Chance Constrained Data Envelopment Analysis The Productive Efficiency of Units with Stochastic Outputs

Transcription

1 Chance Constrained Data Envelopment Analysis The Productive Efficiency of Units with Stochastic Outputs Michal Houda Department of Applied Mathematics and Informatics ROBUST 2016, September 11 16, 2016

2 Data Envelopment Analysis Notation, Efficiency Dominance DMU k k-th decision making unit (k = 1,, K) X := (x ik ) R m K input matrix x k := (x 1k,, x mk ) input vector of DMU k x i := (x i1,, x ik ) values for i-th input (i = 1,, m) Y := (y jk ) R n K input matrix y k := (y 1k,, y mk ) output vector of DMU k y j := (y j1,, y jk ) values for j-th output (j = 1,, n) PPS production possibility set combination of allowed inputs and outputs DMU 0 with (x 0, y 0 ) DMU to be analyzed Definition 1 DMU 1 dominates DMU 2 wrt PPS if x x 0 and y y 0 with at least one (one-dimensional, input or output) inequality strict Definition 2 DMU 0 is efficient wrt PPS if (x, y) PPS dominating (x 0, y 0 ) Michal Houda Chance-constrained DEA 2 / 21

3 Data Envelopment Analysis Notation, Efficiency Dominance DMU k k-th decision making unit (k = 1,, K) X := (x ik ) R m K input matrix x k := (x 1k,, x mk ) input vector of DMU k x i := (x i1,, x ik ) values for i-th input (i = 1,, m) Y := (y jk ) R n K input matrix y k := (y 1k,, y mk ) output vector of DMU k y j := (y j1,, y jk ) values for j-th output (j = 1,, n) PPS production possibility set combination of allowed inputs and outputs DMU 0 with (x 0, y 0 ) DMU to be analyzed Definition 1 DMU 1 dominates DMU 2 wrt PPS if x x 0 and y y 0 with at least one (one-dimensional, input or output) inequality strict Definition 2 DMU 0 is efficient wrt PPS if (x, y) PPS dominating (x 0, y 0 ) Michal Houda Chance-constrained DEA 2 / 21

4 Data Envelopment Analysis 0-1 Model Discrete Production Possibility Set Discrete PPS (BOWLIN, BRENNAN ET AL, 1984): PPS I = { (x k, y k ) } K k=1 Dominance wrt PPS I : additive model with integer constraints max ( s + j + s ) i subject to j i x ik λ k + s i = x i0 i (inputs) k y ik λ k s + i = y j0 j (outputs) k λ k = 1, λ k {0, 1} K, s i, s + j 0 k (1) s slack for Xλ x 0 // s + slack (surplus) for Yλ y 0 DMU 0 is efficient wrt PPS I if no slack is greater than 0 (i e, both inequalities are active) in optimal solution Michal Houda Chance-constrained DEA 3 / 21

5 Data Envelopment Analysis 0-1 Model Discrete Production Possibility Set x y A B C D E F G H Michal Houda Chance-constrained DEA 4 / 21

6 Data Envelopment Analysis BCC Model Continuous Production Possibility Set Continuous (convex) PPS (BANKER, COOPER, CHARNES, 1984): PPS C = { (x, y) x = Xλ, y = Yλ, λ k = 1, λ 0 } Dominance wrt PPS C : BCC (output oriented) model max ϕ + ϵ ( s + j + s ) i subject to j i x ik λ k + s i = x i0 i (inputs) k y ik λ k s + i = ϕy j0 j (outputs) k λ k = 1, λ k 0, s i, s + j k ϵ non-archimedean infinitesimal 0, ϕ unconstrained (2) Michal Houda Chance-constrained DEA 5 / 21

7 Data Envelopment Analysis BCC Model Continuous Production Possibility Set x y A B C D E F G H Michal Houda Chance-constrained DEA 6 / 21

