Performance Measurement of Using VRS Method Dr.G.Thirupati Reddy Professor, Dept of Mechanical Engineering, Sree Visvesvaraya Institute of Technology & Science, Mahabubnagar, Telengana state, INDIA Abstract Data envelopment analysis (DEA) is a linear programming based technique for measuring the relative performance of organizational units where the presence of multiple inputs and outputs makes comparisons difficult. Productivity improvement and cost control have become key objectives of SCCL coal mines in recent years. Selected various coal mines in SCCL and calculated relative efficiency of mines by using Data Envelopment Analysis (DEA) which helps to rank them based on their efficiency score. The efficiency score has been calculated based on approaches Variable Return to Scale (VRS) and comparison is made between ranking of the coal mines. The comparison is made For every inefficient coal mine, DEA identifies a set of corresponding efficient coal mines that can be utilized as benchmarks for improvement of performance and productivity. Methodology As a result many research works have carried out on productivity improvement in coal mines. Operation Research techniques like Linear Programming (LP), Non Linear Programming (NLP), Data Envelopment Analysis (DEA), Fuzzy systems, Stochastic Data Envelopment Analysis (SDEA) and Bench marking etc are very popular tools in productive improvement which can aggregate the input and output components in such situations for obtaining an overall performance measure to improve productivity. Productivity in Coal Mines (OMS) Historically and traditionally productivity in coal mines all over the world is being measured in terms of output per man shift (OMS). Thus, productivity (OMS) in underground mines is = Tonnage of coal produced / Manpower x Shifts For opencast mines also productivity is measured in terms of OMS, but additional efforts put to excavate the overburden is also taken into account, The OMS of opencast mines is determined as The envelopment surface will differ depending on the scale assumptions that underpin the model. Two scale assumptions are generally employed: constant returns to scale (CRS), and variable returns to scale (VRS). The latter encompasses both increasing and decreasing returns to scale. CRS reflects the fact that output will change by the same proportion as inputs are changed (e.g. a doubling of all inputs will double output); VRS reflects the fact that production technology may exhibit increasing, constant and decreasing returns to scale. VRS (Variable to Returns Scale) or BCC Model If the constraint is is adjoined, they are known as BCC (Banker, Cooper, 1984) models. This added constraint introduces an additional variable, into the (dual) multiplier problems. As this extra variable makes it possible to effect returns-to-scale evaluations (increasing, constant and decreasing). So the BCC model is also referred to as the VRS (Variable Returns to scale) model and distinguished form the CCR model which is referred to as the CRS (Constant Returns to Scale) model. The CRS model is designed with the assumption of constant returns to scale. This means that there is no assumption that any positive or negative economies of scale exist. It is assumed is that a small unit should be able to operate as efficiently as a large one that is, constant returns to scale. In order to address this, Banker, Charnes, and Cooper developed the BCC model. It is also referred as VRS model. In all variations of the DEA models, [17] the DMU(s) with the best inherent efficiency in converting inputs X1, X2,,Xn into outputs Y1, Y2,,Ym is identified, and then all other DMUs are ranked relative to that most efficient DMU. For DMU 0, the basic CRS Input Oriented model (so-called CCR after Charnes, Cooper, and Rhodes) is calculated as follows: P+1.4Q/M (1+1.4R) Where P = Production of coal in tones Q = Overburden removed in cubic meter R = Stripping ratio in cubic meter of over burden per tonne of coal VRS Method ISSN: 2231-5381 http://www.ijettjournal.org Page 217
