A New Weight-Restricted DEA Model Based on PROMETHEE II 2 nd - PowerPoint PPT Presentation

A New Weight-Restricted DEA Model Based on PROMETHEE II 2 nd International MCDA workshop on PROMETHEE: Research and case 2 studies Université Libre de Bruxelles-Vrije Universiteit Brussel Belgium Maryam Bagherikahvarin, Yves De Smet 23 January 2015 Keywords: Data Envelopment Analysis, Multi Criteria Decision Aid, PROMETHEE, Stability Intervals, Weight Restrictions

Outline � DEA & MCDA � DEA � MCDA: PROMETHEE II � Synergies � Objective � Methodology � Numerical Examples � The main advantages of this work & further ideas 2

1. DEA & MCDA 2 research areas in OR/MS Inputs + Evaluating Outputs & = Ranking Units Ranking Units Criteria DEA & MCDA DMUs Optimized = & Alternatives Compromised solution 3

1. DEA & MCDA DEA MCDA - A decision making tool in the - Non-parametric and non- presence of conflicting criteria statistical method Combining and absence of optimal several measures of inputs solution: Sorting , Ranking and solution: Sorting , Ranking and and outputs into a single and outputs into a single Choosing alts measure of efficiency - Assigning pre-determined - Generating automated weights to Criteria weights by model -MAUT, AHP, Outranking - CCR, BCC, Additive, FDH, (ELECTRE, PROMETHEE ), Super efficiency, … Interactive 4

1. DEA & MCDA Ranking and Selecting between bank branches, health care centers (Flokou, A. et al., 2010), educational institutions (Salerno, C., 2006), localization of a factory (Vaninsky, A., 2008), proper ways for a project, … � Shanghai ranking (Academic Ranking of World Universities, Shanghai Jiao Tong University, 2007), (Jean-Charles Billaut, Denis Bouyssou, Philippe Vincke, 2009) Bouyssou, Philippe Vincke, 2009) � FIFA world ranking � Country’s ranking in Globalization � Largest producing countries of agricultural commodities, … 5

Outline � DEA & MCDA � DEA � MCDA: PROMETHEE II � Synergies � Objective � Methodology � Numerical Examples � The main advantages of this work & further ideas 6

2. DEA A DEA example: 6 CRS Frontier 5 5 4 4 VRS Frontier Sale 3 3 3 2 2 Production Possibility Set 1 1 0 Store Sale Employee Efficiency 1 1 2 0.5 0 1 2 3 4 5 6 7 2 2 4 0.5 Em plo yee 3 3 3 1 4 4 5 0.8 Figure 1- Efficient frontier 5 5 6 0.83 7

2. DEA BCC Input-Oriented Envelopment model Multiplier model s µ µ µ µ y ∑ ∑ ∑ ∑ m s max ∑ ∑ ∑ ∑ z = = = = + + + + u ∑ ∑ ∑ ∑ s s min ( − − − − + + + + ) θ θ θ θ − − − − ε ε ε ε + + + + o i r r ro i = = = = 1 r = = = = 1 r = = = = 1 . . s t . . s t n ∑ ∑ ∑ ∑ x λ λ λ λ s x − − + + + + − − = = = = θ θ θ θ , = = = = 1 , 2 ,..., ; s m i m µ µ ∑ ∑ ∑ ∑ µ µ y ∑ ∑ ∑ ∑ ν ν ν ν x − − + + ≤ ≤ 0 − − + + u e ≤ ≤ ij j i io 1 j = = = = o i ij n ∑ ∑ y y r rj ∑ ∑ λ λ λ λ s + + + + , 1 , 2 ,..., ; − − − − = = = = r = = = = s = = 1 = = 1 r = = i = = m ∑ ∑ ∑ ∑ ν ν ν ν x j r 1 = = = = rj ro j = = = = 1 n ∑ ∑ ∑ ∑ λ λ λ λ = = = = 1 i ij i = = = = 1 µ µ µ µ ν ν j ν ν , ≥ ≥ ≥ ≥ ε ε ε ε > > > > 0 , u free in sign = = 1 j = = λ λ λ λ s − − s + + , − − , + + ≥ ≥ ≥ ≥ 0 , = = = = 1 , 2 ,..., . j n o i r j i r BCC Output-Oriented BCC Output-Oriented Envelopment model Multiplier model m ∑ ∑ ∑ ∑ ν ν ν ν x min q = = = = − − − − ν ν ν ν m s ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ s − − s + + max φ φ + + ε ε ( − − + + + + ) φ φ + + ε ε + + i io o = = 1 i = = i r . . s t i = = = = 1 r = = = = 1 . . s t m s n ∑ ∑ − − − − ≥ ≥ 0 ∑ ∑ λ λ − − − − ν ν e ≥ ≥ x λ λ s x ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ν ν − − − − , 1 , 2 ,..., ; µ µ + + + + = = = = i = = = = m µ µ yrj ν ν ν ν xij i r = = = = 1 = = = = 1 o ij j i io i r j = = = = 1 s n φ φ φ φ ∑ ∑ ∑ ∑ y y λ λ λ λ s + + − − − − + + = = = = , r = = = = 1 , 2 ,..., s ; 1 = = = = ∑ ∑ ∑ ∑ µ µ µ µ yro r j r rj ro 1 j = = = = = = 1 r = = n ∑ ∑ ∑ ∑ 1 λ λ λ λ = = = = , ≥ ≥ ≥ ≥ 0 , ε ε > > ν ν ε ε > > ν ν free in sign µ µ µ µ ν ν ν ν j i r j = = = = 1 o λ λ λ λ s s , − − − − , + + + + 0 , 1 , 2 ,..., . ≥ ≥ ≥ ≥ j = = = = n j i r Table 1- Different BCC models (Cooper et al. , 2004) 8

