Tricks prof. . Meh ehdi i TOLOO, Ph.D .D. Department of Systems - PowerPoint PPT Presentation

Integer Linear Programming Tricks prof. . Meh ehdi i TOLOO, Ph.D .D. Department of Systems Engineering, Faculty of Economics, VŠB - Technical University of Ostrava, Czech Republic Email: mehdi.toloo@vsb.cz URL: http://homel.vsb.cz/~tol0013/

Contents 1. A variable taking discontinuous values 2. Fixed costs 3. Either-or constraints 4. Conditional constraints 5. Elimination of products of variables 5.1 Two binary variables 5.2 One binary and one continuous variable 5.3 Two continuous variables 6. Applications 2

1. A variable taking discontinuous values 𝑦 = 0 𝑝𝑠 𝑚 ≤ 𝑦 ≤ 𝑣 Indicator variable method 3

1. A variable taking discontinuous values 𝑧𝑚 ≤ 𝑦 ≤ 𝑧𝑣; 𝑧 ∈ {0,1} Validation? 4

Validation 𝒚 constraints 𝒛 0 ≤ 𝑣𝑧 0 ≥ 𝑚𝑧 0 0 𝑧 ∈ {0,1} 𝑦 ≤ 𝑣𝑧 𝑦 ≥ 𝑚𝑧 l ≤ 𝑦 ≤ 𝑣 1 𝑧 ∈ {0,1} 5

2. Fixed cost 6

2. Fixed cost (illustration) 7

Modelling Fixed cost Let 𝑦 ≤ 𝑣 . Consider the following indicator variable y: 1. 𝐷(𝑦, 𝑧) = 𝑙𝑧 + 𝑑𝑦 2. 𝑦 ≤ 𝑣𝑧 If 𝑦 > 0 and 𝑦 ≤ 𝑣𝑧 , then 𝑧 = 1 . If 𝑦 = 0 , how we have 𝑧 = 0 ? 8

The equivalent MBLP model 9

3. Either-or constraints 10

Modeling either-or constraints

Equivalent MBLP model

Other forms 𝑘∈𝐾 𝑏 1𝑘 𝑦 𝑘 ≤ 𝑐 1 𝑘∈𝐾 𝑏 1𝑘 𝑦 𝑘 ≤ 𝑐 1 + 𝑁𝑧 then • or 𝑘∈𝐾 𝑏 2𝑘 𝑦 𝑘 ≥ 𝑐 2 − 𝑁(1 − 𝑧) 𝑘∈𝐾 𝑏 2𝑘 𝑦 𝑘 ≥ 𝑐 2 𝑘∈𝐾 𝑏 1𝑘 𝑦 𝑘 ≤ 𝑐 1 or 𝑘∈𝐾 𝑏 1𝑘 𝑦 𝑘 ≤ 𝑐 1 𝑘∈𝐾 𝑏 2𝑘 𝑦 𝑘 ≤ 𝑐 2 then hence • or 𝑘∈𝐾 𝑏 2𝑘 𝑦 𝑘 = 𝑐 2 and 𝑘∈𝐾 𝑏 2𝑘 𝑦 𝑘 ≥ 𝑐 2 𝑘∈𝐾 𝑏 1𝑘 𝑦 𝑘 ≤ 𝑐 1 + 𝑁𝑧 𝑘∈𝐾 𝑏 2𝑘 𝑦 𝑘 ≤ 𝑐 2 + 𝑁(1 − 𝑧) 𝑘∈𝐾 𝑏 2𝑘 𝑦 𝑘 ≥ 𝑐 2 − 𝑁(1 − 𝑧) 13

4. Conditional constraints 14

4. Conditional constraints Indicator variable 15

5. Elimination of products of variables 5.1 Two binary variables 5.2 One binary and one continuous variable 5.3 Two continuous variables

5.1 Two binary variables Let 𝑐 1 , 𝑐 2 ∈ {0,1} and 𝑨 = 𝑐 1 . 𝑐 2 𝑧 ≤ 𝑐 1 𝑧 ≤ 𝑐 2 𝑨 = 𝑐 1 . 𝑐 2 ⟺ 𝑧 ≥ 𝑐 1 + 𝑐 2 − 1 𝑧 ∈ {0,1} Validation?

