Data Enveloplement analysis (DEA)(part 2)References:•Data Envelopment Analysis, A Comprehensive Text with Models, Applications,ReferencesandDEA-SolverSoftware,Authors:Cooper,WilliamW.,Seiford,Lawrence M.,Tone, Kaoru

One input and two outputs example•7 branch offices•Outputs:–The number of customers (unit = 10)–The sales (unit = 1,000,000 $) per salesman•Input:–The number of employees (salesmen)

One input and two outputs example•The efficient frontier

One input and two outputs example•The efficiency of D:

One input and two outputs example•The efficiency of D:•Its reciprocal:•To be efficient, D, would have to increase both of its outputs by 4/3.

Efficiency •The inefficiency which can be eliminated by decreasinginputs by the same proportion (or increasing outputs by thesame proportion) is calledpurely technical inefficiencyorratio inefficiency. •When only some (but not all) outputs (or inputs) are identified as exhibiting inefficient behavior, the elimination of this inefficiency will alter the proportions in which outputs are produced (or inputs are utilized). This is calledmix inefficiency.•If the unit has no purely technical inefficiency and mix inefficiency, then it has technical efficiency(also namedCCR efficiency).

Purely Technical and Mix Efficiency•Units B, E, F and G form theefficiency frontier.•Units A, C, D: inefficient. •Unit A has purely technical inefficiency. –this inefficiency can be eliminated. –Hypothetical unit Q (with proportionally increased outputs): improved versionof unit A.•Unit Q is on the efficiency frontier but not on the efficient part of the frontier! –It is mixed inefficient(unit B has a better‘Ouput1/Input ratio’ than unit Q) –But it is alsopurely technical efficientorratio efficient(efficiency of Q = 1).

Basic CCR Model•CCR- Charnes, Cooper and Rhodes - 1978•Suppose that•? = 1. . ? ∶service units, •? = 1. . ?: types of inputs,•? = 1. . ?: types of outputs•Input data matrix:Output data matrix:•The Fractional Model:

Basic CCR ModelFractional Model (FP):→ Find?∗and ?∗for unitOsuchthat objective function value𝜃is maximized.•The optimal solution of LP is also optimal for FP with(? = ?∗, ? = ?∗)and sameobjective value𝜃∗.•Model seeks to maximize the efficiency value to make every unit look as good as possible with efficiency rating as close to 100% as possible in comparison with the other service units being evaluated. → (?∗, ?∗)are the set of most favorable weights for the unitO.LP Model :

Basic CCR–Efficiency ExampleDefinition (CCR-Efficiency):The unit is CCR-efficient if•𝜃∗=1, and•there exists at least one optimal (?∗, ?∗)with?∗> 0and?∗> 0.Otherwise, the unit is CCR-inefficient.For example: LP returns:

Checking the results:•Unit C: •v1*= 0.0833, v2*=0.333, u*= 1, θ*=1. •CCR-efficientby definition.•Unit F: •v1*=1, v2*=0, u*= 1, θ*=1.