off Ly (1.000) MMM Random Ly (1.000) MMM Load Ly (1.000) MMM T s Ly (1.000) MMM Exp. Ly (1.000) MMM Gamma Ly (1.000) MMM summary Lyon MMM Nancy Alwz.Off Li (0.206) Ly (0.794) N.. Random Li (0.075) Ly (0.925) N.. Load Li (0.221) Ly (0.779) N.. T s Li (0.186) Ly (0.814) N.. Exp. Li (0.198) Ly (0.802) N.. Gamma Li (0.207) Ly (0.793) N.. Summary Lille and Lyon N.. Orsay Alwz. Off Li (0.415) Ly (0.585) N.. Random Li (0.316) Ly (0.684) N.. Load Li (0.377) Ly (0.623) N.. T s Li (0.410) Ly (0.590) N.. Exp. Li (0.391) Ly (0.609) N.. Gamma Li (0.235) Ly (0.765) N.. Summary Lille and Lyon N.. Rennes Alwz.Off Li (0.527) Ly (0.473) N.. Random Li (0.531) Ly (0.469) N.. Load Li (0.527) Ly (0.473) N.. T s Li (0.529) Ly (0.471) N.. Exp. Li (0.528) Ly (0.472) N.. Gamma Li (0.663) Ly (0.337) N.. summary Lille and Lyon N.. Sophia Alwz. Off Ly (1.000) N.. Random Ly (1.000) N.. Load Li (0.065) Ly (1.000) N.. T s Ly (1.000) N.. Exp. Ly (1.000) N.. Gamma Ly (1.000) N.. Summary Lyon N.. Toulouse Alwz. Off B (0.179) Li (0.089) Ly (0.732) N.. Random B (0.167) Li (0.055) Ly (0.777) N.. Load B (0.179) Li (0.088) Ly (0.733) N.. T s B (0.178) Li (0.086) Ly (0.735) N.. Exp. T (1.000) MMM Gamma T (1.000) MMM Summary Bordeaux, Lille, Lyon, Toulouse N.. A. Fernández-Montes et al. / Expert Systems with Applications 39 (2012) 12061–12070 12067 performing under the Always Off energy policy. The corrective actions recommended for this location and policy are detailed in Table 4. This location presents a CRS technical efficiency of 0.303 and a VRS technical efficiency of 0.567, and hence in order to achieve overall efficiency and to belong to the efficient frontier it must reduce input and increase output. This means that number of bootings and shuttings should be reduced by in 1.4 million (60%), and, most importantly 246 resources (43%) should be removed. In addition, these measures have to be followed by an increase of 77,186 (+134%) in the number of jobs run at this location and a reduction of 8867 kW h (18%) in energy consumption. The peer for this location is Lyon, which belongs to the segment of the production frontier where Sophia has to tend. Within these new dimensions, Sophia will make the most of its resources and will become efficient in the means of production. The other nonefficient locations should be corrected in a similar way. 6.2. Detailed analysis of Random energy policy technical efficiency Orsay is selected to illustrate this energy policy although it is not the least efficient location for this energy policy. The corrective actions recommended for this location and policy are detailed in Table 5. Orsay presents a CRS technical efficiency of 0.303 and a VRS technical efficiency of 0.521, and hence in order to achieve overall efficiency and to belong to the efficient frontier it must reduce input and increase output. This means that the number of bootings and shuttings in must be reduced by 749,696 (65%), and most importantly, 268 resources (39%) should be removed. Table 4 Corrections proposed for Sophia under the Always Off energy policy. Results for firm: Sophia Technical efficiency = 0.567 Scale efficiency = 0.535 (irs) Projection summary Variable Original value Radial movement Slack movement Projected value Output Saved energy 48,848 0 8867 57,715 Output # Jobs 57,533 0 77,186 134,719 Input # Resources 568 246 0 322 Input # Bootings 2,337,336 1,012,296 397,567 927,472 Listing of peers Peer Lambda weight Lyon 1.000 Table 5 Corrections proposed for Orsay under the Random energy policy. Results for firm: Orsay Technical efficiency = 0.608 Scale efficiency = 0.858 (irs) Projection summary Variable Original value Radial movement Slack movement Projected value Output Saved energy 105,075 0 0 105,075 Output # Jobs 89,048 0 