8 Dual problem: Data Envelopment Analysis BCC Model Continuous Production Possibility Set Dual Problem min u T x 0 + q subject to u T x k v T y k + q 0 k (DMUs) v T y 0 = 1 (dual for ϕ) u ϵ1, v ϵ1, q unconstrained (3) q (dual for k λ k = 1) variable returns to scale (VRS) factor BCC (output oriented) DEA problem of fractional programming: min ut x 0 + q v T subject to y 0 u T x k + q v T 1 k (DMUs) y k v T y 0 = 1, u/v T y 0 ϵ1, v/v T y 0 ϵ1, q unconstrained (4) Michal Houda Chance-constrained DEA 7 / 21

9 Data Envelopment Analysis BCC Model Continuous Production Possibility Set DEA Efficiency Definition 3 (DEA Efficiency) DMU 0 is BCC-O (fully) efficient wrt PPS C if 1 ϕ = 1 2 s + = s = 0 Remark weak DEA efficiency: ϕ = 1 but some of s i, s + j are not zero (efficient points which are not extreme points of PPS) two-stage solution procedure: 1 solve the BCC-O problem with ϵ = 0 to obtain ϕ 2 solve the problem max j s+ j + i s i subject to remaining constraints where ϵ = 0 and ϕ = ϕ to obtain maximal possible slacks Michal Houda Chance-constrained DEA 8 / 21

10 Data Envelopment Analysis CCR Model Linear Production Possibility Set Linear PPS (CHARNES, COOPER, RHODES (1978)): PPS L = { (x, y) x = Xλ, y = Yλ, λ 0 } Dominance wrt PPS L : CCR (output oriented) model max ϕ + ϵ ( s + j + s ) i subject to j i x ik λ k + s i = x i0 i (inputs) k y ik λ k s + i = ϕy j0 j (outputs) k λ k 0, s i, s + j 0, ϕ unconstrained (5) Michal Houda Chance-constrained DEA 9 / 21

11 Data Envelopment Analysis CCR Model Linear Production Possibility Set x y A B C D E F G H Michal Houda Chance-constrained DEA 10 / 21

12 Data Envelopment Analysis CCR Model Linear Production Possibility Set Dual Problem Dual problem: min u T x 0 subject to u T x k v T y k 0 k (DMUs) v T y 0 = 1 (dual for ϕ) u ϵ1, v ϵ1 (6) q = 0 constant returns to scale (CRS) CCR (output oriented) DEA problem of fractional programming: min ut x 0 v T y 0 subject to u T x k v T y k 1 k (DMUs) v T y 0 = 1, u/v T y 0 ϵ1, v/v T y 0 ϵ1 (7) Michal Houda Chance-constrained DEA 11 / 21

13 Data Envelopment Analysis Random Data Introducing Randomness Introducing Randomness X, Y are random matrices PPS random production possibility sets α [0; 1) tolerance (risk) level (sufficiently small) LAND, LOVELL, THORE (1993), COOPER, HUANG, LI (1996), COOPER, HUANG ET AL (1998), COOPER, DENG, HUANG, LI (2002), COOPER, HUANG, LI (2004) Definition 4 DMU 0 is not stochastically dominated in its efficiency wrt PPS I if λ {0, 1} K with λ k = 1 we have P {Xλ x 0, Yλ y 0 } α if X, Y are continuous we don t need at least one strict notion Michal Houda Chance-constrained DEA 12 / 21

14 Stochastic 0-1 DEA Model Stochastic extension of the additive 0-1 model β := max P {Xλ x 0, Yλ y 0 } DMU 0 is stochastically dominated β > α Recall PPS I = { (x k, y k ) } K k=1 (random discrete production set) Michal Houda Chance-constrained DEA 13 / 21

15 Data Envelopment Analysis 0-1 Model Discrete Production Possibility Set x y A B C D E F G H Michal Houda Chance-constrained DEA 14 / 21

16 Stochastic Extension to BCC Model Joint Stochastic Efficiency Definition 5 1 (x, y ) PPS C is α-stochastically efficient wrt PPS C if λ 0 with k λ k = 1 we have P {Xλ x, Yλ y } α 2 the set of all α-stochastically efficient points is called α-stochastically efficient frontier of PPS C 3 DMU 0 is α-stochastically efficient wrt PPS C if λ {0, 1} K with λk = 1 we have P {Xλ x 0, Yλ y 0 } α Recall PPS C = { (x, y) x = Xλ, y = Yλ, λ k = 1, λ 0 } (random) Michal Houda Chance-constrained DEA 15 / 21

