Piecewise linear value functions for multi-criteria decision-making Delft University of Technology Piecewise linear value functions for multi-criteria decision-making Rezaei, Jafar DOI 10.1016/j.eswa.2018.01.004 Publication date 2018 Document Version Final published version Published in Expert Systems with Applications Citation (APA) Rezaei, J. (2018). Piecewise linear value functions for multi-criteria decision-making. Expert Systems with Applications, 98, 43-56. https://doi.org/10.1016/j.eswa.2018.01.004 Important note To cite this publication, please use the final published version (if applicable). Please check the document version above. Copyright Other than for strictly personal use, it is not permitted to download, forward or distribute the text or part of it, without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license such as Creative Commons. Takedown policy Please contact us and provide details if you believe this document breaches copyrights. We will remove access to the work immediately and investigate your claim. This work is downloaded from Delft University of Technology. For technical reasons the number of authors shown on this cover page is limited to a maximum of 10. https://doi.org/10.1016/j.eswa.2018.01.004 https://doi.org/10.1016/j.eswa.2018.01.004 Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public. https://www.openaccess.nl/en/you-share-we-take-care Expert Systems With Applications 98 (2018) 43–56 Contents lists available at ScienceDirect Expert Systems With Applications journal homepage: www.elsevier.com/locate/eswa Piecewise linear value functions for multi-criteria decision-making Jafar Rezaei Faculty of Technology Policy and Management, Delft University of Technology, Delft, The Netherlands a r t i c l e i n f o Article history: Received 24 August 2017 Revised 1 January 2018 Accepted 2 January 2018 Available online 3 January 2018 Keywords: Multi-criteria decision-making MCDM Decision criteria Value function Monotonicity a b s t r a c t Multi-criteria decision-making (MCDM) concerns selecting, ranking or sorting a set of alternatives which are evaluated with respect to a number of criteria. There are several MCDM methods, the two core el- ements of which are (i) evaluating the performance of the alternatives with respect to the criteria, (ii) finding the importance (weight) of the criteria. There are several methods to find the weights of the cri- teria, however, when it comes to the alternative measures with respect to the criteria, usually the existing MCDM methods use simple monotonic linear value functions. Usually an increasing or decreasing linear function is assumed between a criterion level (over its entire range) and its value. This assumption, how- ever, might lead to improper results. This study proposes a family of piecewise value functions which can be used for different decision criteria for different decision problems. Several real-world examples from existing literature are provided to illustrate the applicability of the proposed value functions. A numerical example of supplier selection (including a comparison between simple monotonic linear value functions, piecewise linear value functions, and exponential value functions) shows how considering proper value functions could affect the final results of an MCDM problem. © 2018 Elsevier Ltd. All rights reserved. 1 d t i h p p f w a f l U w a l o t a c a U d t w d s t t t l ( c D o C o n h t o t i D h 0 . Introduction Decision theory is primarily concerned with identifying the best ecision. In many real-world situations the decision is to select he best alternative(s) from among a set of alternatives consider- ng a set of criteria. This subdivision of decision-making, which as gained enormous attention, due to its practical value, in the ast recent is called multi-criteria decision-making (MCDM). More recisely, MCDM concerns problems in which the decision-maker aces m alternatives ( a 1 , a 2 , …, a m ), which should be evaluated ith respect to n criteria ( c 1 , c 2 , …, c n ), in order to find the best lternative(s), rank or sort them. In most cases, an additive value unction is used to find the overall value of alternative i, U i , as fol- ows: i = n ∑ j=1 w j u i j , (1) here u ij is the value of alternative i with respect to criterion j , nd w j shows the importance (weight) of criterion j . In some prob- ems, the decision-maker is able to find u ij from external sources as bjective measures, in some other problems, u ij reflects a qualita- ive evaluation provided by the decision-maker(s), experts or users s subjective measures. Price of a car is an objective criterion while omfort of a car is a subjective one. For objective criteria, we usu- E-mail address: j.rezaei@tudelft.nl t [ ttps://doi.org/10.1016/j.eswa.2018.01.004 957-4174/© 2018 Elsevier Ltd. All rights reserved. lly use physical quantities, for instance, ‘International System of nits’ (SI), while for subjective criteria, we do not have such stan- ards, which is why we mostly use pairwise comparison, linguis- ic variables, or Likert scales in order to evaluate the alternatives ith regard to such criteria. In order to find the weights, w j , the ecision-maker might use different tools and methods, from the implest way, which is assigning weights to the criteria intuitively, o use simple methods like SMART (simple multi-attribute rating echnique) ( Edwards, 1977 ), to more structured methods like mul- iple attribute utility theory (MAUT) ( Keeney & Raiffa, 1976 ), ana- ytic hierarchy process (AHP) ( Saaty, 1977 ), and best worst method BWM) ( Rezaei, 2015, 2016 ). While these methods are usually alled ‘multi attribute utility and value theories’ ( Carrico, Hogan, yson, & Athanassopoulos, 1997 ), there is another class of meth- ds, called outranking methods, like ELECTRE (ELimination and hoice Expressing REality) family ( Roy, 1968 ), PROMETHEE meth- ds ( Brans, Mareschal, & Vincke, 1984 ) which do not necessarily eed the weights to select, rank or sort the alternatives. What, owever, is in common in these methods is the way they consider he nature of the criteria. That is to say, in the current literature, ne of the common assumptions about the criteria (most of the ime it is not explicitly mentioned in the literature), is monotonic- ty. efinition 1 ( Keeney & Raiffa, 1976 ). Let u represents a value func- ion for criterion X , then u is monotonically increasing if: x 1 > x 2 ] ⇔ [ u ( x 1 ) > u ( x 2 ) ] . (2) https://doi.org/10.1016/j.eswa.2018.01.004 http://www.ScienceDirect.com http://www.elsevier.com/locate/eswa http://crossmark.crossref.org/dialog/?doi=10.1016/j.eswa.2018.01.004&domain=pdf mailto:j.rezaei@tudelft.nl https://doi.org/10.1016/j.eswa.2018.01.004 44 J. Rezaei / Expert Systems With Applications 98 (2018) 43–56 V Fig. 1. Increasing value function. t a w b s v t c f 2 fi t s w d s s a d t 2 f r i u less efficient fuel energy. 1 It is worth-mentioning that the studies we discuss to support each value func- tion have some theoretical or practical support for the proposed value functions. It does not, however, mean that those studies have used these value functions in their analysis. Definition 2 ( Keeney & Raiffa, 1976 ). Let u represents a value func- tion for criterion X , then u is monotonically decreasing if: [ x 1 > x 2 ] ⇔ [ u ( x 1 ) < u ( x 2 ) ] . (3) A function which is not monotonic is called non-monotonic and may have different shapes. For instance, a value function with the first part increasing and the second part decreasing called non- monotonic, by splitting of which, we have two monotonic func- tions. This assumption – monotonicity – however, is an oversimpli- fication in some real-world decision-making problems. Another simplification is the use of simple linear functions over the en- tire range of a criterion. Considering the two assumptions (mono- tonicity, linearity), we usually see simple increasing and decreas- ing linear value functions for the decision criteria in MCDM prob- lems. The literature is full of such applications. For instance, many of the studies reviewed in the following review papers implic- itly adopt such assumptions: the MCDM applications in supplier selection ( Ho, Xu, & Dey, 2010 ), in infrastructure management ( Kabir, Sadiq, & Tesfamariam, 2014 ), in sustainable energy plan- ning ( Pohekar & Ramachandran, 2004 ), and in forest management and planning ( Ananda & Herath, 2009 ). While in some studies the use of monotonic and/or linear value function might be logical, their use in some other applications might be unfitting. For in- stance, Alanne, Salo, Saari, and