資料包絡分析法 Data Envelopment Analysis-A Comprehensive Text with Models, Applications, References and DEA-Solver Software Second Edition WILLIAM W. COOPER University of Texas at Austin, U.S.A. LAWRENCE M. SEIFORD University of IVIichigan, U.S.A. KAORU TONE National Graduate Institute for Policy Studies, Japan

決策品質的比較 效用函數(Utility Functions) 單位產出下的成本 無法比較，如人飲水，冷暖自知。 多種產出問題 多種投入問題 有些產出或投入無法以金額衡量

多種投入(產出)-以圖書館為例 提供的效果(服務)： 投入成本 圖書或設備的使用次數、諮詢次數等 收購書籍、資料庫或設備，人員數量，空 間大小

總要素生產力 總要素生產力(Total Factor Productivity, TFP ) TFP(i)=Yi/Xi i = 1,…n Yi 代表第i家廠商的產出 Xi 代表第i家廠商的投入 例某行政單位有A,B,C等3個部門 產出(Yi)為某年度i部門的結案公文數 投入(Xi)為某年度i部門的員工數 部門 A B C Y 60 80 90 X 10 20 10 TFP(A)=60/10=6 TFP(B)=80/20=4 TFP(C)=100/10=10

技術效率 技術效率(Technical Efficiency, TE) TE(i)=TFP(i)/TFP* TFP*為所有廠商中最高的TFP(Total Factor Productivity),本例中以部門C的TFP最高 所以 TE(A)=TFP(A)/TFP(C)=6/10 TE(B)=TFP(B)/TFP(C)=4/10 TE(C)=TFP(C)/TFP(C)=10/10=1

Two Inputs and One Output Case Store A B C D E F G H I Employee Xl 4 7 8 2 5 6 5.5 Floor Area X2 3 1 2.5 Sale Y

production possibility set

To measure inefficiency of A D and E are called the reference set for A. P with Input X1= 3.4 and Input X2 = 2.6,

ONE INPUT AND TWO OUTPUTS CASE Store A B C D E F G Employee X 1 Customers Y1 2 3 4 5 6 Sale Y2 7

production possibility set

where d(0, D) and d(0, P) mean "distance from zero to D" and "distance from zero to P," respectively. The ratio is referred to as a "radial measure” .Because we are concerned with output, however, it is easier to interpret (1.5) in terms of its reciprocal(1.33). This result means that, to be efficient, D would have had to increase both of its outputs by 1.33. This kind of inefficiency which can be eliminated without changing proportions is referred to as "technical inefficiency." Another type of inefficiency occurs when only some (but not all) outputs (or inputs) are identified as exhibiting inefficient behavior. This kind of inefficiency is referred to as "mix inefficiency" because its elimination will alter the proportions in which outputs are produced

FIXED AND VARIABLE WEIGHTS Hospital A B C D E F G H I J K L Doctors 20 19 25 27 22 55 33 31 30 50 53 38 Nurses 151 131 160 168 158 255 235 206 244 268 306 284 Outpatients 100 150 180 94 230 220 152 190 250 260 Inpatients 90 72 66 88 80 147 120 (each in units of 100 persons/month)

多投入多產出 人員(X1)與設備(X2)兩種投入 公文(Y1)與專案計畫(Y2)兩種產出 一般的績效評估方式：加權計分（主觀的 給予各投入產出權數） u1×Y1i+u2×Y2i TE(i)= v1×X1i+v2×X2i u1為Y1的權數，u2為Y2的權數 v1為X1的權數，v2為X2的權數

THE CCR MODEL the CCR model, which was initially proposed by Charnes, Cooper and Rhodes in 1978

for each DMU, we formed the virtual input and output by (yet unknown) weights (vi) and (ur)

FP →LP Theorem 2.1 The fractional program (FPo) is equivalent to (LPo).

