Returns to Scale II. S ysteemianalyysin. Laboratorio. Esitelmä 8 Timo Salminen. Teknillinen korkeakoulu

Transkriptio

1 Returns to Scale II

2 Contents Most Productive Scale Size Further Considerations Relaxation of the Convexity Condition

3 Useful Reminder Theorem 5.5 A DMU found to be efficient with a CCR model will also be found to be efficient with the corresponding BCC model and constant returns to scale prevail at DMU CCR frontier BCC Frontier C D E y 3 B A x

4 Most Productive Scale Size A Necessary condition / Let (x, y ) be the coordinates of the point under evaluation and let α and β be positive scalars The point (x, y ) is transformed to (αx, βy ) to achieve MPSS α and β are solved with optimization max s. t. β / α βy αy = n j= n j= n j= λ j y x α, β, λ i j j λ λ j j

5 Most Productive Scale Size A Necessary condition / Theorem 5.6 A necessary condition for a DMU with output and input vectors y and x to be at MPSS is β*/α* = max β/α =, in which case β* = α* and returns to scale will be constant

6 Example point D Optimum : λ B *=, β*=/5, α*=3/8, β*/α*=6/5 Optimum : λ C *=, β*=4/5, α*=3/4, β*/α*=6/ CCR frontier BCC Frontier C D E y 3 B A x

7 Most Productive Scale Size Definition 5. For DMU to be MPSS both of the following conditions need to be satisfied: (i) β*/α*= and (ii) all slacks are zero (in all alternative optima) Step : Optimize β* and α*. Step : Fix β = β* and α = α*. Maximize slacks. If non-zero slacks are found, DMU is not MPSS MPSS point with coordinates y r *, y r * can be obtained with the following formulas: * β y * α x r i + s s + * r * i = = x y * r * i,, r =,..., s i =,..., m

8 Further considerations Theorem 5.7 In the BCC model a reference set to any BCC inefficient DMU does not include both increasing and decreasing returns-to-scale DMUs at the same time 6 y BCC Frontier C CRS D E DRS A B CRS IRS x

9 Characterization of Return to Scale Corollary 5. Let a reference set to a BCC-inefficient DMU (x, y ) be E. Then, E consists of one of the following combinations of BCC-efficient DMUs where IRS, CRS and DRS stand for increasing, constant and decreasing returns-to-scale, respectively.. All DMUs have IRS. Mixture of DMUs with IRS and CRS 3. All DMUs have CRS 4. Mixture of DMUs with CRS and DRS 5. All DMUs have DRS Theorem 5.8 Let the BCC-projected activity of a BCC-inefficient DMU (x, y ) be (x P, y P ) and a reference set to (x, y ) be E. Then (x P, y P ) belongs to. IRS if E consists of DMUs in categories or of Corollary 5., and. DRS, if E consists of DMUs in categories 4 or 5

10 Characterization of Return to Scale Case: E consists of DMUs in category 3 (all CRS) Step : Solve envelopment form of the BCC model for (x, y )

11 Characterization of Return to Scale If DMU is found to be efficient, find an optimal u * of the multiplier form as a value of an optimal dual variable associated with the envelopment model constraint eλ=. If u * = constant returns-to-scale prevails at (x, y ) by Theorem 5. If u * < (>) then solve the following LP max(min) u s. t. vx + uy u vx =, v, u e Let the optimal value be û *. By Theorem 5. we have: If û * (û * ) CRS prevails at (x, y ) If û * < (û * > ) IRS (DRS) prevails at (x, y ) uy If DMU is found to be inefficient Theorem 5.8 can be used to find out returns-toscale characteristics of its projected DMU (x P, y P ) u =

12 Relaxation of the Convexity Condition Convexity condition: eλ = New condition: L eλ U CCR model: L =, U = BCC model: L = U =

13 The Increasing Returns-to-Scale (IRS) Model IRS model: L =, U = Efficient frontier C D E y 3 B A x

14 The Decreasing Returns-to-Scale (DRS) Model DRS model: L =, U = Efficient frontier C D E y 3 B A x

15 The Generalized Returns-to-Scale (GRS) Model GRS model: L =.8, U =. 7 6 Efficient frontier 5 4 C D E y 3 B A x

16 Summary MPSS is achieved when (i) β*/α*= and (ii) all slacks are zero Returns-to-Scale characteristics of a DMU can be solved with a stepwise method Convexity condition can be relaxed to allow up- and downscaling

17 Home Assignment Characterize returns-to-scale of the BCC-projected activity of the BCC-inefficient DMU E. Include all calculation steps in your answer DMU A B C D E x 5 You can also return the solution with to timo.salminen@evli.com x 3 y 3

18 References Introduction to Data Envelopment Analysis and its uses, Cooper William W, 5