Gustafsson (2007) , for evaluation of residential energy supply systems use monotonic-linear value functions for all the selected evaluation criteria including “global warming potential (kg CO 2 m −2 a −1 )”, and “acidification potential (kg SO 2 m −2 a −1 )”. Considering a monotonic-linear value function for such criteria implies that the decision-maker accepts any level of such harmful environmental criteria for an energy supply sys- tem. However, if the decision-maker does not accept some high levels of such criteria (which seems logical), a piecewise linear function might better represent the preferences of the decision- maker (see the decrease-level value function in the next section). Some authors have discussed nonlinear monotonic value functions (e.g., exponential value functions by Kirkwood, 1997; Pratt, 1964 ). Others use qualitative scoring to address the non- monotonicity ( Brugha, 20 0 0; Kakeneno & Brugha, 2017; O’Brien & Brugha, 2010 ). We can also find some forms of eliciting piecewise linear value function in Jacquet-Lagreze and Siskos (2001 ), and Stewart and Janssen (2013 ). Some other value function construc- tion or elicitation frameworks can be found in Herrera, Herrera- iedma, and Verdegay (1996 ), Lahdelma and Salminen (2012 ), Mustajoki and Hämäläinen (20 0 0 ), Stewart and Janssen (2013 ), and Yager (1988 ). Although in PROMETHEE we use different types of piecewise functions for pairwise comparisons ( Brans, Mareschal, & Vincke, 1984 ), the functions are not used to evaluate the decision criteria. So, despite some effort s in literature, there is no a library of some standard piecewise linear value functions which can be used in different methods like AHP or BWM. It is also important to note that while in many studies value functions are elicited ac- cording to the preference data we have from the decision-maker(s), in MCDM, usually we use the value function as a subjective input. This implies that, in MCDM methods (except a few methods, such as UTA), the value function is not elicited, but an approximation is used. This also suggests that the rich literature on determining and eliciting value functions is not actually helping MCDM methods in this area. In this paper, first, a number of piecewise linear value functions with different shapes are proposed to be considered for the decision criteria. It is then shown, with some real-world ex- amples, how such consideration might change the final results of a decision problem. A comparison between simple monotonic linear value functions, piecewise linear value functions, and exponential value functions is conducted, which shows the effectiveness of the proposed pricewise value functions. This is a significant contribu- ion to this field and it is expected to be widely used by MCDM pplications. In the next section, some piecewise linear value functions along ith some real-world examples are presented, which is followed y some remarks in Section 3 . In Section 4 , some numerical analy- es are used to show the applicability of considering the proposed alue functions in a decision problem. In Section 5 , the determina- ion of the value functions is discussed. In Section 6 , the paper is oncluded, some limitations of the study are discussed, and some uture research directions are proposed. . Piecewise linear value functions In this section, a number of piecewise value functions are de- ned for decision criteria. We provide some example cases from he existing literature or practical decision-making problems to upport 1 each value function. In all the following value functions e consider [ d l j , d u j ] as the defined domain for the criterion by the ecision-maker; x ij shows the performance of alternative i with re- pect to criterion j ; and u ij shows the value of alternative i with re- pect to criterion j . For instance, if a decision-maker wants to buy car considering price as one criterion, if all the alternatives the ecision-maker considers are between €17,0 0 0 and €25,0 0 0, then he criterion might be defined for this range [17,0 0 0, 25,0 0 0] . .1. Increasing Increasing value function is perhaps the most commonly used unction in MCDM applications. It basically shows that as the crite- ion level, x ij , increases, its value, u ij , increases as well. It is shown n Fig. 1 and formulated as follows: i j = ⎧ ⎨ ⎩ x i j − d l j d u j − d l j , d l j ≤ x i j ≤ d u j , 0 , otherwise . (4) For this function we can think of: • Product quality in supplier selection ( Xia & Wu, 2007 ). Con- sidering a set of suppliers, a buyer may always prefer a sup- plier with a higher product quality compared to a supplier with lower product quality. • Energy efficiency in alternative-fuel bus selection ( Tzeng, Lin, & Opricovic, 2005 ). Considering a set of buses, a bus with more efficient fuel energy might always be preferred to a bus with J. Rezaei / Expert Systems With Applications 98 (2018) 43–56 45 Fig. 2. Decreasing value function. Fig. 3. V-shape value function. 2 i l u 2 c a i u r t Fig. 4. Inverted V-shape value function. 2 x a F u 2 i t l u .2. Decreasing Decreasing value function shows that as the criterion level, x ij , ncreases, its value, u ij , decreases. It is shown in Fig. 2 and formu- ated as follows: i j = ⎧ ⎨ ⎩ d u j − x i j d u j − d l j , d l j ≤ x i j ≤ d u j , 0 , otherwise . (5) For this function we can think of: • Product price in supplier selection ( Xia & Wu, 2007 ). Consid- ering a set of suppliers, a supplier with a lower product price might always be preferred to a supplier with higher product price. So, a higher product price has a lower value. • Maintenance cost in alternative-fuel bus selection ( Tzeng et al., 2005 ). Considering a set of buses, a bus with less maintenance cost might be preferred to a bus with higher maintenance cost. So, a higher maintenance cost is associated with a lower value. .3. V-shape V-shape value function shows that as the criterion level, x ij , in- reases up to a certain level, d m j , its value, u ij , decreases gradually, nd after that certain level, d m j , its value, u ij , increases gradually. It s shown in Fig. 3 and formulated as follows: i j = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ d m j − x i j d m j − d l j , d l j ≤ x i j ≤ d m j , x i j − d m j d u j − d m j , d m j ≤ x i j ≤ d u j , 0 , otherwise . (6) For this function, we could not find many examples, and it may epresent a small number of very particular decision criteria. For his function we can think of: • Relative market share in selecting a firm for investment ( Wilson & Anell, 1999 ). Wilson and Anell (1999) found that for invest- ment decision-making, firms with low and high market share are more desirable to the investors. This implies that the value of a firm decreases while its market share increases up to a cer- tain level, d m , and after that its value increases again. • Firm size in R&D productivity ( Tsai & Wang, 2005 ). Tsai and Wang (2005) found that both small and large firms have higher R&D productivity compared to medium-sized firms. This is true for both high-tech and traditional industries. This means that the relationship between size and value (measured by R&D pro- ductivity) is V-shape with a minimum level of value assigned to a certain size of d m j . .4. Inverted V-shape Inverted V-shape value function shows that as the criterion level, ij , increases up to a certain level, d m j , its value, u ij , increases, and fter that certain level, d m j , its value, u ij , decreases. It is shown in ig. 4 and formulated as follows: i j = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ x i j − d l j d m j − d l j , d l j ≤ x i j ≤ d m j , d u j − x i j d u j − d m j , d m j ≤ x i j ≤ d u j , 0 , otherwise . (7) For this function we can think of: • Commute time in selecting a job ( Redmond & Mokhtar- ian, 2001 ). For many people, the ideal commute, d m j , is larger than zero. This implies that commute times between zero and the optimal commute time, and between the optimal commute time and larger times, have lower value than the optimal com- mute time for such individuals. This suggests an inverted V- shape value function. • Cognitive proximity in innovation partner selection ( Nooteboom, 20 0 0 ). For a company there is an optimal cognitive distance to the partner they are working on innova- tion ( d m j ). This implies that any distance less than d m j or larger than d m j has less value. .5. Increase-level Increase-level value function shows that as the criterion level, x ij , ncreases up to a certain level, d m j , its value, u ij , increases, and after hat certain level, d m j , its value, u ij , will remain at the maximum evel. It is shown in Fig. 5 and formulated as follows: i j = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ x i j − d l j d m j − d l j , d l j ≤ x i j ≤ d m j , 1 , d m j ≤ x i j ≤ d u j , 0 , otherwise . (8) For this function we can think of: 46 J. Rezaei / Expert Systems With Applications 98 (2018) 43–56 Fig. 5. Increase-level value function. Fig. 6. Level-decrease value function. Fig. 7. Level-increase value function. Fig. 8. Decrease-level value function. 