Theorem 2.2 (Units Invariance Theorem) The optimal values of max θ = θ * in (2.3) and (2.7) are independent of the units in which the inputs and outputs are measured provided these units are the same for every DMU. one person can measure outputs in miles and inputs in gallons of gasoline and quarts of oil while another measures these same outputs and inputs in kilometers and liters. They will nevertheless obtain the same efficiency value from (2.3) or (2.7)

Definition 2.1 (CCR-Efiiciency) 1. DMUo is CCR-efficient if θ * = 1 and there exists at least one optimal (v*,u*), with V* > 0 and u* > 0. 2. Otherwise, DMUo is CCR-inefficient has θ * < 1 (CCR-inefiicient). Then there must be at least one constraint (or DMU) in (2.9) for which the weight {v*,u*) produces equality between the left and right hand sides since, otherwise,θ * could be enlarged Thus, CCR-inefRciency means that either (i) 9* < 1 or (M) 6* = 1 and at least one element of {v*,u*) is zero for every optimal solution of (LPo).

2.6.1 Example 2.1 (1 Input and 1 Output Case) DMU A B C D E F G H Input 2 3 4 5 6 8 Output 1 We can evaluate the efficiency of DMU A, by solving the LP problem below The optimal solution, easily obtained by simple ratio calculations, is given by(v* = 0.5, u* = 0.5, e* = 0.5).

DEA-Solver Pro5.0- CCR-I No. DMU Score V(1) Input U(1) Output 1 A 0.5 2 B 0.3333333 3 C 0.6666667 4 D 0.75 0.25 5 E 0.8 0.2 6 F 0.4 7 G 0.1666667 8 H 0.625 0.125 The θ * values in Table 2.2 show what is needed to bring each DMU onto the efficient frontier. For example, the value of 6* = 1/2 applied to A's input will bring A onto the efficient frontier by reducing its input(2) 50% while leaving its output at its present value(1).

No. DMU Score Rank Reference set (lambda) 1 A 0.5 6 B 0.3333333 2 3 C 0.6666667 4 D 0.75 5 E 0.8 1.3333333 F 0.4 8 7 G H 0.625 1.6666667

Example 2.2 DMU A B C D E F Input Xl 4 7 8 2 10 X2 3 1 Output y

Example -2.2DEA-Solver Pro5.0/ CCR(CCR-I) No. DMU Score V(1) Xl V(2) X2 U(1) y 1 A 0.8571429 0.14285714 0.1428571 2 B 0.6315789 5.26E-02 0.2105263 3 C 8.33E-02 0.3333333 4 D 0.16666667 0.1666667 5 E 0.5 6 F The (unique) optimal solution is (v*= 0.0526, v* = 0.2105, u* = 0.6316, 0* =0.6316), the CCR-efRciency of B is 0.6316, and the reference set is EB ={C, D}.

Example -2.2DEA-Solver Pro5.0/ CCR(CCR-I) No. DMU Score Rank Reference set (lambda) 1 A 0.8571429 5 D 0.7142857 E 0.2857143 2 B 0.6315789 6 C 0.1052632 0.8947368 3 4 F

Now let us observe the difference between the optimal weights v. = 0 Now let us observe the difference between the optimal weights v* = 0.0526and v^ = 0.2105. The ratio v^/v^ = 0.2105/0.0526 = 4 suggests that it is advantageous for B to weight Input X2 four times more than Input xi in order to maximize the ratio scale measured by virtual input vs. virtual output. Therefore a reduction in Input X2 has a bigger effect on efficiency than does a reduction in Input X1

The optimal solution for F is (v1. = 0, v2. = 1, u. =1, 0 The optimal solution for F is (v1*= 0, v2* = 1, u* =1, 0* =1) and with 0* = 1, F looks efficient. However, we notice that v* = 0. Furthermore, let us examine the inefficiency of F by comparing F with C. C has Input x1 = 8 and Input x2 = 1, while F has Input xi = 10 and Input X2 =1. F has 2 units of excess in Input X1 compared with C. This deficiency is concealed because the optimal solution forces the weight of Input Xi to zero (v^ = 0). C is in the reference set of F and hence by direct comparison we can identify the fact that F has used an excessive amount of this input.