The VRS model is closely related to the standard CRS model as is evident in the dual of the BCC model: Data collection For our analysis, we have chosen four input variables namely, 1.Wage Cost ( In Lakhs rupees per year), 2.Store Cost (In Lakhs rupees per year), 3.OBR Cost (In Lakhs rupees per year), 4.Other cost (In Lakhs rupees per year) and one output variable namely 5.Production (in Lakh Tonnes per year), The difference compared to the CRS model is the The general Input oriented maximization using VRS introduction of the convexity condition. This DEA model is used to obtain efficiency score. TORA additional constraint gives the frontiers piecewise linear and concave characteristics. package and Data Envelopment Analysis Programme (DEAP) has been used to solve the model. Table 1: Normalized Data for Open-Cast mines Normalized data of OC mines Mines(DMU) Wage Cost Store Cost OBR Cost Other Cost Production 1.4159 1.3481 1.6260 1.5881 1.4980 0.4178 0.2750 1.1271 0.6606 1.0283 0.8347 0.3747 0.2395 0.2439 0.4547 0.2877 0.0429 0.0886 1.4318 0.9398 2.2116 2.7843 1.0544 1.9245 1.6182 0.1794 0.3421 0.5946 0.3132 0.6900 OCM7 0.0900 0.0640 0.1193 0.0033 0.1348 0.8788 0.6435 2.3050 0.6806 1.2584 0.4472 0.3099 1.5266 0.3449 0.7523 0.3140 0.1812 0.5095 0.1531 0.4167 0.2761 0.0975 0.4884 0.2727 0.4347 0.8668 0.4730 1.9179 0.5059 1.3427 2.5188 3.8545 1.5713 2.2644 2.1494 1.7423 1.7183 0.7791 0.7015 0.8720 2.5188 2.4909 1.0527 3.9112 1.4102 This paper attempts in Benchmarking of 15 Open Cast Mines for improving productivity (Output) using the Variable Return to Scale (VRS) Model (Input-oriented) DEA with Variable Returns to Scale (VRS) Model Unlike CRS model, variation in inputs may not lead to the same level of variation in the output in some situations. In order to address this issue, an extension of CRS model, popularly known as VRS model is used and compared with CRS model (Banker et al., 1984). ISSN: 2231-5381 http://www.ijettjournal.org Page 218
The methodology was explained and the analysis carried out by using TORA and DEA software s. Analysis of using Input-oriented VRS model The analysis carried out using same 15 OC Mines used for CRS model analysis. The efficiency scores, shadow values and peer groups shown in the table 2 after solving Input-oriented VRS Model. Table 2: Efficiency Scores, Shadow values and peer groups for OC mines after solving Input oriented VRS model DMU Efficiency Shadow Values Benchmark or Peer Group 96.50% 0.1407, 0.5965, 0.2628 4,12,13 100% 1.0000 2 100% 1.0000 3 100% 1.0000 4 94.20% 0.3302, 0.0903, 0.5795 4,6,13 100% 1.0000 6 OCM7 100% 1.0000 7 84.90% 0.2143, 0.042, 0.7437 2,4,12 90.20% 0.1237, 0.1983, 0.3494, 0.3286 2,6,7,12 93.30% 0.0378, 0.1459, 0.6779, 0.1383 2,6,7,12 100% 1.0000 11 100% 1.0000 12 100% 1.0000 13 86.50% 0.0636, 0.0825, 0.1138 4,6,13 63.20% 0.6111, 0.3889 4,13 From the table 2 eight mines,,,, OCM7,, and out of 15 have emerged benchmarking units for other coal mines. than CRS model. The efficiency comparisons of OC Mines are shown in fig:1. ISSN: 2231-5381 http://www.ijettjournal.org Page 219