2 x i I u 2 x a I u • Fill rate in supplier selection ( Chae, 2009 ). Although a buyer prefers suppliers with higher fill rate, which implies that as the fill rate increases its value increases, the buyer might be indif- ferent to any increase after a certain level, d m j , as usually buy- ers pre-identify a desirable service level which is satisfied by a certain minimum level of supplier’s fill rate. • Diversity of restaurants in hotel location selection ( Chou, Hsu, & Chen, 2008 ). In order to find the best location for an international hotel, a decision-maker prefers locations with more divers restaurants. However, reaching a level, d m j , might fully satisfy a decision-maker implying that the decision-maker might not be sensitive to any increase after that certain level. 2.6. Level-decrease Level-decrease value function shows that as the criterion level, x ij , increases up to a certain level, d m j , its value, u ij , remains at max- imum level, and after that certain level, d m j , its value, u ij , decreases gradually. It is shown in Fig. 6 and formulated as follows: u i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 , d l j ≤ x i j ≤ d m j , d u j − x i j d u j − d m j , d m j ≤ x i j ≤ d u j , 0 , otherwise . (9) For this function we can think of: • Distance in selecting a university ( Carrico et al., 1997 ). While a student prefers a closer university to a farther university, this preference might start after a certain distance, d m j , implying that any distance between [ d l j , d m j ] is optimal and indifferent for the student. • Lead time in supplier selection ( Çebi & Otay, 2016 ). Although a supplier with a shorter lead time is preferred, if the lead time is in a limit such that it does not negatively affect the com- pany’s production, the company might then be indifferent to that range. .7. Level-increase Level-increase value function shows that as the criterion level, ij , increases up to a certain level, d m j , its value, u ij , remains at min- mum level, and after that certain level, d m j , its value, u ij , increases. t is shown in Fig. 7 and formulated as follows: i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0 , d l j ≤ x i j ≤ d m j , x i j − d m j d u j − d m j , d m j ≤ x i j ≤ d u j , 0 , otherwise . (10) For this function we can think of: • Level of trust in making a buyer–supplier relationship ( Ploetner & Ehret, 2006 ). Trust increases the level of partnership between a buyer and a supplier, however it is only effective after a cer- tain threshold, d m j . • Level of relational satisfaction in evaluating quality commu- nication in marriage ( Montgomery, 1981 ). Below a minimum level of relational satisfaction, d m j , quality communication can- not take place thus results in minimum value. .8. Decrease-level Decrease-level value function shows that as the criterion level, ij , increases up to a certain level, d m j , its value, u ij , decreases, and fter that certain level d m j , its value, u ij , will remain at minimum. t is shown in Fig. 8 and formulated as follows: i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ d m j − x i j d m j − d l j , d l j ≤ x i j ≤ d m j , 0 , d m j ≤ x i j ≤ d u j , 0 , otherwise . (11) For this function we can think of: • Carbon emission in transportation mode selection ( Hoen, Tan, Fransoo, & van Houtum, 2014 ). In selecting a transportation mode, the more the carbon emission by the mode, the less the value of that mode. A decision-maker, however might assign J. Rezaei / Expert Systems With Applications 98 (2018) 43–56 47 Fig. 9. Increasing stepwise value function. 2 l a j I u w h t t s r v u w Fig. 10. Decreasing stepwise value function. 2 l a u s u w m c u w 3 s u R m e d i c b zero value to a mode with carbon emission higher than a cer- tain level, d m j . • Distance when selecting a school ( Frenette, 2004 ). It has been shown that the longer the distance to the school the less the preference to attend that school. It is also clear that, for some people, there is no value after a certain distance, d m j . .9. Increasing stepwise Increasing stepwise value function shows that as the criterion evel, x ij , increases up to a certain level, d m j , its value remains at certain level, u 0 , and after that certain level, d m j , its value, u ij , umps to a higher level (maximum) and remains at the maximum. t is shown in Fig. 9 and formulated as follows: i j = ⎧ ⎨ ⎩ u 0 , d l j ≤ x i j ≤ d m j , 1 , d m j ≤ x i j ≤ d u j , 0 , otherwise . (12) here 0 < u 0 < 1. For this function we can think of: • Suppliers capabilities in supplier segmentation ( Rezaei & Ortt, 2012 ). Suppliers of a company are evaluated based on their capabilities, and then segmented based on two levels (low and high) with respect to their capabilities. As such a supplier scored between d l j and d m j is considered as a low-level capa- bilities supplier while a supplier scored between d m j and d u j is considered as a high-level capabilities supplier. • Symmetry in selecting a close type of partnership ( Lambert, Emmelhainz, & Gardner, 1996 ). In order to have a successful relationship between supply chain partners, there should be some demographical similarities (for instance, in terms of brand image, productivity) between them. So, more symmetry means closer relationship. However, if we consider two levels of closeness, it is clear that for some level of symmetry the value of closeness remains the same. For the increasing stepwise value function, a criterion might ave more than one jump. For instance, if a decision-maker wants o consider three levels low, medium, and high when segmenting he suppliers with respect to their capabilities, then an increasing tepwise function with two jumps should be defined for this crite- ion. The following value function is a general increasing stepwise alue function with k jumps. i j = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ u 0 , d l j ≤ x i j ≤ d m 1 j , u 1 , d m 1 j ≤ x i j ≤ d m 2 j , . . . 1 , d mk j ≤ x i j ≤ d u j , 0 , otherwise . (13) here 0 < u < u < … < 1. 0 1 .10. Decreasing stepwise Decreasing stepwise value function shows that as the criterion evel, x ij , increases up to a certain level, d m j , its value, u ij , remains t a the maximum level, and after that certain level, d m j , its value, ij , jumps down to a lower level, u 0 , and remains at that level. It is hown in Fig. 10 and formulated as follows: i j = ⎧ ⎨ ⎩ 1 , d l j ≤ x i j ≤ d m j , u 0 , d m j ≤ x i j ≤ d u j , 0 , otherwise . (14) here 0 < u 0 < 1. For this function we can think of: • Considering supply risk in portfolio modeling ( Kraljic, 1983 ). For a company, an item with a higher level of risk results in less value, however, due to portfolio modeling, there is no dif- ference between all levels of risk in the domain [ d l j , d m j ] . Simi- larly, all levels of risk in the domain [ d m j , d u j ] result in the same value. • Delay in logistics service provider selection ( Qi, 2015 ). Some companies consider stepwise value function for delay in de- livering the items by a logistics service provider, which means that the value of that provider decreases when delay increases, however it is constant within certain intervals. For the decreasing stepwise function, a criterion might have ore than one jump. The following value function is a general de- reasing stepwise function with k jumps. i j = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 1 , d l j ≤ x i j ≤ d m 1 j , u 1 , d m 1 j ≤ x i j ≤ d m 2 j , . . . u k , d mk j ≤ x i j ≤ d u j , 0 , otherwise . (15) here 0 < u k < … < u 1 < 1. . Some remarks on the value functions Here, a number of remarks are discussed, shedding light on ome aspects of the proposed value functions, which might be sed in real-world applications. emark 1. Shape and parameters of a value function is decision- aker-dependent, implying that (i) while a decision-maker consid- rs, for instance, a level-increasing function for the size of gar- en when buying a house, another decision-maker considers an ncreasing stepwise function, and (ii) while two decision-makers onsider increasing stepwise function for the size of garden when uying a house, the parameters they consider for their functions ( d l j , d m j , d u j ) might be different. 