11. Models With Restricted Multipliers Assurance Region Method

Transkriptio

1 . Models With Restricted Mltipliers Assrance Region Method Kimmo Krki Esitelmä - Kimmo Krki

2 Contents Introdction to Models With Restricted Mltipliers (Ch 6.) Assrance region method (Ch 6.2) Formlation (Ch 6.2.) AR-Efficiency Example (Ch 6.2.2) Change of Efficient Frontier (Ch 6.2.3) On Determining Lower and Upper Bonds (Ch 6.2.4) Home assignment Esitelmä - Kimmo Krki

3 Introdction to Models With Restricted Mltipliers DEA models corresponding to different prodction possibility sets, which originate from technical aspects of organizational activities We may need a priori information or information otside the data This leads to the imposition of additional constraints Two approaches Assrance region method Cone-ratio method Esitelmä - Kimmo Krki

4 Assrance region method Developed to help in choosing a best site for the location of a high-energy physics laboratory when other approaches proved to be deficient in evalating otpts like contribtions to fndamental knowledge In assrance region method we impose an additional constraint where L,2 and U,2 are lower and pper bonds for the ratio Axiliary information sch as prices, nit costs, etc, is often sed when choosing the bonds v L = = 2,2 U,2 v, Esitelmä - Kimmo Krki v 2 v

5 Formla for the Assrance Region Method (/4) AR method is formlated adding constraints for pairs of items if needed L v v U Generally we can think of an inpt nmeraire and otpt nmeraire Then we can constrain all the vales of inpt and otpt variables as follows v l L, i We can set v = =, 2 2, 2v, r vi v r, i U, r ( i = 2,, m) ( r = 2,, s) Esitelmä - Kimmo Krki

6 Esitelmä - Kimmo Krki, ) (.. max, + = v Q vp AR Y vx vx t s y v Formla for the Assrance Region Method (2/4) We get the CCR-AR model Where = l l P = U L U L Q

7 Esitelmä - Kimmo Krki Formla for the Assrance Region Method (3/4),, ) (.. min,,, + + τ π λ τ λ π λ θ θ τ π λ θ DAR y Q Y P X x t s τ λ π λ θ Q Y y s P X x s + + = + = + Beacse the comptations are sally carried ot on the dal side we formlate the dal problem Let s assme that both (AR ) and (DAR ) have a finite positive optimm and let an optimal soltion of (DAR ) be (θ, λ, π, τ, s -, s + ), where the slack variables are defined as

8 Formla for the Assrance Region Method (4/4) - AR-Efficiency Based on the soltion of the (DAR ) we define AR-efficiency The DMU associated with (x, y ) is AR-efficient if and only if it satisfies (Definition 6.) θ =, An improvement of an AR-inefficient (x, y ) obtained by sing (DAR ) can be represented xˆ yˆ = θ = y The activity x, y ) is AR-efficient (Theorem 6.) ( x + s s s + = and + P π Q τ s + = ( = X λ ) ( = Yλ ) Esitelmä - Kimmo Krki

9 General Hospital Example (/4) Let s consider a case of 4 hospitals Two inpts (Doctors and Nrses) Two otpts (Otpatients and Inpatients) First we solve the basic CCR model to find the efficient hospitals Then we add the following assrance region constraint,2 v2 2 5,,2 v 5 Inpt Otpatient Otpt Inpatient Hos. Doctors Nrses s s H H H H H H H H H H H H H H Esitelmä - Kimmo Krki

10 General Hospital Example (2/4) The basic CCR soltion Hospital CCR Eff. Doctor Nrse Otpatient Inpatient v v 2 2 H,955 3,32E-4 9,59E-6,67E-7 H2 2,42E-4,4E-6 7,4E-6 2,25E-7 H3,4E-4 2,4E-5 3,39E-6 3,25E-6 H4,72 2,82E-4,28E- 6,99E-6 H5,827 2,45E-5 3,84E-6 H6 7,84E-4,54E-5 2,56E-6 2,45E-6 H7,844,33E-4,2E-5 5,5E-6 H8 2,9E-4 4,58E-8,E-8 5,6E-6 H9,995 2,33E-5 4,26E-6,36E-7 H 3,9E-5,49E-5 2,5E-6,43E-6 H,93,62E-4 7,39E-8 4,2E-6 H2,969 7,93E-5 6,E-6 3,E-6 H3,786,47E-5 2,3E-6 H4,974,28E-5 2,E-6 Esitelmä - Kimmo Krki