22,811 111,859 Input # Resources 684 268 0 415 Input # Bootings 1,141,004 447,675 302,021 391,307 Listing of peers Peer Lambda weight Lille 0.316 Lyon 0.684 Table 6 Corrections proposed for Nancy under the Load energy policy. Results for firm: Nancy Technical efficiency = 0.675 Scale efficiency = 0.687 (irs) Projection summary Variable Original value Radial movement Slack movement Projected value Output Saved energy 74,616 0 22,948 97,564 Output # Jobs 73,934 0 44,823 118,757 Input # Resources 574 186 0 387 Input # Bootings 1,176,234 382,423 0 793,810 Listing of peers: Peer Lambda weight Lille 0.221 Lyon 0.779 12068 A. Fernández-Montes et al. /Expert Systems with Applications 39 (2012) 12061–12070 In addition, these measures have to be followed by an increase of 22,811 (+25%) in jobs run at this location. The peers for this location are Lyon and Lille, which both belong to the segment of the production frontier where Orsay has to tend. Within these new dimensions, Orsay will make the most of its resources and will become efficient in the means of production. The other non-efficient locations should be corrected in a similar way. 6.3. Detailed analysis of Load energy policy technical efficiency Nancy is selected to illustrate this energy policy although it is not the least efficient location for this energy policy. The corrective actions for this location and policy are detailed in Table 6. Nancy presents a CRS technical efficiency of 0.464 and a VRS technical efficiency of 0.675, and hence in order to achieve overall efficiency and to belong to the efficient frontier it must reduce input and increase output. This means that the number of bootings and shuttings must be reduced by 382,423 (32%), and, most importantly 186 resources (32%) should be removed. In addition, these measures have to be followed by an increase of 44,823 (+60%) in the jobs run at this location and a reduction of 22,948 kW h (+30%) in energy consumption. The peers for this location are Lyon and Lille, which both belong to the segment of the production frontier where Nancy has to tend. Within these new dimensions, Nancy will make the most of its resources and will become efficient in the means of production. The other non-efficient locations should to be corrected in a similar way. Table 7 Corrections proposed for Toulouse under the T s energy policy. Results for firm: Toulouse Technical efficiency = 0.937 Scale efficiency = 0.999 (irs) Projection summary Variable Original value Radial movement Slack movement Projected value Output Saved energy 85,250 0 0 85,250 Output # Jobs 165,995 0 0 165,995 Input # Resources 434 27 0 406 Input # Bootings 876,026 55,393 0 820,632 Listing of peers Peer Lambda weight Lille 0.086 Lyon 0.735 Bordeaux 0.178 Table 8 Corrections proposed for Rennes under the Exponential energy policy. Results for firm: Rennes Technical efficiency = 0.670 Scale efficiency = 0.868 (irs) Projection summary Variable Original value Radial movement Slack movement Projected value Output Saved energy 152,141 0 0 152,141 Output # Jobs 57,987 0 38,556 96,543 Input # Resources 714 235 0 478 Input # Bootings 1,770,858 584,429 832,549 353,879 Listing of peers Peer Lambda weight Lille 0.528 Lyon 0.472 Table 9 Corrections proposed for Nancy under the Gamma energy policy. Results for firm: Nancy Technical efficiency = 0.667 Scale efficiency = 0.608 (irs) Projection summary Variable Original value Radial movement Slack movement Projected value Output Saved energy 18,832 0 39,073 57,906 Output # Jobs 73,934 0 45,857 119,791 Input # Resources 574 190 0 383 Input # Bootings 156,116 51,909 0 104,206 Listing of peers Peer Lambda weight Lyon 0.793 Lille 0.207 A. Fernández-Montes et al. / Expert Systems with Applications 39 (2012) 12061–12070 12069