48 J. Rezaei / Expert Systems With Applications 98 (2018) 43–56 Fig. 11. Increasing-level-decreasing value function. Table 1 Suppliers performance with respect to different decision criteria ( x ij ). Criteria Supplier Quality Price ( €/item) Trust CO 2 (g/item) Delivery (day) 1 85 27 4 10 0 0 3 2 90 28 2 1500 4 3 80 26 5 20 0 0 3 4 75 25 5 10 0 0 2 5 95 29 7 1700 3 6 99 30 6 20 0 0 1 w w v e h a d t a r m n t t t d m o U w s T w w T v 2 4 v e u p a s b fi w a e g 2 Please note that we report some weights for the criteria as the aim of the study is not the weighing part. Remark 2. A decision-maker might consider a hybrid value func- tion for a criterion. For instance, a criterion might be character- ized with an increasing-level-decreasing, which is a combination of increasing-level and level-decreasing. This function can also be considered as a special form of inverted V-shape function. It is shown in Fig. 11 and formulated as follows: u i j = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x i j − d l j d m j − d l j , d l j ≤ x i j ≤ d m 1 j , 1 , d m 1 ≤ x i j ≤ d m 2 , d u j − x i j d u j − d m j , d m 2 ≤ x i j ≤ d u j , 0 , otherwise . (16) For instance, a decision-maker has to select the best R&D part- ner among 10 partners. One of the criteria is distance and the com- pany gives less preference to very close or very distant partners, which are distributed in the range [10 km, 20 0 0 km]. The com- pany considers an optimal distance of [20 0 km, 50 0 km]. This im- plies that distance follows an ‘increasing-level-decreasing’ function for this decision-maker: [ d l j , d m 1 j , d m 2 j , d u j ] = [ 10 , 200 , 500 , 2000 ] . 4. Numerical and comparison analyses In this section, we show how to incorporate the proposed piecewise value functions into account when applying an MCDM method, and we show that the results might be different when we consider the proposed piecewise value functions. We consider an MCDM problem, where a buyer should select a supplier from among six qualified suppliers considering five crite- ria: quality which is measured by 1 −α, where α shows the lot-size average imperfect rate; price (euro) per item; trust, which is mea- sured by a Likert scale (1: very low to 7: very high); CO 2 (gram) per item; delivery (day), the amount of time which takes to deliver items from the supplier to the buyer (all the criteria are continuous except trust). Table 1 shows the performance of the six suppliers with respect to the five criteria. The buyer has used an elicitation method 2 to find the weights hich are as follows: ∗ quality = 0 . 20 , w ∗price = 0 . 30 , w ∗trust = 0 . 27 , w ∗CO 2 = 0 . 08 , w ∗ delivery = 0 . 15 . And we assume that the decision-maker considers piecewise alue functions for these criteria (see, Table 2 ). So, as can be seen from Table 2 , the decision-maker consid- rs a level-increase linear function for quality with the lowest and ighest values of 0 and 100, respectively. For the decision-maker ny number below 85 has no value at all. For criterion price, the ecision-maker gives the highest value to any price below 15 (al- hough in the existing set of suppliers there is no supplier with price within this range), after which the value decreases till it eaches to the maximum price of 30. For criterion trust which is easured using a Likert scale (1: very low to 7: very high), any umber less than 3 has no value for the decision-maker, while he value gradually increases between 3 and 7. For CO 2 emission, here is a decreasing value function from 0 to 1500 g per item, af- er which till 20 0 0 g, all the numbers have zero value. Finally, for elivery there is a simple decreasing function with minimum and aximum values of 0 and 5 days. By using the following equation, we can find the overall value f each supplier and then rank them to find the best supplier. i = n ∑ j=1 w j u i j (17) here, u ij is the value of the performance of supplier i with re- pect to criterion j (using the equations in Table 2 for the data in able 1 ), and w j is the weight of criterion j as follows: ∗ qual ity = 0 . 20 , w ∗price = 0 . 30 , w ∗trust = 0 . 27 , ∗ CO 2 = 0 . 08 , w ∗delivery = 0 . 15 . The value scores and the aggregated values are presented in able 3 (see also Fig. 13 for the final results). As can be seen from Table 3 , supplier 6 with the greatest overall alue of 0.51 is ranked as the first supplier. Suppliers 5, 4, 3, 1, and are ranked in the next places. .1. Comparing with the simple linear value functions In existing literature, considering the nature of the criteria, the alues are calculated, for instance, using the following simple lin- ar value function: i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ x i j − d l j d u j − d l j , if more x i j is more desirable ( such as quality ) , d u j − x i j d u j − d l j , if more x i j is less desirable ( such as price ) . (18) Eq. (18) is used to find the values of the criteria for each sup- lier using the data in Table 1 . Considering the criteria weights, nd u ij ( Eq. (18) for the data in Table 1 ) using Eq. (17) the value cores and also the aggregated overall score of each alternative can e calculated which are shown in Table 4 (see also Fig. 13 for the nal results). In Table 4 it is assumed that quality and trust are criteria for hich the higher the better, while for the other criteria (price, CO 2 , nd delivery), the lower the better. In fact, we consider simple lin- ar functions (increasing and decreasing respectively) for the two roups of criteria. J. Rezaei / Expert Systems With Applications 98 (2018) 43–56 49 Table 2 Piecewise value functions. Shape Value function u i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0 , d l j ≤ x i j ≤ d m j , x i j − d m j d u j − d m j d m j ≤ x i j ≤ d u j , 0 , otherwise . = ⎧ ⎪ ⎨ ⎪ ⎩ 0 , 0 ≤ x i j ≤ 85 , x i j − 85 100 − 85 , 85 ≤ x i j ≤ 100 , 0 , otherwise . u i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 , d l j ≤ x ≤ d m j , d u j − x i j d u j − d m j , d m j ≤ x ≤ d u j , 0 , otherwise . = ⎧ ⎪ ⎨ ⎪ ⎩ 1 , 13 ≤ x i j ≤ 15 , 30 − x i j 30 − 15 , 15 ≤ x i j ≤ 30 , 0 , otherwise . u i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0 , d l j ≤ x i j ≤ d m j , x i j − d m j d u j − d m j , d m j ≤ x i j ≤ d u j , 0 , otherwise . = ⎧ ⎪ ⎨ ⎪ ⎩ 0 , 1 ≤ x i j ≤ 3 , x i j − 3 7 − 3 , 3 ≤ x i j ≤ 7 , 0 , otherwise . u i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ d m j − x i j d m j − d l j , d l j ≤ x i j ≤ d m j , 0 , d m j ≤ x i j ≤ d u j , 0 , otherwise . = ⎧ ⎪ ⎨ ⎪ ⎩ 1500 − x i j 1500 − 0 , 0 ≤ x i j ≤ 1500 , 0 , 1500 ≤ x i j ≤ 2000 , 0 , otherwise . u i j = ⎧ ⎨ ⎩ d u j − x i j d u j − d l j , d l j ≤ x i j ≤ d u j , 0 , otherwise . = { 5 − x i j 5 − 0 , 0 ≤ x i j ≤ 5 , 0 , otherwise . r ( a c i 3 t s v e t According to Table 4 , the best supplier is supplier 4, which is anked as the 3rd one considering the piecewise value functions Table 3 ). The ranking of the other suppliers is also different. So, s can be seen, such differences are associated to the way we cal- ulate the value of the criteria. If we look at the criterion trust, for nstance ( Table 3 ), we can see that only the numbers greater than can be used for compensating the other criteria. In other words, he values 1, 2 and 3 for this criterion have no selection power. No upplier can compensate its weakness in other criteria by having a alue between 1 and 3 for trust. However, such important issue is ntirely ignored in the simple way of determining the value func- ions which is very popular in existing studies. This consideration 50 J. Rezaei / Expert Systems With Applications 98 (2018) 43–56 Table 3 Value scores, u ij , and the aggregated overall score considering the proposed piecewise value functions. Supplier Quality Price Trust CO 2 Delivery Aggregated value Rank 1 0.00 0.20 0.25 0.50 0.40 0.23 5 2 0.33 0.13 0.00 0.00 0.20 0.14 6 3 0.00 0.27 0.50 0.00 0.40 0.28 4 4 0.00 0.33 0.50 0.50 0.60 0.37 3 5 0.67 0.07 1.00 0.00 0.40 0.48 2 6 0.93 0.00 0.75 0.00 0.80 0.51 1 Table 4 Value scores, u ij , and the aggregated overall scores considering the simple linear value func- tions. Supplier Quality Price Trust CO 2 Delivery Aggregated value Rank 1 0.42 0.60 0.40 1.00 0.33 0.50 4 2 0.63 0.40 0.00 0.50 0.00 0.29 6 3 0.21 0.80 0.60 0.00 0.33 0.49 5 4 0.00 1.00 0.60 1.00 0.67 0.64 1 5 0.83 0.20 1.00 0.30 0.33 0.57 2 6 1.00 0.00 0.80 0.00 1.00 0.57 3 Fig. 