11 General Hospital Example (3/4) Soltion of Assrance Region Method Hospital AR Eff. Doctor Nrse Otpatient Inpatient v v 2 v 2 /v 2 2 / H,926,39E-4 2,78E-5,2 7,28E-6 2,E-6,29 H2,E-4 2,9E-5,2 5,75E-6,67E-6,29 H3,3E-4 2,6E-5,2,2E-6 5,2E-6 5 H4,634,6E-4 2,32E-5,2,6E-6 5,79E-6 5 H5,82 4,7E-6 2,35E-5 5 6,53E-7 3,27E-6 5 H6 7,76E-5,55E-5,2 2,8E-6 2,24E-6,8 H7,83 8,62E-5,72E-5,2 8,59E-7 4,29E-6 5 H8,872 7,94E-5,59E-5,2 7,9E-7 3,95E-6 5 H9,982 4,49E-6 2,24E-5 5 3,55E-6 7,E-7 5 H 5,9E-6,4E-5,2 2,84E-6 8,63E-7,3 H,849 6,56E-5,3E-5,2 6,53E-7 3,27E-6 5 H2,93 5,6E-5,3E-5,2 5,3E-7 2,57E-6 5 H3,74 2,83E-6,4E-5 5 3,93E-7,97E-6 5 H4,929 2,42E-6,2E-6 5 3,36E-7,68E-6 5 Esitelmä - Kimmo Krki

12 General Hospital Example (4/4) Efficiency scores obtained by the AR method are lower than those of the CCR method Esitelmä - Kimmo Krki

13 Change of Efficient Frontier by Assrance Region Method (/2) Imposition of new constraints change the efficient frontier Let s consider the case of Example 2.2 from the book, where we have two inpts and one otpt. DMU x x 2 y CCR(θ ) Reference Set x x A 4 3,857 D E,429,429,857 B 7 3,636 C D,526,25,636 C 8 C,833,3333 D 4 2 D,667,667 E 2 4 E,243,429 F F Now we add a new constraint on the weights v and v 2 v,5 2 2 v That is, we reqire,5( v ) ( v2 ), ( v2 ) 2( v ) Esitelmä - Kimmo Krki

14 Change of Efficient Frontier by Assrance Region Method (2/2) Efficient frontier is bonded by the new constraints The line segment C is otside the efficient frontier We move the line segment C in parallel to toch P5. This is given by the dotted line 8 2 ( v ) + ( v ) =,8 Ths the efficiency of DMU C drops to,8 Esitelmä - Kimmo Krki

15 On Determining the Lower and Upper Bonds Assme all inpt and otpt items are technological We defined virtal inpts and otpts for (x k, y k ) We can intepret v i ( i ) as the nit cost (price) of x ik (y ik ) Let c ik be the actal nit cost of the inpt x ik. Now we set the lower and pper bonds as follows These can be intepreted as bonds for prices (or nit costs), which are sefl in a case when actal prices are not known exactly C = v x k k l ij c = min c + vmxmk and Pk = y k jk ik, ij c = max c jk ik ( k =,, n) Esitelmä - Kimmo Krki s y sk

16 Smmary Assrance Region Method pts constraints on the ratio of the inpt or otpt weights AR method helps to get rid of zero weights in the optimal soltion and ths the efficiency score is redced from the original CCR soltion Lower and pper bonds of the constraints needs to be chosen careflly often sing axiliary information abot the problem Esitelmä - Kimmo Krki

17 Home Assignment (/2) Consider the following problem with the given CCR soltions. DMU x x 2 y CCR(θ ) Reference Set x x A 4 3,857 D E,429,429,857 B 7 3,636 C D,526,25,636 C 8 C,833,3333 D 4 2 D,667,667 E 2 4 E,243,429 F F We impose an additional assrance region constraint to the problem v 3,5 2 5 v Esitelmä - Kimmo Krki

18 Home Assignment (2/2) Solve graphically in the (v,)-space which of the CCR-efficient DMUs are AR-efficient Solve graphically the efficiency redction for the CCR-efficient DMU(s) which are fond ARinefficient Finally formlate and solve the CCR-AR problem for the AR-inefficient DMU(s) (same DMU(s) as above). Inclde also the dal variables in yor soltion Esitelmä - Kimmo Krki