17 Stochastic Extension to BCC Model Necessary and Sufficient Conditions for Stochastic Efficiency Testing efficiency of DMU 0 Proposition 6 (sufficient condition) If P{1 T (Xλ x 0 ) < 1 T (Yλ y 0 )} α then DMU 0 is α-stochastically efficient Proposition 7 (necessary condition) If DMU 0 is α-stochastically efficient then λ 0 with λ k = 1 and P{x i λ < x i0 } 1 ϵ i (inputs) P{y j λ > y i0 } 1 ϵ j (outputs) then P{1 T (Xλ x 0 ) < 1 T (Yλ y 0 )} α Michal Houda Chance-constrained DEA 16 / 21

18 Stochastic Extension to BCC Model Joint Chance-Constrained Efficiency Almost 100% Confidence Chance-Constrained Problem β := max P{1 T (Xλ x 0 ) + 1 T (y 0 Yλ) < 0} subject to P{x i λ < x i0 } 1 ϵ i (inputs) P{y j λ > y i0 } 1 ϵ j (outputs) λ 0, λ k = 1 Proposition 8 1 If DMU 0 is α-stochastically efficient then β α 2 If β > α then DMU 0 is not α-stochastically efficient Michal Houda Chance-constrained DEA 17 / 21

19 Stochastic Extension to BCC Model Joint Chance-Constrained Efficiency Example: normally distributed outputs let Y follow multivariate normal distribution; denote X := EX, Ȳ := EY, and σ2 j (λ) := var(yt j λ y j0) β := min 1 T (Xλ x 0 ) + 1 T (ȳ 0 Ȳλ) + σ 0(λ)Φ ( 1) (α) subject to x i λ x i0 i (inputs) ȳ j λ + σ k (λ)φ ( 1) (ϵ) ȳ j j (outputs) λ 0, λ k = 1 Proposition 9 1 If DMU 0 is α-stochastically efficient then β 0 2 If β < 0 then DMU 0 is not α-stochastically efficient Michal Houda Chance-constrained DEA 18 / 21

20 Marginal Chance-Constrained DEA E-model for BCC COOPER, DENG, HUANG, LI (2002, 2003): E-model for BCC max ϕ + ϵ ( j s + j + i s i ) subject to P{x i λ + s i x i0 } 1 α i (inputs) P{y j λ s + j ϕy i0 } 1 α j (outputs) λ 0, λ k = 1 (8) Definition 10 DMU 0 is α-stochastically DEA efficient if 1 ϕ = 1 2 s + = s = 0 Michal Houda Chance-constrained DEA 19 / 21

21 Marginal Chance-Constrained DEA Normally Distributed Outputs let Y follow multivariate normal distribution; let Ȳ := EY, and σ2 j (ϕ, λ) := var(yt j λ ϕy j0) Then the (nonlinear) DEA problem to solve is max ϕ + ϵ ( j s + j + i s i ) subject to x i λ + s i = x i0 i (inputs) ȳ T j λ + Φ ( 1) (α)σ j (ϕ, λ) s + j = ϕȳ j0 j (outputs) λk = 1, λ k 0, s i, s + j 0 (9) (DEA problem with adjusted outputs) We can use second-order cone programming approximation scheme to find the upper and lower bounds for the problem (cf CHENG, HOUDA, LISSER (2014)) Michal Houda Chance-constrained DEA 20 / 21

22 References Cooper, W W, Seiford, L M and Zhu, J, eds (2004) Handbook on Data Envelopment Analysis Kluwer, New York Cooper, W W, Deng, H, Huang, Z and Li, S X (2002) Chance constrained programming approaches to technical efficiencies and inefficiencies in stochastic data envelopment analysis Journal of the Operational Research Society, 53(12), Cheng, J, Houda, M and Lisser, A (2014) Second-order cone programming approach for elliptically distributed joint probabilistic constraints with dependent rows Optimization Online, Paper No 4363, May 2014 Michal Houda Chance-constrained DEA 21 / 21