12. Exponential value functions. s u f c F f v d M v ( f t 3 To see how these value functions are elicited considering the decision-maker is even of a higher importance for compensatory methods such as AHP and BWM. 4.2. Comparing with the exponential value functions Another important way to approximate the value functions in practice is the use of exponential value functions ( Kirkwood, 1997; Pratt, 1964 ). The exponential value functions can specifically be used when the preferences are monotonically increasing or de- creasing. Although this approach is not popular in MCDM do- main, and we were not able to find any application of these value functions particularly in MCDM field, we would like to compare our results to the results of applying these functions, which are, to some extent, close to some of our proposed piecewise value functions (such as level-increase, level-decrease, increase-level, and decrease-level). Using the same notations as before and consider- ing a shape parameter ρ which is called ‘risk tolerance’, a mono- tonically increasing exponential value function can be shown as follows: u i j = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 − exp [ − ( x i j − d l j ) /ρ ] 1 − exp [ − ( d u j − d l j ) /ρ ] , ρ � = Infinity x i j − d l j d u j − d l j , otherwise . (19) r A monotonically decreasing exponential value function can be hown as follows: i j = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1 − exp [ − ( d u j − x i j ) /ρ ] 1 − exp [ − ( d u j − d l j ) /ρ ] , ρ � = Infinity d u j − x i j d u j − d l j , otherwise . (20) Fig. 12 shows the monotonically increasing exponential value unctions (for different values of ρ) (a), and the monotonically de- reasing exponential value functions (for different values of ρ) (b). Risk-averse decision-makers have ρ > 0 (hill-like functions in ig. 12 ), while risk-seeking decision-makers have ρ < 0 (bowl-like unctions in Fig. 12 ). ρ = Infinity (straight-line in Fig. 12 ) shows the alue for the risk neutral decision-makers. In fact, ρ = Infinity pro- uces the simple linear value functions which are very popular in CDM field. In order to do the comparison analysis, we use exponential alue functions for the criteria of the aforementioned example Table 1 ) to check the similarities and differences. To make a air comparison, we try to generate 3 the corresponding exponen- ial value functions of the piecewise value functions ( Table 2 ) as isk tolerance, refer to Kirkwood (1997) . J. Rezaei / Expert Systems With Applications 98 (2018) 43–56 51 c t a t t p l ρ f s o p lose as possible. For quality, a monotonically increasing exponen- ial value function with negative ρ would be appropriate. For price monotonically decreasing exponential value function with a posi- ive ρ, for trust, a monotonically increasing exponential value func- ion with a negative ρ, for CO 2 , a monotonically decreasing ex- onential value function with a negative ρ, and, finally, for de- ivery a monotonically decreasing exponential value function with Table 5 Exponential value functions. Shape = Infinity would be suitable. Table 5 shows the functions, where unctions with different ρ′ s are shown and a more suitable one is hown in bold. Using the exponential value functions of Table 5 , for the data f Table 1 , we get the value scores and the aggregated values as resented in Table 6 (see also Fig. 13 for the final results). Function u i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 − exp [ −( x i j − d l j ) /ρ] 1 − exp[ −( d u j − d l j ) /ρ] , ρ � = Infinity x i j − d l j d u j − d l j , otherwise . = ⎧ ⎨ ⎩ 1 − exp [ −( x i j − 0 ) /ρ] 1 − exp[ −( 100 − 0 ) /ρ] , ρ � = Infinity x i j − 0 100 − 0 , otherwise . u i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 − exp [ −( d u j − x i j ) /ρ] 1 − exp[ −( d u j − d l j ) /ρ] , ρ � = Infinity d u j − x i j d u j − d l j , otherwise . = ⎧ ⎨ ⎩ 1 − exp [ −( 30 − x i j ) /ρ] 1 − exp[ −( 30 − 0 ) /ρ] , ρ � = Infinity 30 − x i j 30 − 0 , otherwise . u i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 − exp [ −( x i j − d l j ) /ρ] 1 − exp[ −( d u j − d l j ) /ρ] , ρ � = Infinity x i j − d l j d u j − d l j , otherwise . = ⎧ ⎨ ⎩ 1 − exp [ −( x i j − 1 ) /ρ] 1 − exp[ −( 7 − 1 ) /ρ] , ρ � = Infinity x i j − 1 7 − 1 , otherwise . u i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 − exp [ −( d u j − x i j ) /ρ] 1 − exp[ −( d u j − d l j ) /ρ] , ρ � = Infinity d u j − x i j d u j − d l j , otherwise . = ⎧ ⎨ ⎩ 1 − exp [ −( 20 0 0 − x i j ) /ρ] 1 − exp[ −( 20 0 0 − 1500 ) /ρ] , ρ � = Infinity 20 0 0 − x i j 20 0 0 − 0 , otherwise . ( continued on next page ) 52 J. Rezaei / Expert Systems With Applications 98 (2018) 43–56 Table 5 ( continued ) Shape Function u i j = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 − exp [ −( d u j − x i j ) /ρ] 1 − exp[ −( d u j − d l j ) /ρ] , Infinity d u j − x i j d u j − d l j , otherwise . = ⎧ ⎨ ⎩ 1 − exp [ −( 5 − x i j ) /ρ] 1 − exp[ −( 5 − 0 ) /ρ] , ρ � = Infinity 5 − x i j 5 − 0 , otherwise . Table 6 Value scores, u ij , and the aggregated overall scores considering the exponential value func- tions. Supplier Quality Price Trust CO 2 Delivery Aggregated value Rank 1 0.85 0.45 0.05 0.08 0.40 0.22 5 2 0.90 0.33 0.00 0.02 0.20 0.16 6 3 0.80 0.55 0.13 0.00 0.40 0.26 4 4 0.75 0.63 0.13 0.08 0.60 0.32 3 5 0.95 0.18 1.00 0.00 0.40 0.46 1 6 0.99 0.00 0.37 0.00 0.80 0.38 2 Fig. 13. Final results using three types of value functions. v o t p t l c c a As can be seen from Table 6 , supplier 5 with the greatest over- all value of 0.46 is ranked as the first supplier, which is different from what we get from the proposed piecewise value functions ( Table 3 ). While supplier 5 was ranked the 2nd based on our pro- posed value functions, using the exponential value functions, this supplier becomes number 2. Other suppliers (1, 2, 3, 4) have the same ranking based on the two approaches. The differences are ob- viously associated to the way we get the values of the criteria. We also checked some other close ρ values for the exponential value functions. There are some changes in the aggregated values, yet, the ranking is the same. As it can be seen, the results of the two approaches (piecewise alue functions and exponential functions are much closer to each ther than to the results of the regular simple linear value func- ions). Our observation is that the exponential value functions can lay a role close to a number of proposed value functions in his study such as increase-level, decrease-level, level-increase, and evel-decrease. In order to make an exponential value functions lose to one of the mentioned proposed value functions we should hoose ρ values close to zero. If we try to make the ρ as close s we perfectly make the “level” part of the criterion, then the J. Rezaei / Expert Systems With Applications 98 (2018) 43–56 53 Fig. 14. Fitting an exponential value function to a level-decrease value function. o h s i e 1 c i t p O c t i c t i o t c v f o l r a t a t w e e ( e e p e a i 5 fi h l n a v m fi s 1 & t s e fl n r u m T p m f t m s t c t 6 m i i v ther part becomes very steep and not representative. On the other and, if we want to choose a ρ value which better represents the lope of the function for the “increase” or the “decrease” part, then t is impossible to cover the “level” part of the value function prop- rly. For example, let us consider the criterion price again. In Fig. 4 , it can be seen that, if we consider ρ = 1, the level part is fully overed, but the decrease part of the exponential value function s very much different from the decrease part of the linear func- ion. Even if we choose ρ = 3, which does not fully cover the level art of the proposed function, the decrease part is really different. n the other hand, we can find ρ = 8 as a close one to the de- rease part of the linear function, but this time it is not possible o cover the level part properly. So, although a very good approx- mation, the exponential value functions might not be suitable for ases in which a decision-maker has a clear value-indifference in- erval (a level part) for a criterion. However, these functions are ndeed suitable when the decision-maker has different preferences n the lower and on the upper parts of the criterion measure. From the figure we can also see that while we could make the wo piecewise and exponential value functions, to some degree, lose to each other, they are too different from the simple linear alue function. It is also clear that none of the simple linear value unctions or the exponential value functions can represent the V- r inverted