3 THE CCR MODEL AND PRODUCTION CORRESPONDENCE

We have been dealing with the pairs of positive input and output vectors(xj , yj) (j =1,…,n) of n DMUs. In this chapter, the positive data assumption is relaxed. All data are assumed to be nonnegative but at least one component of every input and output vector is positive The set of feasible activities is called the production possibility set and is denoted by P

Properties of P (the Production Possibility Set) 所有滿足以下四項假設的集合： P = {(x, y)∣x>Xλ , y<Yλ , λ > 0}, where λ is a semipositive vector in P (Al) The observed activities {xj,yj) (j = 1,... ,n) belong to P. (A2) If an activity {x, y) belongs to P, then the activity [tx, ty) belongs to P for any positive scalar t. We call this property the constant returns-to-scale assumption. (A3) For an activity {x,y) in P, any semipositive activity ( , ) with >x and < y is included in P. That is, any activity with input no less than x in any component and with output no greater than y in any component is feasible. (A4) Any semipositive linear combination of activities in P belongs to P

THE CCR MODEL AND DUAL PROBLEM D(dual problem of)LP0 :(3.6)-(3.9) real variable θ and a nonnegative vector λ = (λ1,..., λn )T {DLPo) has a feasible solution θ = 1, λo = 1, λj = 0 {j ≠ o). Hence the optimal θ, denoted by θ *, is not greater than 1 (3.8)forces λ to be nonzero because yo ≧ 0 and y ≠ 0, Hence, from (3.7), θ must be greater than zero.

The constraints of (DLPo) require the activity {θxo, y0) to belong to P, while the objective seeks the minimum θ that reduces the input vector xo radially to θxo while remaining in P it can be said that {Xλ, Yλ) outperforms {θxo , y0) when θ * < 1.

Phase II=Farrell Efficiency we define the input excesses s¯ and the output shortfalls s+ and identify them as “slack” vectors by(3.10)： where e = ( 1 , . . . , 1) (a vector of ones) so that The objective of Phase II is to find a solution that maximizes the sum of input excesses and output shortfalls while keeping θ = θ*. Phase II=Farrell Efficiency

Definition 3.1 and 3.2

Definition 3.1 and 3.2 The first of these two conditions is referred to as “radial efficiency.” It is also referred to as “technical efficiency” because a value of θ* < 1 means that all inputs can be simultaneously reduced without altering the mix (=proportions) in which they are utilized. Hence the inefficiencies associated with any nonzero slack identified in the above two-phase procedure are referred to as "mix inefficiencies.“ "weak efficiency" is sometime used when attention is restricted to (i) in Definition 3.2. The conditions (i) and (ii) taken together describe what is also called "Pareto-Koopmans" or "strong" efficiency,

Theorem 3. 1 The CCR-efficiency given in Definition 3 Theorem 3.1 The CCR-efficiency given in Definition 3.2 is equivalent to that given by Definition 2.1.

Computational Procedure for the CCR Model

Using the usual LP notation, we now rewrite [DLPo] as follows

Example 3.1 DMU (I)Xl (I)X2 (O)y A 4 3 1 B 7 C 8 D 2 E F 10 G

No. DMU Score Rank Reference set (lambda) 1 A 0.8571429 5 D 0.7142857 E 0.2857143 2 B 0.6315789 7 C 0.1052632 0.8947368 3 4 6 F G 0.6666667

Excess Shortage No. DMU Score Xl X2 y S-(1) S-(2) S+(1) 1 A 0.8571429 2 B 0.6315789 3 C 4 D 5 E 6 F 7 G 0.6666667

θ* =0.8571429, And = 0.7143, = 0.2857 show the proportions contributed by D and E to the point used to evaluate A. Hence A is technically inefficient. No mix inefficiencies are present because all slacks are zero. Thus removal of all inefficiencies is achieved by reducing all inputs by 0.1429 or, approximately, 15% of their observed values 0.8571 X (Input of A) = 0.7143 x (Input of D) + 0.2857 x (Input of E) (Output of A) = 0.7143 X (Output of D) + 0.2857 x (Output of E).

projection Thus, as seen in Worksheet"Sample-CCR-I.Projection," the CCR-projection of (3.32) and (3.33) is achieved by

multiplier The optimal solution for the multiplier problem {LPA) can be found in Worksheet "Sample-CCR-I.Weight" as follows, This solution satisfies constraints (3.3)-(3.5) and maximizes the objective in (3.2), i.e., u*y = 0.8571 x 1 = 0.8571 = θ*

Weighted Data The Worksheet "Sample-CCR-I. Weighted Data" includes optimal weighted inputs and output, i.e., The sum of the first two terms is 1 which corresponds to the constraint (3.3).

Moving from DMU A to DMU B, θ* Reference set (lambda) Slack Multiplier Weighted Data constrain

The optimal solution of the LP problem for F Although F is "radial-efficient," it is nevertheless "CCR-inefficient" due to this excess (mix inefficiency) associated with .

Pareto-Koopmans efficiency Thus the performance of F can be improved by subtracting 2 units from Input 1. This can be accomplished by subtracting 2 units from input 1 on the left and setting s^* = 0 on the right without worsening any other input and output. Hence condition {ii) in Definition 3.2 is not satisfied until this is done, so F did not achieve Pareto-Koopmans efficiency in its performance

The optimal solution of the LP problem for G Considering the excess in Input 2 (s^* = 0.6667), G can be expressed by: 0.6667 X (Input 1 of G) = (Input 1 of E) (33.33% reduction) 0.6667 X (Input 2 of G) = (Input 2 oi E) + 0.6667 (42.86% reduction (Output of G) = (Output of E).

圖解