Two binary variables (Validation) 𝒄 𝟐 𝒄 𝟑 𝒜 = 𝒄 𝟐 𝒄 𝟑 constraints 𝒛 𝑧 ≤ 0 𝑧 ≤ 0 0 0 0 0 𝑧 ≥ −1 𝑧 ≤ 0 𝑧 ≤ 1 0 1 0 0 𝑧 ≥ 0 𝑧 ≤ 1 𝑧 ≤ 0 1 0 0 0 𝑧 ≥ 0 𝑧 ≤ 1 𝑧 ≤ 1 1 1 1 1 𝑧 ≥ 1

5.2 One binary and one continuous variable Let 𝑐 ∈ 0,1 , 0 ≤ 𝑦 ≤ 𝑉, and 𝑨 = 𝑐𝑦 𝑧 ≤ 𝑉𝑐 𝑧 ≤ 𝑦 𝑨 = 𝑐𝑦 ⟺ 𝑧 ≥ 𝑦 − 𝑉(1 − 𝑐) 𝑧 ≥ 0 Validation?

Validation 𝒄 𝒚 𝒜 = 𝒄𝒚 constraints 𝒛 𝑧 ≤ 0 𝑧 ≤ 𝑦 0 0 ≤ 𝑦 ≤ 𝑉 0 0 𝑧 ≥ 𝑦 − 𝑉 𝑧 ≥ 0 𝑧 ≤ 𝑉 𝑧 ≤ 𝑦 1 0 ≤ 𝑦 ≤ 𝑉 0 ≤ 𝑨(= 𝑦) ≤ 𝑉 0 ≤ 𝑧(= 𝑦) ≤ 𝑉 𝑧 ≥ 𝑦 𝑧 ≥ 0

5.3 Two continuous variables Let 𝑦 1 , 𝑦 2 ∈ ℝ and 𝑨 = 𝑦 1 𝑦 2 1 𝑧 1 = 2 𝑦 1 + 𝑦 2 , 𝑧 1 ∈ ℝ Step 1. 1 𝑧 2 = 2 𝑦 1 − 𝑦 2 , 𝑧 2 ∈ ℝ 2 − 𝑧 2 Step 2. 𝑨 = 𝑦 1 𝑦 2 = 𝑧 1 2 Step 3. solve the non-linear problem via separable programming (see Taha 2005)

Bounds on 𝑦 1 and 𝑦 2 If L 1 ≤ 𝑦 1 ≤ 𝑉 1 , 𝑀 2 ≤ 𝑦 2 ≤ 𝑉 2 Then 1 2 𝑀 1 + 𝑀 2 ≤ 𝑧 1 ≤ 1 2 𝑁 1 + 𝑁 2 1 2 𝑀 1 − 𝑁 2 ≤ 𝑧 2 ≤ 1 2 𝑁 1 − 𝑀 2

5.3 Two continuous variables (Special case) The product 𝑦 1 𝑦 2 can be replaced by a single variable y whenever: 1. The lower bounds 𝑀 1 and 𝑀 2 are nonnegative 2. One of the variables, say 𝑦 1 , is not referenced in any other term except in products of the above form. Then substituting for 𝑧 and adding the following constraint 𝑀 1 𝑦 2 ≤ 𝑧 ≤ 𝑉 1 𝑦 2

Applications in DEA • Flexible measures • Selective measures • Best efficient unit

Multiplier form of the CCR model x 11 y 11 DMU 1 y s 1 x m 1 x 1 n y 1 n DMU n y sn x mn 25

Inputs and Outputs in DEA • Smaller input amounts are preferable. • Larger output amounts are preferable. • I/O should reflect an analyst's or a manager's interest in the components that will enter into the relative efficiency evaluations of the DMUs. 26

Flexible & Selective measures • Data play an important and critical role in DEA and selecting input and output measures is an essential issue in this method. • The input versus output status of the chosen performance measures is known. • In some situations, certain performance measures can play either input or output roles, which are called flexible measures . • selective measures deal with the rule of thumb. 27

Mostafa (2009) 28

Flexible Measures • Beasley (1990, 1995) firstly faced with a data selection issue in DEA. He found that research income measure in the evaluation of research productivity by universities can be considered either as input or output. • Some other such measures are: • Uptime measure in evaluating robotics installations (Cook et al., 1992) • Outages measure in the evaluation of power plants (Cook et al., 1998) • Deposits measure in the evaluation of bank efficiency (Cook and Zhu 2005) • medical interns have a similar interpretation in the evaluation of hospital efficiency. 29

Flexible measures 30

Flexible Measure Cook & Zhu (2007) ∗ = 0 presents an input status There are 2 𝑀 various cases (combinations) 𝑒 𝑚 ∗ = 1 presents an output status 𝑒 𝑚 for 𝑀 flexible measures. 31