V-shape value functions. As a general conclusion, we think that the proposed piecewise inear functions have two salient features: (i) simplicity; and (ii) epresentativeness. That is, it is easy to work with linear functions, nd it is easy for a practitioner to find a more representative func- ion from the proposed library of the pricewise value functions for particular criterion. The cut-off points can also be estimated by he decision-maker. The simple monotonic-linear value functions, hich are dominant in existing literature, are very simple. How- ver, they might not be representative in some cases. Finally, the xponential value functions might have a better representativeness compared to the simple monotonic-linear value functions). How- ver, they are not simple. Working with non-liner functions is not asy for practitioner, and, more importantly, it is very difficult for a ractitioner to estimate a value for ρ (the shape parameter of the xponential value functions), as it cannot be easily interpreted by practitioner (please note that we consider a value function as an nput for an MCDM problem in this study). s I t l . Determining the value functions One of the big challenges in real-world decision-making is to nd a proper value function for a decision criterion. This, per- aps, has been one of the main reasons why the use of simple inear value functions in multi-criteria decision-making is domi- ant. The linear value functions are easy for modeling purposes nd can, to some extent, represent the reality. More complicated alue functions, although might be closer to reality of the decision- aker’s preferences, are more difficult to be elicited and are dif- cult for modeling purposes. We refer the interested readers to ome existing procedures for identifying value functions ( Fishburn, 967; Keeney & Nair, 1976; Kirkwood, 1997; Pratt, 1964; Stewart Janssen, 2013 ). We think that the proposed value functions in his paper do not have the disadvantage of nonlinearity and at the ame time have the advantages of being closer to the real pref- rences of the decision-maker as they provide some diversity and exibility in modeling the functions. As we do not consider the onlinearity of the value functions we do not use the concept of isk tolerance in determining the value functions as it has been sed by others. We rather propose a simple procedure, which is ore practical. A decision analyst, could first show the value functions in able 2 to the decision-maker to see which one most suits the reference structure of the decision-maker. Once the decision- aker selects a particular value function, the other details of the unction, such as the lower bound, the upper bound and the hresholds can be determined. We should highlight again that in ost MCDM methods, the value function is not elicited. It is rather imply assumed to have a particular shape, and this is why we hink having a pre-specified set of standard value functions which an be used as subjective approximation of the real preferences of he decision-maker can make a significant impact on the results. . Conclusion, limitations and future research This study proposes a set of piecewise value functions for ulti-criteria decision-making (MCDM) problems. While the exist- ng applications of MCDM methods usually use two general simple ncreasing and decreasing linear value functions, this study pro- ides several real-world examples to support the applicability of ome other forms of value functions for the criteria used in MCDM. t is also explicated how, in some decision problems, a combina- ion of two or more value functions can be used for a particu- ar decision criterion. The proposed functions can be used for dif- 54 J. Rezaei / Expert Systems With Applications 98 (2018) 43–56 Ç C E F F H J K K K K L M O P P P Q R R R R R S T ferent MCDM methods in different decision problems. A numeri- cal example of supplier selection problem (including a comparison between simple monotonic linear value functions, piecewise lin- ear value functions, and exponential value functions) showed how the use of the proposed value functions could affect the final re- sults. Considering these value functions could better represent the real preferences of the decision-maker. It can also help reduce the inappropriate compensations of the decision criteria, for instance, through using a level-increasing function which assigns zero value to any value of the criterion below a certain threshold. The pro- posed value functions are presented in a general form such that they can be tailor-made for a specific decision-maker. That is to say, not only it is possible for two different decision-makers to use two different value functions for a single criterion. It is also pos- sible to use different domain (e.g. min and max) values for that particular value function. Despite the advantages of the proposed value functions, they have some limitations. Although the proposed value functions con- sider some real-world features of the decision criteria, they are lin- ear which might be, to some degree, a simplification. We think that the bigger problem in existing literature of MCDM is mono- tonicity assumption and not linearity assumption. Nevertheless, more research needs to be conducted to empirically find the share of each. Furthermore, to formulate the decision criteria one should pay enough attention to check the real contribution of the decision criterion into the ultimate goal of the decision-making problem. For instance, if a criterion contributes to another criterion which has a real role in making the decision, one should exclude the ini- tial one. For a detailed discussion on this matter, interested read- ers are referred to Brugha (1998) . One interesting future direction would be to apply the proposed value functions in some real-world MCDM problems and compare their fitness to the other value func- tions. In this regard, finding a more systematic approach to deter- mine the value functions in practice would be also very interest- ing. It would be also interesting to study the cases in which there are more than one decision-maker. As different decision-makers may choose different value functions, different domains, and dif- ferent thresholds for a single criterion, proposing a way to find the final output of the MCDM problem for the group would be an interesting future research. Finally, finding a sensitivity anal- ysis for the proposed value functions is recommended. Consider- ing the studies of Bertsch and Fichtner (2016 ), Bertsch, Treitz, Gel- dermann, and Rentz (2007 ), Insua and French (1991 ), Wulf and Bertsch (2017 ) could give interesting ideas to make such sensitivity analysis framework. References Alanne, K. , Salo, A. , Saari, A. , & Gustafsson, S.-I. (2007). Multi-criteria evaluation of residential energy supply systems. Energy and Buildings, 39 , 1218–1226 . Ananda, J. , & Herath, G. (2009). A critical review of multi-criteria decision making methods with special reference to forest management and planning. Ecological Economics, 68 , 2535–2548 . Bertsch, V. , & Fichtner, W. (2016). A participatory multi-criteria approach for power generation and transmission planning. Annals of Operations Research, 245 , 177–207 . Bertsch, V. , Treitz, M. , Geldermann, J. , & Rentz, O. (2007). Sensitivity analyses in multi-attribute decision support for off-site nuclear emergency and recovery management. International Journal of Energy Sector Management, 1 , 342–365 . Brans, J. P. , Mareschal, B. , Vincke, P. , & Brans, J. P. (1984). PROMETHEE: A new fam- ily of outranking methods in multicriteria analysis. In Proceedings of the 1984 conference of the international federation of operational research societies (IFORS) (pp. 477–490). Amsterdam: North Holland. 84 . Brugha, C. M. (1998). Structuring and weighting criteria in multi criteria deci- sion making (MCDM). In Trends in multicriteria decision making (pp. 229–242). Springer . Brugha, C. M. (20 0 0). Relative measurement and the power function. European Jour- nal of Operational Research, 121 , 627–640 . Carrico, C. S. , Hogan, S. M. , Dyson, R. G. , & Athanassopoulos, A. D. (1997). Data envel- opment analysis and university selection. The Journal of the Operational Research Society, 48 , 1163–1177 . ebi, F. , & Otay, İ. (2016). A two-stage fuzzy approach for supplier evaluation and order allocation problem with quantity discounts and lead time. Information Sci- ences, 339 , 143–157 . Chae, B. (2009). Developing key performance indicators for supply chain: An indus- try perspective. Supply Chain Management: An International Journal, 14 , 422–428 . hou, T.