Data Envelopment Analysis

George Easaw Ph.D.

Alliance University

Bangalore

An IIT Bombay Departmental performance management problem ..

A problem in IIT Bombay of measuring Dept. performance

IIT Bombay has been in the educational service field for the past 60 years .. It has at least 14 depts..

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Each dept has got a varying number of inputs - students who join for BTech, MTech and PhD. Each dept has diff nos of Labs, faculty members.

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These depts have their own outputs - pass percentages, papers published, patents registered and average placement salary details.

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How to find which dept is doing best or worst ?

What is Data Envelopment Analysis?

• It is an Operations Research tool used to empirically measure the "relative" productive efficiencies of Decision Making Units (DMUs)
• Uses Linear Programming tool ..
• Multiple inputs and outputs are employed
• Plots the efficient frontier of the DMUs considered
• DMUs on the efficiency frontier are considered to be efficient while those within the frontier are candidates for improvement

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How efficiency is measured ?

• DMUs on the efficiency frontier are considered to be 100% efficient (relatively). These DMUs are considered not to need any intervention (in local terms, not global) to improve their performance
• DMUs not on the efficiency frontier are less efficient and has potential for improvement
• How much improvement is possible is found by measuring the radial distance of the line joining the DMU with the origin, from the 100% efficiency frontier

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For example, E-E’ is the improvement possible for DMU E to reach the efficient frontier ..

LP formulation

• LP formulation is done for each DMU
• The decision variables are the weights attached to the different inputs and outputs
• The objective function of efficiency is actually a ratio - non-linear
• The constraints also work on the premise that the efficiency for any DMU cannot be more than 1, at most equal to 1.
• As usual, all the decision variables are assumed to be non-negative

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LP formulation (contd..)

• To make the objective function linear, the denominator, ie. the weighted sum of all the input variables for the DMU in question is made equal to 1.
• The expression for all the inputs for the ‘DMU in question’ equated to unity (number 1) now becomes an equality constraint
• The constraints are made linear, the denominator of constraints are subtracted from numerator as per the inequality and made into linear inequality constraints

Sample problem ..

5 plants of Canon company at different geographies produce the same item camera A. The inputs are materials and labour hours. Data for all the five plants is given here.

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Plant prodn/day material cost labour hours

1 100 10 2

2 80 8 4

3 120 12 1.5

4 100 8 4

5 110 10 8

Find which plants are producing at high efficiency and which plants need to improve ?

Formulation..

• Let us take weights for the output be u1 and weights for the inputs be v1 and v2. ( weights are the decision variables)
• Obj. fn. is

Max Eff = (u1 × 100) / (v1 × 10 + v2 × 2)

sub to constraints

efficiency of plant 1: (u1 × 100) / (v1 × 10 + v2 × 2) ≤ 1

efficiency of plant 2: (u1* 80) / (v1 * 8 + v2 * 4) ≤ 1

efficiency of plant 3: (u1* 120) / (v1 * 12 + v2 * 1.5) ≤ 1

efficiency of plant 4: (u1 * 100) / (v1 * 8 + v2 * 4) ≤ 1

efficiency of plant 5: (u1 * 110) / (v1 * 10 + v2 * 9) ≤ 1

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and non-negativity constraints all u and v ≥ 0.

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• New (refined) formulation (for plant 1)

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Maximize Efficiency = u1 * 100

subject to

efficiency of plant 1: (u1 * 100) - (v1 * 10 + v2 * 2) ≤ 0 - A

efficiency of plant 2: (u1 * 80) - (v1 * 8 + v2 * 4) ≤ 0 - B

efficiency of plant 3: (u1 * 120) - (v1 * 12 + v2 * 1.5) ≤ 0 - C

efficiency of plant 4: (u1 * 100) - (v1 * 8 + v2 * 4) ≤ 0 - D

efficiency of plant 5: (u1 * 110) - (v1 * 10 + v2 * 9) ≤ 0 - E

v1 * 10 + v2 * 2 = 1

all u and v ≥ 0.

Similarly we need to prepare LP formulation for all the other four plants, keeping the obj fn and last constraint corr to the plant under consideration, retaining A,B,C,D and E

ie. there will be a total of five Linear Programming formulations to be solved if there are five plants to be analysed.

Ie. ‘n’ LP formulations if there are n plants

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The solver will give us the solution for the values of u1, u2 (output variables) and the value of the v1 (input variable) for each of the five LP formulations.

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The objective function value will tell us whether the corresponding DMU is efficient (on efficiency contour) or how much it is inefficient (within contour)

How to solve Linear Programming problems on any spreadsheet package ..

In Excel, we have LP solver. This demo will be on Excel LP Solver ..

Open a new worksheet and enter data in rows as given in the next slide..

a. obj function

b. Decision Variables

c Obj function coefficients

d. values of the decision variables and

e. constraints (columns giving total value and limits of constraints)

=sumproduct(c7:d7,c8:d8) means entry in c7 will be multiplied with c8, similarly entry in d7 will be multiplied with d8, and entered in that cell ..

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If you highlight this cell and pull the +ve sign at bottom right of cell, all cells below will also change (unless when you insert a $ sign, which means that cell value remains fixed across any cells thru which it is pulled.)

(see how constraint cells depend on the fixed decision variable values)

Preparation on LP solver ..

If LP solver is not installed, click on

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Excel 2010 Home --> options --> Addins --> Solver .. Install ...

there you are ..!! (ready to solve any LP problem) ..

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Alternatively you can also go to Google Drive (Cloud) and use the Solver under Tools ..

Click on the LP solver, Solver window appears-

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Enter all solver parameters asked, for eg :

a. target cell (cell co-ordinates where the final obj fn value is calculated and displayed),

b. Variable cells to change (cell co-ordinates where value of decision variables are computed and provided)

c. Max or Min,

d. Constraints, (give LHS cell, RHS cell c-ordinates and the inequality / inequality sign)

e. LP simplex method ..

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Click on Solve .. Go to Options for sensitivity analysis ..(to get shadow prices for the input and output variables)

Each formulation solution gives the efficiency value for one DMU. The shadow prices of the corresponding inputs and outputs can also be got.

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Similarly do the exercise for all the DMUs.

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It is now possible to identify the 100% relative efficiency DMUs on the efficiency frontier and the less efficient ones inside the frontier.

(being maximisation, the efficiency frontier will be concave to origin)

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