120.00% 100.00% Efficiency 80.00% 60.00% 40.00% 20.00% 0.00% Efficiency OCM7 Fig.1: Vs Efficiency Score for Input oriented VRS model Table 3: Slacks assigned to input parameters of after solving Input oriented VRS model DMU Wage Cost Store Cost OBR Cost Other Cost Production 0.147 0 0 0.434 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.513 0.345 0 0 0 0 0 0 0 0 OCM7 0 0 0 0 0 0 0.134 0.286 0 0 0 0 0.448 0 0 0.07 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.054 0.763 0 0 0 0.436 0.049 0 0.716 0 The table 3 shows the assignment of Slack variables to the Inputs of the in-efficient coal mines. The slack variable values of all inputs are zero in case of efficient DMUs i.e. shown efficiency is 100%. ISSN: 2231-5381 http://www.ijettjournal.org Page 220
Table 4 : Ranking and Peer count of after solving Input oriented VRS model DMU Efficiency Benchmark or Peer Group Ranking Peer Count 96.50% 4,12,13 6 0 100% 2 3 4 100% 3 5 1 100% 4 1 6 94.20% 4,6,13 7 0 100% 6 2 5 OCM7 100% 7 4 3 84.90% 2,4,12 11 0 90.20% 2,6,7,12 9 0 93.30% 2,6,7,12 8 0 100% 11 5 1 100% 12 2 5 100% 13 2 5 86.50% 4,6,13 10 0 63.20% 4,13 12 0 For example, having efficiency score of 94.20% can refer, and can assign a weightage of 0.3302 to, 0.0903 to and 0.5795 to to become a benchmark unit. One DMU (e.g. ) have become the peer unit six times while, and become the referring institute for five times, respectively. and OCM7 becomes the referring institute for four and three times Virtual Efficient (Target) Inputs of From the results, it is clear that 8 mines (,,,, OCM7,, and ) have an efficiency score of 100%. The usage of combinations of efficient DMUs is called virtual producers corresponding to the inefficient ones. The shadow values and peer groups are helpful in constructing the virtual producers. For example mine whereas and for one time respectively. ranked as 1 due to maximum number of peer count (6) among the efficient mines and, and are ranked as 2.,, and are called as dominant peers which are the Benchmarking for the other mines for improving production. has got efficiency score less than 1(0.9420)., and are in the peer group of and their corresponding shadow values are 0.3302, 0.0903 and 0.5795. Its virtual producer is a linear combination of inputs of efficient mines of, and (peer group which have a relative efficiency 1 with respect to ). The virtual efficient wage cost for is: 0.2877*0.3302+ 0.1794* 0.0903+ 2.5188*0.5795. ISSN: 2231-5381 http://www.ijettjournal.org Page 221
Table 5: Improvements in Inputs and Output of after solving Input oriented VRS model DMU Wage Cost Store Cost OBR Cost Other Cost Production Actual to Actual to Target Target Actual to Target Actual to Target Actual to Target 1.4159 to 1.3481 to 1.2194 1.3011 1.626 to 1.5694 1.5881 to 1.0984 1.498 to 1.498 0.4178 to 0.4178 0.275 to 0.275 1.1271 to 1.1271 0.6606 to 0.6606 1.0283 to 1.0283 0.8347 to 0.3747 to 0.8347 0.3747 0.2395 to 0.2395 0.2439 to 0.2439 0.4547 to 0.4547 0.2877 to 0.0429 to 0.2877 0.0429 0.0886 to 0.0886 1.4318 to 1.4318 0.9398 to 0.9398 2.2116 to 2.7843 to 1.5708 2.2787 1.0544 to 0.9935 1.9245 to 1.8133 1.6182 to 1.6182 0.1794 to 0.3421 to 0.1794 0.3421 0.5946 to 0.5946 0.3132 to 0.3132 0.69 to 0.69 OCM7 0.09 to 0.09 0.064 to 0.064 0.1193 to 0.1193 0.0033 to 0.0033 0.1348 to 0.1348 0.8788 to 0.7463 0.6435 to 0.4125 2.305 to 1.6716 0.6806 to 0.5779 1.2584 to 1.2584 0.4472 to 0.4035 0.3099 to 0.2796 1.5266 to 0.9292 0.3449 to 0.3112 0.7523 to 0.7523 0.314 to 0.2229 0.1812 to 0.1691 0.5095 to 0.4756 0.1531 to 0.1429 0.4167 to 0.4167 0.2761 to 0.2761 0.0975 to 0.0975 0.4884 to 0.4884 0.2727 to 0.2727 0.4347 to 0.4347 0.8668 to 0.8668 0.473 to 0.473 1.9179 to 1.9179 0.5059 to 0.5059 1.3427 to 1.3427 2.5188 to 2.5188 3.8545 to 3.8545 1.5713 to 1.5713 2.2644 to 2.2644 2.1494 to 2.1494 1.7423 to 0.4526 1.7183 to 0.7228 0.7791 to 0.6736 0.7015 to 0.6065 0.872 to 0.872 2.5188 to 1.1554 2.4909 to 1.5252 1.0527 to 0.6652 3.9112 to 1.7556 1.4102 to 1.4102 The comparisons are made between Actual Inputs and Target Inputs are shown in figs 2,3 & 4. 