-Y. , Hsu, C.-L. , & Chen, M.-C. (2008). A fuzzy multi-criteria decision model for international tourist hotels location selection. International Journal of Hospi- tality Management, 27 , 293–301 . dwards, W. (1977). How to use multiattribute utility measurement for social deci- sion making. IEEE Transactions on Systems, Man and Cybernetics, 7 , 326–340 . ishburn, P. C. (1967). Methods of estimating additive utilities. Management Science, 13 , 435–453 . renette, M. (2004). Access to college and university: Does distance to school mat- ter. Canadian Public Policy/Analyse de Politiques, 30 , 427–443 . Herrera, F. , Herrera-Viedma, E. , & Verdegay, J. (1996). Direct approach processes in group decision making using linguistic OWA operators. Fuzzy Sets and Systems, 79 , 175–190 . o, W. , Xu, X. , & Dey, P. K. (2010). Multi-criteria decision making approaches for supplier evaluation and selection: A literature review. European Journal of Oper- ational Research, 202 , 16–24 . Hoen, K. , Tan, T. , Fransoo, J. , & van Houtum, G. (2014). Effect of carbon emission reg- ulations on transport mode selection under stochastic demand. Flexible Services and Manufacturing Journal, 26 , 170–195 . Insua, D. R. , & French, S. (1991). A framework for sensitivity analysis in discrete multi-objective decision-making. European Journal of Operational Research, 54 , 176–190 . acquet-Lagreze, E. , & Siskos, Y. (2001). Preference disaggregation: 20 years of MCDA experience. European Journal of Operational Research, 130 , 233–245 . abir, G. , Sadiq, R. , & Tesfamariam, S. (2014). A review of multi-criteria decision– making methods for infrastructure management. Structure and Infrastructure En- gineering, 10 , 1176–1210 . akeneno, J. R. , & Brugha, C. M. (2017). Usability of nomology-based methodologies in supporting problem structuring across cultures: The case of participatory de- cision-making in Tanzania rural communities. Central European Journal of Oper- ations Research, 25 , 393–415 . Keeney, R. L., & Nair, K. (1976). Evaluating potential nuclear power plant sites in the Pacific Northwest using decision analysis, IIASA Professional Paper. IIASA, Laxenburg, Austria: PP-76-001. Keeney, R. L. , & Raiffa, H. (1976). Decisions with multiple objectives: preferences and value tradeoffs . USA: John Wiley & Sons, Inc . irkwood, C. W. (1997). Strategic decision making . California, USA: Wadsworth Pub- lishing Company . raljic, P. (1983). Purchasing must become supply management. Harvard Business Review, 61 , 109–117 . ahdelma, R. , & Salminen, P. (2012). The shape of the utility or value function in stochastic multicriteria acceptability analysis. OR Spectrum, 34 , 785–802 . Lambert, D. M. , Emmelhainz, M. A. , & Gardner, J. T. (1996). Developing and imple- menting supply chain partnerships. The International Journal of Logistics Manage- ment, 7 , 1–18 . Montgomery, B. M. (1981). The form and function of quality communication in mar- riage. Family Relations, 30 , 21–30 . ustajoki, J. , & Hämäläinen, R. P. (20 0 0). Web-HIPRE: Global decision support by value tree and AHP analysis. INFOR: Information Systems and Operational Re- search, 38 , 208–220 . Nooteboom, B. (20 0 0). Learning and innovation in organizations and economies . Ox- ford: University Press . ’Brien, D. B. , & Brugha, C. M. (2010). Adapting and refining in multi-criteria deci- sion-making. Journal of the Operational Research Society, 61 , 756–767 . loetner, O. , & Ehret, M. (2006). From relationships to partnerships—New forms of cooperation between buyer and seller. Industrial Marketing Management, 35 , 4–9 . ohekar, S. , & Ramachandran, M. (2004). Application of multi-criteria decision mak- ing to sustainable energy planning—a review. Renewable and Sustainable Energy Reviews, 8 , 365–381 . ratt, J. W. (1964). Risk aversion in the small and in the large. Econometrica, 32 , 122–136 . i, X. (2015). Disruption management for liner shipping. In Handbook of ocean con- tainer transport logistics (pp. 231–249). Springer . edmond, L. S. , & Mokhtarian, P. L. (2001). The positive utility of the commute: Modeling ideal commute time and relative desired commute amount. Trans- portation, 28 , 179–205 . ezaei, J. (2015). Best-worst multi-criteria decision-making method. Omega, 53 , 49–57 . ezaei, J. (2016). Best-worst multi-criteria decision-making method: Some proper- ties and a linear model. Omega, 64 , 126–130 . ezaei, J. , & Ortt, R. (2012). A multi-variable approach to supplier segmentation. In- ternational Journal of Production Research, 50 , 4593–4611 . oy, B. (1968). Classement et choix en présence de points de vue multiples (la méthode ELECTRE). RIRO, 2 , 57–75 . Saaty, T. L. (1977). A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology, 15 , 234–281 . tewart, T. J. , & Janssen, R. (2013). Integrated value function construction with appli- cation to impact assessments. International Transactions in Operational Research, 20 , 559–578 . sai, K.-H. , & Wang, J.-C. (2005). Does R&D performance decline with firm size?—A re-examination in terms of elasticity. Research Policy, 34 , 966–976 . http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0001 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0001 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0001 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0001 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0001 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0001 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0002 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0002 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0002 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0002 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0003 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0003 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0003 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0003 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0004 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0004 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0004 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0004 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0004 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0004 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0005 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0005 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0005 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0005 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0005 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0005 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0006 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0006 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0007 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0007 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0008 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0008 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0008 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0008 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0008 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0008 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0009 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0009 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0009 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0009 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0010 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0010 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0011 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0011 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0011 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0011 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0011 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0012 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0012 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0013 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0013 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0014 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0014 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0015 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0015 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0015 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0015 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0015 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0016 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0016 