Schedule

• On Thursdays we will be here in SOS180 for:– 19-02-2009 (today)– 26-02-2009– 05-03-2009

• Homework 1 is on the web, due to next Friday (17:00 27-02-2009).

Today

• We will write and solve the corresponding LP to analyze a simple DEA problem.

• Here is our data set:

DMU A B C D E F G

Input I1 4 7 8 4 2 10 3

I2 3 3 1 2 4 1 7

Output O 1 1 1 1 1 1 1

Graphically...

Mathematical modelling

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1) and 0(between DMU of efficiency

output on weight

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DMU from output of level observed 0

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Model variables

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Linear Program (LP):

Obtain file(s):

• Copy and paste the excel file to your own accounts:F:\COURSES\UGRADS\INDR\INDR471\SHARE\Labs\ lab1-LP-

raw.xls

• Also open the power point file:F:\COURSES\UGRADS\INDR\INDR471\SHARE\labs\DEA-Lab-18-

02-09.ppt

1

Solve the corresponding LP for DMUs A in Excel. Keep the sensitivity report.

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SUMPRODUCT

Multiplies corresponding components in the given arrays,

and returns the sum of those products.

Syntax

SUMPRODUCT(array1,array2,array3, ...)

Array1, array2, array3, ...   are 2 to 30 arrays whose

components you want to multiply and then add.

Remarks•The array arguments must have the same dimensions. •If they do not, SUMPRODUCT returns the #VALUE! error value. •SUMPRODUCT treats array entries that are not numeric

as if they were zeros.

Excel solver

• Tools Add-in Excel solver

• Options International Decimal and Thousands (for later use, but…)

1.a

What is the efficiency of DMU A? What are the optimal input-output weights for DMU A? Does the solution show whether DMU A has purely technical (or ratio) and/or mix inefficiencies? How? Why? Explain.

1.b

What is the level of input for DMU A to be ratio (or purely technical) efficient?

Note its meaning: If the DMU becomes ratio efficient, it can produce the same level of output with a decreased amount of input, while keeping the input ratios the same.

iee IE

inputs for A* efficiency of A:4*0.8517 = 3.4286 (input 4’tu 3.42 olmalı)3*0.8517 = 2.5714

1.c

1. Which constraints are binding? 2. What are the shadow prices of the constraints? 3. Use this information to determine the

reference set for each of the DMUs. 4. Can we use the above information to find

target input/output levels for DMU A?

Note: A binding constraint have a slack value of 0. The shadow prices of non-binding constraints are always 0 (Why?). The values of slacks for the constraints are given in “Final Value” of “Constraints” section in the

sensitivity report.

Definition: Reference Set

For an inefficient DMUe, we define its reference set

Re,

based on the max-slack solution as obtained in phases

one and two, by (binding olanlar reference set oluyor.)

Re= {kє{1, …, K} | k*>0}

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Dual variable vs its constraint in LP

Hypothetical composite unit (HCU) and target setting for inefficient for

DMU e

e. DMUfor set reference the: where

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Input targets:

Output targets:

CCR projections.

For DMU A to be efficient:

(Lamda=shadow price)0.7143*inputs for unit D + 0.2857 * inputs for unit E

=0.7143* (4) + 0.2857 * (2) (2) + (4)

= 3.42862.5714

current inputs: 4, 3

inputs for A* efficiency of A:4*0.8517 = 3.42863*0.8517 = 2.5714

Meaning of shadow prices in DEA

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LPe

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LPe with an increase in the RHS

1.d

Do you observe binding constraint(s) with a shadow price of 0 for any of these DMUs? If yes, which one? Is it efficient or inefficient?

Binding constraints with a shadow price of 0 indicate the existence of multiple optima.

2.a

What is the efficiency of DMU G? What are the optimal input-output weights for DMU G? Does the solution show whether DMU C has purely technical (or ratio) and/or mix inefficiencies? How? Why? Explain.

2.b

What is the level of input for DMU G to be ratio (or purely technical) efficient?

Note its meaning: If the DMU becomes ratio efficient, it can produce the same level of output with a decreased amount of input, while keeping the input ratios the same.

iee IE

inputs for G* efficiency of G:3*0.6667 = 27*0.6667 = 4.6667

2.c

1. Which constraints are binding? 2. What are the shadow prices of the constraints? 3. Use this information to determine the

reference set for each of the DMUs. 4. Can we use the above information to find

target input/output levels for DMU G?

Note: A binding constraint have a slack value of 0. The shadow prices of non-binding constraints are always 0 (Why?). The values of slacks for the constraints are given in “Final Value” of “Constraints” section in the

sensitivity report.

For DMU G to be efficient:

1 * inputs for unit E

(2)= (4)

current inputs: 3, 7

inputs for G* efficiency of G:3*0.6667 = 27*0.6667 = 4.6667

2.d

Do you observe binding constraint(s) with a shadow price of 0 for any of these DMUs? If yes, which one? Is it efficient or inefficient?

Binding constraints with a shadow price of 0 indicate the existence of multiple optima.

Our goal

• To compute the efficiency of each DMU (identify both ratio and mix inefficiency, if present)

• To set target levels for each DMU, which will make them efficient

• To identify efficient DMUs which have similar input-output levels to an inefficient DMU e, so that DMU e can learn from these specific efficient DMUs.

Does the LP formulation fulfill these goals?

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LPe for CCR and its dual DLPe:

edunrestrict:

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Linear combinationof observedinput values.

Linear combinationof observedoutput values.

Output of DMU e

Effectiveinput of DMU e