3 2.5 Wage Cost 2 1.5 1 0.5 0 Actual Wage Cost Target Wage Cost OCM7 Fig 2: Actual Wage Cost Vs Target Wage Cost for Input oriented VRS model ISSN: 2231-5381 http://www.ijettjournal.org Page 222
International Journal of Engineering Trends and Technology (IJETT) Volume17Number5 Nov2014 4 3 2 1 Actual Store Cost 0 Target Store Cost OCM7 Store Cost 5 Fig 3: Actual Store Cost Vs Target Store Cost for Input oriented VRS model 2.5 1.5 1 0.5 Actual OBR Cost 0 Target OBR Cost OCM7 OBR Cost 2 Other Cost Fig 5.5 Actual OBR Cost Vs Target OBR Cost for Input oriented VRS model 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 Actual Other Cost Target Other Cost Fig 4: Actual Other Cost Vs Target Other Cost for Input oriented VRS model ISSN: 2231-5381 http://www.ijettjournal.org Page 223
Conclusions The efficiency score of these DMU s approaches unity while that of DEA-inefficient DMU s is less than unity. The corresponding shadow values assigned to the each inefficient mines to improve their performance with reference to the concerned allotted efficient peer group. and allotted 4 peer groups and,,, allotted 3 peer groups and allotted 2 peer groups and remaining have one peer group or bench marking. The efficiency scores of mines increased in VRS model. There is no adjustment of slack required for those efficient mines but adjustment of slack required in case of in-efficient mines. For example the efficiency of is 96.50% and the slacks assigned are 0.147 to wage cost and 0.434 to other cost respectively for improving the efficiency DEA efficiency ranking finds that 8 DMUs out of 15 DMUs have emerged as benchmarking units for the other 7 DMUs. The benchmarking units are listed as,,,, OCM7,, and as shown in Table 4.8. The efficiency score for these DMUs approaches unity while that of DEAinefficient DMUs is less than unity. The inefficient units can refer the peer groups given with the corresponding weightage shown in table1 for improvement in productivity. However, there is a scope for improvement of Open cast mines because mean efficiency score for all DMUs shows 0.9392 (93.92%) and it is greater than Inputoriented CRS model mean efficiency score of 0.8178 (81.78%). References 1. Banker, R.D. and R.C. Morey, 1986b, The use of categorical variables in data envelopment analysis, Management Science 32, No. 2. Srinivas Talluri, Data Envelopment Analysis: Models and Extensions, 2000, the journal of Production/Operations management. 3. Farrell, M.J. (1957) The measurement of productive efficiency, Journal of Royal Statistical Society, Vol. 120, No. 3. 4. R. Dattakumar and R. Jagadeesh, A review of literature on benchmarking Benchmarking: An International Journal, vol. 10, iss. 3, pp. 176-209, 2003. 5. Charnes, A. and W.W. Cooper, 1961, Management Models and Industrial Applications of Linear Programming, 2 vols. with A. Charnes (New York: John Wiley and Sons, Inc.). 6. Doyle, J., & Green, R. (1994). Efficiency and cross-efficiency in DEA: Derivations, meanings and uses. Journal of the Operational Research Society, 45(5), 7. www.scclmines.com 8. www.deazone.com 9. www.deaos.com 10. www.books.google.co.in ISSN: 2231-5381 http://www.ijettjournal.org Page 224