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0016 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0016 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0016 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0017 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0017 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0017 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0017 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0017 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0017 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0018 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0018 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0018 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0018 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0019 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0019 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0019 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0019 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0020 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0020 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0020 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0020 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0020 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0021 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0021 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0021 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0021 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0022 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0022 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0022 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0022 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0023 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0023 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0024 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0024 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0025 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0025 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0025 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0025 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0026 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0026 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0026 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0026 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0026 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0028 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0028 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0029 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0029 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0029 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0029 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0030 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0030 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0031 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0031 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0031 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0031 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0032 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0032 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0032 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0032 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0033 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0033 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0033 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0033 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0034 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0034 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0035 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0035 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0036 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0036 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0036 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0036 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0037 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0037 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0038 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0038 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0039 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0039 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0039 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0039 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0040 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0040 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0041 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0041 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0042 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0042 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0042 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0042 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0043 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0043 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0043 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0043 J. Rezaei / Expert Systems With Applications 98 (2018) 43–56 55 T W W X Y zeng, G.-H. , Lin, C.-W. , & Opricovic, S. (2005). Multi-criteria analysis of alternative– fuel buses for public transportation. Energy Policy, 33 , 1373–1383 . ilson, T. L. , & Anell, B. I. (1999). Business service firms and market share. Journal of Small Business Strategy, 10 , 41–53 . ulf, D. , & Bertsch, V. (2017). A natural language generation approach to sup- port understanding and traceability of multi-dimensional preferential sensitivity analysis in multi-criteria decision making. Expert Systems with Applications, 83 , 131–144 . ia, W. , & Wu, Z. (2007). Supplier selection with multiple criteria in volume dis- count environments. Omega, 35 , 494–504 . ager, R. R. (1988). On ordered weighted averaging aggregation operators in multi- criteria decision making. IEEE Transactions on Systems, Man, and Cybernetics, 18 , 183–190 . http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0044 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0044 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0044 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0044 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0044 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0045 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0045 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0045 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0045 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0046 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0046 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0046 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0046 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0047 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0047 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0047 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0047 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0048 http://refhub.elsevier.com/S0957-4174(18)30004-6/sbref0048 56 J. Rezaei / Expert Systems With Applications 98 (2018) 43–56 t Delft University of Technology, the Netherlands, where he also obtained his Ph.D. One of ) analysis. He has published in various academic journals, including International Journal arketing Management, Applied Soft Computing, Applied Mathematical Modelling, IEEE Journal of Operational Research, Information Science, Omega, and Expert Systems with Jafar Rezaei is an associate professor of operations and supply chain management a his main research interests is in the area of multi-criteria decision-making (MCDM of Production Economics, International Journal of Production Research, Industrial M Transactions on Engineering Management, Journal of Cleaner Production, European Applications. Piecewise linear value functions for multi-criteria decision-making 1 Introduction 2 Piecewise linear value functions 2.1 Increasing 2.2 Decreasing 2.3 V-shape 2.4 Inverted V-shape 2.5 Increase-level 2.6 Level-decrease 2.7 Level-increase 2.8 Decrease-level 2.9 Increasing stepwise 2.10 Decreasing stepwise 3 Some remarks on the value functions 4 Numerical and comparison analyses 4.1. Comparing with the simple linear value functions 4.2. Comparing with the exponential value functions 5 Determining the value functions 6 Conclusion, limitations and future research References