Charnes & Cooper (1962)’s transformation

Linearization ∗ = 0 → 𝜀 𝑚 ∗ = 0 → 𝑨 𝑚 presents an input status 𝑒 𝑚 ∗ = 1 → 𝑥 𝑚 ∗ = 0 → 𝑨 𝑚 presents an output status 𝑒 𝑚

overall input v.s. output status ∗ = 0 ; 𝐾 𝑝𝑣𝑢 𝑚 = 𝑝: 𝑒 𝑚 ∗ = 1 Let 𝐾 𝑗𝑜 𝑚 = 𝑝: 𝑒 𝑚 1. If 𝐾 𝑗𝑜 𝑚 > |𝐾 𝑝𝑣𝑢 𝑚 | , then flexible measure 𝑚 must be selected as input. 2. If 𝐾 𝑗𝑜 𝑚 < |𝐾 𝑝𝑣𝑢 𝑚 | , then flexible measure 𝑚 must be selected as output. 3. = |𝐾 𝑝𝑣𝑢 𝑚 | ? 𝐾 𝑗𝑜 𝑚 34

Aggregated v.s. Individual 35

Toloo (2009, 2014) 36

Toloo (2009) 37

Toloo (2014) ∗ = max{𝑓 𝑙 Theorem: 𝑓 𝑝 ∗ : 𝑙 = 1, … , 2 𝑀 } 38

University evaluation ( Beasley1990 ) • 50 universities • 2 inputs: General Expenditure (GE) and Equipment Expenditure (EE) • 3 outputs: Under Graduate Students (UGS), Post Graduate Teaching (PGT), and PG Research (PGR) • 1 flexible measure: Research Income (RI)

Cook and Zhu (2007): 20 out of the 50 universities treat the research income measure as an output, i.e., the majority of 30 treat it as an input.

41

Envelopment form of the CCR model

Flexible measure in envelopment form 𝑜 𝑘=1 𝜇 𝑘 𝑨 𝑚𝑘 ≤ 𝜄𝑨 𝑚𝑝 𝑨 𝑚 as input 𝑜 𝑘=1 𝜇 𝑘 𝑨 𝑚𝑘 ≥ 𝑨 𝑚𝑝 𝑨 𝑚 as output 𝑜 𝜇 𝑘 𝑨 𝑚𝑘 ≤ 𝜄𝑨 𝑚𝑝 + 𝑁 𝑘=1 𝑒 𝑚 𝜇 𝑘 𝑨 𝑚𝑘 ≥ 𝑨 𝑚𝑝 − 𝑁(1 − 𝑜 𝑘=1 𝑒 𝑚 )

Toloo (2012) 44

Toloo (2012)

Cook & Zhu (2207) Toloo (2012) 46

Selective measures The rule of thumb in DEA: How to meet the rule?

illustration • consider the problem of evaluating 50 branches of a bank with 25 inputs and 30 outputs. • The total number of measures, i.e. 55, and DMUs do not satisfy the rule of thumb and we subsequently encounter many efficient units. • To make the problem easier, suppose that the manager pre-selected three inputs, e.g. employees, expenses and space, and three outputs, e.g. loans, profits and deposits. • With this assumptions, if the manager wants to select 2 out of 22 remaining inputs and 1 out of 27 remaining outputs and also consider all possible combinations of performance measures, then an optimization problem must be solved at most 196350(= 50 × 22 × 27 1 ) 2 times, which is illogical.

Toloo et al. (2015) 49

Assumptions • Let 𝑡 1 and 𝑡 2 denote subsets of outputs corresponding to fixed-output and selective- output measures, respectively. Similarly, assume that 𝑛 1 and 𝑛 2 are the parallel subsets of inputs.

Toloo et al. (2015)

Linearization Theorem: The selecting model meets the rule of thumb

Special case: (𝑡 1 = 𝑛 1 = 𝜚)

Toloo & Tichy(2015) 54

Envelopment form of selecting model Toloo & Tichy (2015)

Linearization 𝑦 for ∈ 𝑛 2 . 𝑦 = 𝑒 𝑗 𝑦 𝑡 𝑗 • Let 𝑢 𝑗 • Impose the following restrictions on the model: 𝑦 ≤ 𝑁𝑒 𝑗 𝑦 0 ≤ 𝑢 𝑗 𝑦 − 𝑁 1 − 𝑒 𝑗 𝑦 ≤ 𝑢 𝑗 𝑦 ≤ 𝑡 𝑗 𝑦 𝑡 𝑗

Equivalent MBLP model