2. DEA Some difficulties in DEA � No common set of weights � No strict bounding for weights (probability of having non- realistic answers): - Some inputs or outputs can be characterized by low or high weight values; - Some inputs or outputs can be characterized by low or high weight values; - Contradiction with a priori information offered by the Decision Maker (DM). � DMUs can not be ranked with such a weights, which may vary from unit to unit 9

2. DEA Weight Restricted DEA models � Thompson et al. (1986): assessing the efficiency of physics laboratories (AR), � Dyson and Thanassoulis (1988): eliminating use of zero weights (RA), � Wong and Beasley (1990): introducing virtual weights DEA models, � Roll and Golany (1993): using generated weights of DEA model, � Takamura and Tone (2003): using the judgments of people, � Ueda (2000,2007): suggesting a canonical correlation analysis, � Dimitrov and Sutton (2012): proposing a symmetric weight assignment technique. 10

2. DEA Using MCDA in DEA to determine bounds � DEA and AHP: � Shang et Sueyoshi (1995): using subjective AHP results in DEA to rank and select between flexible manufacturing systems: the pareto solutions of DEA and the subjectivity of AHP � Sinuany-Stern et al. (2000): suggesting two stage AHP/DEA ranking model: removing the pitfalls of Shang et Sueyoshi but does not incorporate the DM preferences incorporate the DM preferences � Takamura and Tone (2003): integrating AR and AHP : 1. providing criteria weights for each DM by AHP , 2. employing AR to limit them: more than one DM � Liu (2003): Combining DEA and AHP to integrate two objective and subjective weight restrictions method � Han-Lin Li and Li-Ching Ma (2008): Developing an iterative method of ranking DMUs by integrating DEA , AHP and Gower plot 11

2. DEA Some unwillingness of AHP � Lack of undeniable foundations on the utility preferences of the DM (Saati, 1986, Barzilai et al., 1987, Dyer, 1990, Winkler, 1990) ; � No special graphical tool; � � Subjectivity: constructing a pair wise comparison matrix based on DM's preferences. From the view point of a DM: easier to use some models with less subjectivity to evaluate different alternatives (Sinuany-Stern et al., 2000) . 12

2. DEA � DEA and MACBETH: � Junior (2008): Employing MACBETH as a MCDA tool to produce the bounds of the weights and adding these restrictions to a virtual weight DEA model to evaluate the alternatives/DMUs. MACBETH : a MCDA approach to help an individual or a group, quantifying MACBETH : a MCDA approach to help an individual or a group, quantifying the relative attractiveness of options by qualitative judgements about differences in value (Bana e Costa et al., 1993) � Causing a contradicted result with MACBETH ranking. To avoid this weakness: adding some extra constraints to the virtual weight restrictions 13

Outline � DEA & MCDA � DEA � MCDA: PROMETHEE II � Synergies � Objective � Methodology � Numerical Examples � The main advantages of this work & further ideas 14

4. PROMETHEE II PROMETHEE II � J. P. Brans (1982): based on pair wise comparisons : allowing a DM to rank completely a finite set of n actions that are evaluated over a set of k criteria: • For each criterion f j , j=1,2,…,k: P j (a,b) – Preference function P j 1 j – Weight w j 0 q j p j d j (a,b) • Preference degree of a over b : k ( ) ( ) = ∑ , , π a b w P a b j j 1 = j 15

4. PROMETHEE II • Net flow score k ( ) ( ) ∑ φ = ⋅ φ a w a j j = 1 j with with 1 ( ) − ∑ ( ) ( ) , , φ = −   a P a b P b a   1 n j j j b A ∈ • Unicriterion net flow score 16

4. PROMETHEE II Weight Stability Intervals (Mareschal, B. (1988)) � what is the impact of changing a given weight value in a computed ranking? Determination of exact weight values is often a cognitive complex task for the DM. DM. Purpose of WSI: Preserve the preference ranking of a subset of alternatives: automated generation of intervals limits (confirming the robustness of PROMETHEE II outputs, typically the first alternative). 17

Outline � DEA & MCDA � DEA � MCDA: PROMETHEE II � Synergies � Objective � Methodology � Numerical Examples � The main advantages of this work & further ideas 18