### DATA of various DMUs. The basic principle of DEA’s

DATA ENVELOPMENT ANALYSISINTRODUCTION:DEA was developed by charnes, cooper and Rhodes using the linear programming method. Earlier it was introduced by Farrell. This method is used to compare the efficiencies of different profitable, non profitable organisations, industry and service departments such as schools, hospitals, manufacturing firm etc.DEA is used to compare:1. Different firms of a same industry.2. Different departments of a same firm.3. Different branches of a same firm etc.HOW DOES DEA WORK?The firms or branches which are being compared are called Decision Making Units (DMU). The best DMU(s) are made as a benchmark and other DMUs are compared with these best DMUs to achieve the best efficiency. Efficiency= output / input. There may be many output and many inputs. If there is a single input and single output, then finding the efficiency is quite easy and then comparing them with other DMUs is also an easy task. However if there are a number of inputs and number of outputs, then finding the efficiency requires assigning some weights to each of the constraints ( inputs and outputs ) and thereby bringing them to a common comparable platform.Eg: Let us suppose that there are managers, engineers and workers working in a branch of a firm manufacturing cricket balls, footballs and basket balls. There may be a number of production units of this firm manufacturing varying number of cricket balls, foot balls and basket balls. They may be different number of managers, engineers and workers. In this case how can we assess the efficiency and how can we compare various branches? The answer is to assign different weights to different constraints such as salary of each type of manpower and resources required to produce each type of ball. We can take a weighted average of inputs and outputs and compare them with the formula given above. In this paper I will discuss the step by step method of finding the weights, efficiency and comparison of various DMUs. The basic principle of DEA’s working can be summarized as:• Comparing the DMUs with a target on the frontier.• The frontier is the best practice frontier based on the current set of data available.• The input oriented model sees if the DMUs can reduce their inputs to achieve the same output as currently being achieved to meet the frontier. • The output oriented DEA sees if the same inputs can be used to increase the output. One input- one output example:Bank branch A B C D E F G HTransactions/month(output) 1000 850 900 950 930 680 750 700No. of Employees (input) 10 9 10 11 12 10 8 9Efficiency (output/ input) 100.0 94.4 90.0 86.4 77.5 68.0 93.8 77.8 The dark black line indicates the frontier. This means that branches A and H lie on the frontier and these act as a bench mark for the other branches. All other branches can follow either of the A and H to become efficient. The ultimate aim is to be as close to the frontier lie as possible. Each DMU can choose which branch to follow (either A or H) as per their convenience. For example the firm C can choose to be on the horizontal frontier and F can choose to be on the vertical frontier. It is most logical for the branch G to follow the branch A and be on the horizontal frontier. • The slope of the line connecting each point to the origin corresponds to transactions per employee.• The highest slope is attained by the line from the origin through A• This line is called efficient frontier.• All points are on or below this line.• The name Data Envelopment Analysis is comes from this property because in mathematical terms, such a frontier is said to “envelope” these points• It is assumed that the efficient frontier is effective in the range of interest and is called the “constant returns–to-scale” assumption.• In order to know which DMU to follow and keep as a bench mark, draw a line for the origin through the DMUs, and this DMU has to keep the DMU corresponding to that frontier line as its benchmark. (F has to follow H).• The ratio of (the length to the DMU from origin) to the ( length of the line connecting the frontier to the origin passing through the DMU) gives the DMU efficiency score. • Eg: efficiency score of DMU –E: (OE)/ (OZ).• For an efficient DMU, the efficient score is always 1. Calculation of the target to be obtained:1. Find the equation of the best frontier line. (Slope from the line from origin to the DMU point. Coordinates’ of each DMU are known).2. Find the equation of the line through the DMU under consideration (similar procedure).3. Find the interaction of both the lines. This gives the input and output to be obtained. (Either increase the output or decrease the inputs to the match the target DMU).This single input and single output can easily be analysed using a graphical representation. But as the number of inputs and outputs increase, it will be very difficult to analyse through a graphical representation. Therefore we use linear programming for such higher end problems.Two input – three output example: Consider four universities where the inputs are funds received from public and fees, and professors available. The outputs maybe the number of students, average salary package of the graduates passed from the university and the number of academic publications per year. university Input-1(income in crores) Input-2( number of professors) Output-1(students in 1000) Output-2 (average salary in rupees,1000s) Output-3 (publications in 1000s)A 100 2500 5 50 12B 80 2200 4 40 8C 60 1500 3 55 6D 100 2750 4 40 8Fixed weights : Let the weights of input 1= u1, input 2= u2, output1= v1, output2= v2, output3= v3.The efficiency is given by : u1*input1 + u2*input2v1* input1+v2*input2+v3*input3As we can see this is a weighted average of the inputs and the outputs. As given by a fixed weights. For instance let u1= 5, u2= 3 and v1= 4, v2=5, v3=7, then the efficiency is given as: ip-1 ip-2 op-1 op-2 op-3 efficiencyA 100 2500 5 50 12 4.425B 80 2200 4 40 8 3.886C 60 1500 3 55 6 6.854D 100 2750 4 40 8 3.109WEIGHTS 5 3 4 5 7 As we can see the best university is university-C and the worst is the university-D. We have taken the weighted average to find out the weights. I have taken these weights as an arbitrary numbers but they can be realistic as per the situation.Limitations of fixed weights:• The justification of the weights assigned to the different constraints is debatable. • The change is efficiency ratings due to these weights are also questionable.• Linear regression methodology allows some of the points to lie above the frontier line also. Therefore it violates the basic assumption that all values must lie below a benchmarking line and it is not possible to create a benchmark in the first place.• The universities (as in this case) maybe situated in various different locations where the standard fee structure and standard salary of the graduates are different. Therefore assigning common weights is not recommendable.Variable weights: