Logistics provider selection for omni-channel environment with fuzzy axiomatic design and extended regret theory

Applied Soft Computing 71 (2018) 353–363

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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

Logistics provider selection for omni-channel environment with fuzzy axiomatic design and extended regret theory Weijie Chen a,∗ , Mark Goh b , Yan Zou a a b

School of Economics and Management, Chongqing Normal University, Chongqing 400047, PR China NUS Business School and The Logistics Institute-Asia Pacific, National University of Singapore, Singapore

a r t i c l e

i n f o

Article history: Received 25 January 2018 Received in revised form 29 June 2018 Accepted 10 July 2018 Keywords: Omni-channel Logistics provider selection Axiomatic design Regret/Rejoice theory Linguistic variables

a b s t r a c t As e-commerce marketplaces proliferate, omni-channels will become the new engine of growth. Omnichannel retailers need to optimally determine how to select suitable logistics providers (LSPs) to help maintain their competitive advantage. Although there are many methods to solve the problem of LSP selection, most of them overlook the decision maker’s psychology. Most importantly, previous studies paid little attention to the probability of success for each candidate under each criterion. To compensate for these shortcomings, this study proposes a new method of logistics provider selection in an omnichannel environment. We present the model in three phases. The first phase involves computing the probability of success of each LSP with respect to each criterion through axiomatic design method. The second phase uses the perspective of the extended regret aversion/rejoice preference to develop a bounded rational decision making model for determining the criteria weights. In this phase, the regret/rejoice levels are treated as continuous parameters, whereby decision makers can regret and rejoice simultaneously. The final phase computes the expected perceived utility values to select the best LSP. To validate the capability of the proposed model, LSPs of six from a case study are ranked based on the proposed model, and the results are compared with the traditional regret and TOPSIS. The findings suggest that the proposed method provides more reasonable and reliable results, which are in line with the psychological behavior of human beings. © 2018 Elsevier B.V. All rights reserved.

1. Introduction With the growing adoption of the Internet, smartphones, and handheld devices worldwide, consumer behavior is changing drastically. Already, the rapid expansion of e-commerce marketplaces is creating enormous needs for online shopping [1]. At the same time, the rise in the number of third-party logistics providers (LSP) has pushed the growth of the overall industry. In this context, the omni-channel, as the new engine of growth, is driving the next wave of retail logistics development. Retailers are moving to create an omni-channel approach to sales, where customers have a seamless shopping experience that integrates online shopping from a desktop, mobile device or even in a physical store. However, online shopping is one thing but delivery is another. As the omni-channel retail integrates online services, offline experience, and modern logistics, many other factors determine success in the omni-channel marketplace. Already, retailers are discovering that retail logstics play a key role in the omni-channel. There-

∗ Corresponding author. E-mail address: [email protected] (W. Chen). https://doi.org/10.1016/j.asoc.2018.07.019 1568-4946/© 2018 Elsevier B.V. All rights reserved.

fore, determining suitable logistics providers in anomni-channel environment has become a key consideration for retailers, which involves multi-criteria. Urban consumers today have growing expectations for flexible delivery/pick-up and /or faster delivery options in a day or hours. This naturally means that retailers must now find ways to solve the “last mile” of delivery in an omni-channel envrionment. As speed to market is a now competitive imperative for omni-channel retailers, logically then, the assessment and selection of the LSP should also be managed accordingly. However, due to limited time, ability, or incomplete information collection, decision makers often face situations of uncertain or imprecise information [2]. For instance, the exact information about the criteria weights for LSP selection cannot be immediately obtained. There may be a need to use imprecise information, for example, feasible ranges or weights or the ranking of criteria in the order of their importance [3]. In our paper, we consider the value of the criteria with linguistic terms and the weights of the criteria with incomplete or unknown information, to ensure realism of the problem. Put simply, the LSP selection problem in an omni-channel environment can be considered as a MCDM problem. This selection process includes the decision maker, operational factors, and meth-

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ods. In practice, the decision maker’s attitude has a very important bearing on the decision outcomes. Although many models have applied to LSP selection [4–7], they based mainly on the assumptions of complete information and the rational behavior of the decision makers. It is difficult to explain many phenomena in real situations based on the hypothesis of completely rational decision. Moreover, these methods did not consider the probability of success for each candidate under each criterion. Therefore, there is a need to better structure the problem based on fuzzy design of the functional requirements. Further, Simon [8] suggested that decision makers exercised bounded rationality during the decision process, and thus he proposed the principle of bounded rationality. This led research to explore the actual decision making process [9]. Based on bounded rationality, Loomes [10] and Bell [11] proposed regret theory, whose core idea is that the decision makers not only focus on the results obtained by the choice of the alternative, but also pay heed to the outcome of the other alternatives. Regret theory is now widely applied in various fields, such as route choice [12], insurance [13], and MCDM [14,15]. However, scant attention has paid to regret/rejoice with uncertain and incomplete information. Due to the above analysis, the core focus of this paper is to propose a novel approach for LSP selection in an omnichannel environment based on the axiomatic design method and regret/rejoice theory. Another focus is to identify relevant decision criteria that are important to the LSP selection problem. The novel aspects of the paper can be summarized as follows: (1) To best of the authors’ knowledge, this paper first presents an evaluation system of logistics provider selection for omnichannel environment, which is able to not merely capture the bounded rationality of decision maker under uncertain conditions, but also can avoid the mutual complementarities among attributes. (2) This study quantifies the behavior of the decision maker in a fuzzy and incomplete information environment using a regret aversion (or rejoice preference) and determines the probability of success for each candidate under each criterion through the axiomatic design method. (3) The paper analyzes sensitivity and compares the proposed method with the traditional regret and TOPSIS, which show that the proposed method is more in line with the psychological behavior of human beings. The rest of this paper is organized as follows. Section 2 reviews the literature. Section 3 introduces some concepts and theory related to triangular fuzzy numbers, axiomatic design method, and regret theory. Section 4 presents the study framework based on the axiomatic design method and regret/rejoice theory for LSP selection in an omni-channel environment. Section 5 discusses a numerical example on LSP selection to show the feasibility of our proposed method, and applies sensitivity analysis to validate the proposed integrated model. Finally, the conclusions are given in Section 6. 2. Literature review We will first provide an overview of the evaluation criteria for logistics services in an omni-channel environment. Next, we focus on the methods used in the selection of an LSP. The third subsection positions this paper in the context of the existing research gaps. 2.1. Logistics evaluation criteria in omni-channel environment Omni-channel retail deep integrates online services, offline experience, and modern logistics. According to a UPS Pulse of the

Online Shopper survey (2016), consumers want more flexibility in shipping and fulfillment, including the ability to select delivery dates and times, and to re-route packages based on personal preferences. While the LSPs are familiar with bulk orders for delivery, in an omni-channel mode, the storing and packing have become smaller. Today, logistics in the omni-channel allows a retailer to tailor how a product is purchased and delivered. For example, a consumer can walk into a store, find a product, purchase that same item online, and have it delivered to the home the next day. This expectation of instant consumer gratification in turn has retailers scrambling to shore up their supply chain to ensure on time delivery. This place a higher premium on logistics flexibility in procuring, storing, delivery, and last mile service. Logistics is thus a key enabler for the omni-channel. Thus, the LSP needs to combine reliability, performance, agility, and productivity to maintain retailer competitiveness and margins. From prior studies [16–20] and the characteristics of logistics development in an omni-channel environment, we identify some key criteria relevant to this environment, as follows: 2.1.1. Flexibility and reliability To ensure enhanced customer service experience, the LSPs need to provide more flexibility to geographic distribution and may offer a larger variety of services to its customers, particularly on special or non-routine requests as reported by EfT [21]. Reliability is the ability of the system to perform its required functions under stated conditions [20]. In this sense, the LSP’s service must be seen and accepted as reliable, without the customer having to return the goods the next day for refund or seek refunds for missing delivery consistently. 2.1.2. Service quality In an omni-environment, consumer demand is the driver. Therefore, service quality reflects a service encounter, which includes the accuracy of order fulfilment, on-time delivery, pre and post-sale services to customers, promptness in attending to complaints. This level of service quality encompasses the product purchase stage and the return stage for unhappy customers [16]. 2.1.3. Reputation It refers to market beliefs of an LSP e.g. how good the LSP is in satisfying customer needs. There are actually no studies done on reputation of LSPs in an omni-channel [20]. 2.1.4. Financial record It is reflected in the firm’s return on investment, return on assets, and value added services. Sound financial performance of the logistics provider ensures that the services used in the logistics operations can be continuously upgraded [16,20]. 2.1.5. Information system strength An LSP can collect, aggregate and analyse transactional data. A provider with a robust information system can help to not only increase visibility for the client by way of continuous status updates via dispatch management, but also to continually control the target criteria and to react responsively to exceptional situations in an omni-channel environment [17,18]. 2.1.6. Expected cost The omni-channel retailers need to weigh the cost of the outsourcing of logistics services, which may comprise elements such as contract price, expected leasing cost, cost saving, and operational cost [18,19].

W. Chen et al. / Applied Soft Computing 71 (2018) 353–363

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2.1.7. Business growth potential Increasing the e-commerce shipments with smaller packages and more stock keeping units may introduce a different level of logistics complexity in the omni-channel. This presents even greater opportunities for the LSPs to accommodate growing demand and to generate more profit for themselves [21]. 2.1.8. Impact on environment The omni-channel retail models, under greater pressure to outsource logistics services, may lose direct control over their environmental footprint and performance. Thus, LSP must be able to meet the same high standards for vehicular emissions reduction and noise pollution.

Fig. 1. Membership function of TFNF = (a,b,c).

3. Triangular fuzzy numbers (TFN), axiomatic design method, and regret theory 3.1. TFN and defuzzification

2.2. Current methods for LSP selection The literature is replete with models on the LSP selection problem. The methods to select the LSPs can be divided into two categories: MCDM and mathematical programming. The MCDM techniques include AHP, ANP, TOPSIS, VIKOR, TODIM and DEMATEL. Jharkharia and Shankar [20] used the analytic network process (ANP) approach to select an LSP. Sasikumar and Haq [4] presented a fuzzy MCDM model based on the VIKOR method to select the best third-party reverse logistics provider. Govindan et al. [22] adopted an interpretive structural modeling (ISM) to analyze the linkages among the attributes of the third party reverse logistics provider. Govindan et al. [5] used the grey Decision Making Trial and Evaluation Laboratory (DEMATEL) method to analyze the interdependent relationships between the third-party LSP selection criteria for an automotive manufacturer. Liu and Wang [6] integrated fuzzy Delphi, fuzzy inference, and fuzzy linear assignment to resolve uncertain and imprecise decision situations when choosing third-party logistics services. Kannan et al. [7] proposed the ISM and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) to select a reverse logistics provider. The methods applied within mathematical programming include Data Envelopment Analysis [23], multi-objective programming [24,25], and dynamic programming [26]. 2.3. Gap analysis and research highlights Reviewing the methods mentioned earlier, while each category with their own characteristic features seem to be effective and applicable for assessing and selecting the LSPs, they have, however, two main disadvantages. One major issue is that there exists the problem of complementarity among attributes when deals with the aggregation of the attributes. There may be such a situation where the LSP we choosed performs is poor in some attribute, and well in others. This can result in huge losses for retailers due to poor performance of a certain attribute. These approaches suffer from this disadvantage. Another important limitation of the existing models is that they assume the decision makers are completely rational and seldom include the psychology of the decision makers. In practice, the decision maker is bounded rationality. To address the above gaps, in this study, we consider the LSP selection problem in an omni-channel environment. Thus, the research highlights of our work can be stated as follows: • We present a new evaluation system for LSP selection in an omnichannel environment. • Determine the probability of success for each candidate under each criterion through the axiomatic design method. • Through bounded rationality, we quantify the behavior of the decision maker in a fuzzy and incomplete information environment using a regret aversion/rejoice preference.

Zadeh [27] introduced fuzzy set theory to deal with vague and uncertain information and the fuzzy set is characterized by a membership function, which assigns to each object a grade of membership ranging between zero and one. Van [28] employed triangular membership functions to define the TFNs as follows. Definition 1 ([28] (TFN)). A fuzzy number F on R is a TFN if its membership function F : R → [0, 1] is given by:

F (x) =

⎧ x−a ⎪ , x ∈ [a, b] ⎪ ⎪ b−a ⎪ ⎪ ⎨ x−c b−c ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

, x ∈ [b, c]

0, else

where a ≤ b ≤ c, and a and c denote the lower and upper values of the support of F,i.e. the set of elements {x ∈ R|a < x < c} respectively, and b is the modal value, as shown in Fig. 1. ˜ 1 = (al , bm , c u ) and A ˜ 2 = (al , bm , c u ) and any Given two TFNs A 1 1 1 2 2 2 real number , let ⊕ denote the fuzzy number additive operation and ⊗ be the fuzzy number multiplicative operation. Thus, the arithmetic operation laws are described as follows [29]: ˜1 ⊕ A ˜ 2 = (al + al , bm + bm , c u + Addition of two TFNs: A 1 2 1 2 1 u l l c2 ), a1 , a2 ≥ 0, ˜1 ⊗ A ˜ 2 = (al · al , bm · bm , c u · Multiplication of two TFNs: A 1 2 1 2 1 c2u ), al1 , al2 ≥ 0, ˜ = (al , bm , c u ), al ≥ Multiplication of any real number  : A 0,  ≥ 0. Yao and Chiang [30] compared defuzzification to the TFNs by the centroid and signed distance, and found it better to use the signed distance to defuzzify. So, we follow as such. ˜ be TFN, denoted as A ˜ = (a, b, c). The Definition 2 ([31]). Let A signed distance or the defuzzified value of TFN is ˜ = v(A)

1 (2b + a + c) 4

(3.1)

3.2. Fuzzy axiomatic design Axiomatic Design(AD), initiated by [32], can help designers to structure design problems under a suitable design requirement. AD is widely applied to fields such as product design [33], office design [34], and MCDM [35]. In our paper, the design problem pertains to the LSP selection and the design requirement refers to the omnichannel environment under MCDM. AD has two design axioms, which can be stated as follows [36]: The independence axiom: maintain the independence of the functional requirements (FRs). The information axiom: minimize the information content.

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In this paper, based on the fuzzy AD methodology, the design range can be regarded as the performance level expected of the LSP by the retailer and the system range can be regarded as the actual performance level of the LSP. The functional requirement FRj can be regarded as criterion ej . The independence axiom states that the independence of FRs must always be maintained. In the context of the decision process, the FRs represent the independent criteria. Each potential LSP is then evaluated with respect to each criterion for the decision goals. 3.3. Regret and rejoice theory

Fig. 2. Design, system, common ranges, and probability density function of FRs.

Fig. 3. Common area of system and design ranges of TFNs.

The FRs represent the smallest set of independent functional requirements, which define the ultimate design goals of the best LSP selection. To resolve a complex problem, a designer can decompose it into smaller problems and then strive to keep their independence amongst each other. Similarly, in MCDM, we divide the decision goal into a number of indicators, so the FRs can be defined as the various criteria. The information axiom in AD theory is used to select the “best design”, among those designs that satisfy the independence axiom. It states that the design with the smallest information content (Ij ) is the best design [36]. The information content is defined as:

 

Ij = log2

1 pj

= −log2 (pj )

(3.2)

where pj is the probability of fulfilling FRj and is expressed as Eq. (3.3). pj =

CommonRange . SystemRange

(3.3)

The common range, which is the acceptable solution, is the intersection of the “design range” and the “system range”, and the information content can be expressed as a uniform probability distribution function, as shown in Fig. 2. In a complex socio-economic environment, there is incomplete or fuzzy information on the system and design ranges, which cannot be expressed using crisp values, but they can be expressed as linguistic terms, fuzzy set or fuzzy numbers. To handle fuzzy information, [37,38] proposed a fuzzy AD based on TFN and AD. The common area is the intersection of the TFN of the design and system ranges, as shown in Fig. 3. For a uniform probability distribution function, pj can be rewritten as Eq. (3.4). pj =

commonarea TFNofsystemrange

(3.4)

In practice, decision makers often need to make the best of whatever data or information they have on-hand to make decisions, notwithstanding the complexity, ambiguity, nor the timing of the information flow in the decision making environment. So, during the decision making progress, the human response of regret aversion/rejoice preferences is often experienced. When decision makers make a decision, they will feel regret if they have made a wrong decision. Likewise, they will rejoice if they have made the right decision. It is reasonable for a decision maker to include regret/rejoice in their choice and we can safely assume that they all desire to eliminate or reduce the regret possibility. Regret/rejoice theory, as developed by [10], is based on the intuition that when a decision maker chooses among various uncertain choices, they are concerned not only about their payoff but also about their loss. Following [11] and [10], let xi and xk be the consequences of choosing LSPs i and k, respectively, then the perceived utility function of achieving xi can be defined as U(xi , xk ) = u(xi ) + R(u(xi ) − u(xk ))

(3.5)

where u(·) expresses the utility value that the decision maker chooses LSP i and it is a von Neumann-Morgenstern utility function with u (·) > 0 and u"(·) < 0. The power function u(x) = xϕ , (0 ≤ ϕ ≤ 1) [39] is usually used to simulate the utility of the decision maker, where ϕ denotes the risk aversion coefficient of the decision maker. The rejoice/ regret function R captures the regret or rejoice by transforming the differences in utility between choosing LSP I and k, and is concave with R(0) = 0, R (·) > 0 and R"(·) < 0 [10]. R(u(xi ) − u(xk )) > 0 represents the gain in value to the omnichannel retailer of choosing LSP i over the forgone LSP k, and vice versa. When two LSPs perform equally well, i.e. u(xi )=u(xk ), then R(0) = 0. In this case, the omni-channel retailer experiences neither regret nor rejoice. The regret/rejoice function is thus R(x) = 1 − exp(−(x)),  ≥ 0 [11] that satisfies these requirements, with  as the regret aversion coefficient. The regret theory was originally derived for pair-wise selection choices. Quiggin [40] extended regret theory to general choice sets. For notational purposes, we now let xi (i = 1, . . ., m) represent the consequence of choosing LSP hi (i = 1, . . ., m), respectively. Then, the decision maker’s perceived utility for consequence xi is Ui = ui (xi ) + R(u(xi ) − u(x∗ )) x∗





= max xi |i = 1, 2, ..., m , and R(u(xi where the regret value, which is always non-positive.

(3.6) ) − u(x∗ ))

denotes

4. Extended regret theory for MCDM problems under incomplete information We propose a hybrid method using fuzzy axiomatic design and extended regret/rejoice theory based on linguistic variables under incomplete information for the LSP selection problem. Suppose a retailer wants to choose a LSP for omni-channel retail. Let U = {h1 , h2 , ..., hm }, m ≥ 2 be a set of m potential LSPs, E = (e1 , e2 , ..., en ) be a parameter set and each LSP is assessed on n

W. Chen et al. / Applied Soft Computing 71 (2018) 353–363

criteria, w = (w1 , w2 , ..., wn ) be the weight vector of the n criteria, n

with wj ≥ 0 and

wj = 1, (j = 1, 2, ..., n). The decision is to select

j=1

an LSP from set U to realize omni-channel retail. In this paper, the assessment of the LSP hi (1 ≤ i ≤ m) against criterion ej (1 ≤ j ≤ n) is given by fuzzy linguistic variables. 4.1. Determine probability of fulfilling FRi As mentioned, the functional requirement FRj can be regarded as criterion ej , the design range can be regarded as the performance level expected of the LSP by the omni-channel retailer and the system range can be regarded as the actual performance level of the LSP. Step 1: Determine the design and system ranges. With a high degree of information fuzziness and uncertainty, decision makers often give linguistic assessments rather than exact numerical values to express their opinions. Here, the system and design ranges for a criterion can be expressed as linguistic variables. The decision makers decide on the intangible criteria using terms such as “Very Poor (VP)”, “Poor (P)”, “Somewhat Poor (MP)”, “Neutral (M)”, “Somewhat Good (MG)”, “Good (G)”, and “Very Good (VG)”. The expert panel offers the design range FRs according to the intended state. The system range A is the features of the LSPs, which can be evaluated by the expert panel in the light of the bids received from the LSPs. Step 2: Convert linguistic terms into triangular fuzzy numbers. The system range A is as follows:

Definition 4. Let hi be the ith LSP, and let a˜ ij be the fuzzy evaluation of LSP hi for criteria j. The positive ideal solution (PIS) and negative and a˜ − , respecideal solution (NIP) of criteria j are denoted as a˜ + j j tively. Let u(v(·)) be a fuzzy utility function as defined as Definition 3. Then an expect fuzzy regret utility is defined as Uij = pij [u(v(˜aij )) + R(u(v(˜aij )) − u(v(˜a+ )))] j

11

⎜ a˜ 21 ⎜ A = (˜aij )m×n = ⎜ ⎝ ..

a˜ 12

···

a˜ 1n

Uij = pij [u(v(˜aij )) + R(u(v(˜aij )) − u(v(˜a− )))] j

a˜ 22

···

a˜ 2n

.

. . .

..

. . .

a˜ m1

a˜ m2

···

.

⎟ ⎟, ⎟ ⎠



ϕ



ϕ

Uij = pij {(v(˜aij ))ϕ + 1 − exp[−((v(˜aij ))ϕ − (v(˜a+ )) )]}, j Uij = pij {(v(˜aij ))ϕ + 1 − exp[−((v(˜aij ))ϕ − (v(˜a− )) )]}. j It is fair to assume that any decision maker wants to make a decision outcome that is free of regret (minimize) and full of rejoice (maximize). This can be expressed mathematically as:

MaximizeUi = where a˜ ij = (˜aijl , a˜ ijm , a˜ iju ) .

According to the Eq. (3.6), we present the extended expected perceived utility as: (4.1)

∗ where  Pi denotes the probability of choosing hi and x = max xi |i = 1, 2, ..., m . As the decision maker wants to rejoice or avoid regret when making a decision, we present the expected perceived utility from the perspective of rejoice.

x−



ϕ



pij [(v(˜aij )) + 1 − exp(−((v(˜aij )) − (v(˜a+ )) ))] j ϕ

ϕ

n



wj



ϕ

pij [(v(˜aij )) + 1 − exp(−((v(˜aij )) − (v(˜a− )) ))] j ϕ

ϕ

j=1

4.2. Model to determine criteria weights



wj

j=1

Step 3: Determine the common and system ranges. System and design ranges are considered as TFNs in the LSP selection problem. Therefore, the common area between the design range and system range is as shown in Fig. 3. Step 4: Find the probability pij of achieving design range FRj for LSP hi , using Eq. (3.4). Based on the results of pij , any LSP hi with pij = 0 should be eliminated.

Ui = Pi [ui (xi ) + R(u(xi ) − u(x− ))]

n





a˜ mn

Ui = Pi [ui (xi ) + R(u(xi ) − u(x∗ ))]

(4.4)

which is the gain in value of having chosen LSP hi in terms of crite. rion ej rather than NIS a˜ − j Next, for the utility of the decision maker, we use the power function u(x) = xϕ , (0 < ϕ < 1) [38]. The regret/rejoice function can be thought of as R(x) = 1 − exp(−(x)) [11]. So Eqs. (4.3) and (4.4) can be re-written as

MinimizeUi =

⎞

(4.3)

where pij is the probability of LSP hi meeting functional requirement FRj . Eq. (4.3) states the loss in value of having chosen LSP hi in terms of criterion ej rather than PIS a˜ + . j Similarly, the expected fuzzy rejoice utility is defined as



⎛ a˜

357

During the MCDM process, the different weights on the attributes reflect their importance in choosing the optimal LSP. However, in reality, because of time pressure, lack of data, complexity and uncertainty of the actual decision making problem or limited ability of the decision maker, it is usually impossible or infeasible to have precise values of the criteria weights, leaving us with only incomplete information about the criteria weights. Generally, the incomplete attribute weight information can be expressed in the following forms [41]: (1) (2) (3) (4) (5)

A weak ranking: {wj1 ≥ wj2 }, j1 = / j2 ; A strict ranking: {wj1 − wj2 ≥ εj1 j2 }, j1 = / j2 , εj1 j2 > 0; A ranking with multiples: {wj1 ≥ ˛j1 j2 wj2 }, 0 ≤ ˛j1 j2 ≤ 1, j1 = / j2 ; An interval form: {ˇj ≤ wj ≤ ˇj + εj }, 0 ≤ ˇj < ˇj + εj ≤ 1; A ranking of differences: {wj1 − wj2 ≥ wj3 − wj4 }, j1 = / j2 = / j3 = / j4 .

Using the regret/ rejoice perspectives, we present a model to find the weights of the attributes. Model (M-1)

MinimizeUi =

n



(4.2)

= min xi |i = 1, 2, ..., m . with To apply the extended regret/ rejoice theory, we call on the following definitions. Definition 3. Let A be a triangular fuzzy set, and v(T ) be a defuzzified value of a TFN T (T ∈ A). Suppose u(x) is classical utility function with u (x) > 0 and u"(x) < 0 for any real x. The fuzzy utility function is defined as u(v(·)) : A → R, T → u(v(T )).

wj

ϕ

ϕ

ϕ

j=1



MaximizeUi =

n



wj



pij [(v(˜aij )) + 1 − exp(−((v(˜aij )) − (v(˜a+ )) ))] j

ϕ



pij [(v(˜aij )) + 1 − exp(−((v(˜aij )) − (v(˜a− )) ))] j

j=1 ⎧ T w = (w1 , w2 , ..., wm ) ⎪ ⎪ ⎪ n ⎨

s.t. wj = 1 ⎪ ⎪ ⎪ j=1 ⎩

wj ≥ 0, j = 1, 2, ..., m

ϕ

∈ H

ϕ

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The multi-objective problem in Model (M-1) can be solved using techniques such as the linear weight method, which has been widely used because of its simplicity. In the actual decision making process, the decision makers do not always fully regret or rejoice. Rather, they tend to experience partial regret and rejoice simultaneously. Thus, we set  as the degree of regret and  as the degree of rejoice, with 0 ≤  ≤ 1, 0 ≤  ≤ 1,  +  ≤ 1. Without loss of generality, we can change the multi-objective (M-1) into a single objective using the linear weighted method. Model (M-2)

Model (M-3) max z = − n m



 i=1

s.t



Maximizezi = (−Ui ) + Ui

s.t.

⎧ w = (w1 , w2 , ..., wm )T ∈ H ⎪ ⎪ ⎪ ⎪ n ⎪ ⎪ ⎨ w = 1

⎪ ⎩

i=1

T

(i)

(i) T

(i)

from the optimal solutions w(i) = (w1 , w2 , ..., wn ) (i =

wj ≥ 0

···

(n) w1

(2)

wm

···



(1)

w1

w2

(1)

w2

.. .

.. .

w1

ϕ

ϕ (˜a− )) ))]} j

wj {pij [(v(˜aij )) + 1 − exp(−((v(˜aij )) − (v

n

 + ( 2

(4.5) wj2

− 1)

j=1

where  is the Lagrange multiplier. Solving Eq. (4.5), we obtain an expression for the attribute weights as follows:

  m 

   tij − sij     i=1 wj =   2  n m 

   tij − sij

(4.6)

i=1 ϕ

sij = pij [(v(˜aij ))ϕ + 1 − exp(−((v(˜aij ))ϕ − (v(˜a+ )) ))] j ϕ

and

ϕ − (v(˜a− )) ))]. j

 m  

    (tij −sij )  i=1    2 . We note that the weight vector does  n m 

 (tij −sij )

(n)

T



(2)

···

w1

(2)

···

w2

.. .

ϕ

If  = , then Eq. (4.6) can be transformed to wj =



of the matrix {[(−Ui ) + Ui ]W } {[(−Ui ) + Ui ]W }. Then we can construct a combined weight vector as follows:

=(

ϕ

tij = pij [(v(˜aij )) + 1 − exp(−((v(˜aij ))

wm



ϕ

j=1

ϕ

and we can find the normalized eigenvector ω = (ω1 , ω2 , ..., ωn )T

w = Wω

i=1

ϕ

wj {pij [(v(˜aij )) + 1 − exp(−((v(˜aij )) − (v(˜a+ )) ))]}+ j

j=1

where

⎞

⎜ (1) ⎟ ⎜ w2 w2(2) · · · w2(n) ⎟ ⎜ ⎟ W =⎜ ⎟ .. . ⎜ .. .. .. ⎟ ⎝ . ⎠ . . (1)

n m



j=1

(2) w1

ϕ

ϕ

j

1, 2, ..., m) as:

wm

ϕ

j=1



tor w(i) = (w1i , w2i , ..., wni ) corresponding to LSP hi , which is a function of the parameter pair (, ). However, to find the final weight vector w = (w1 , w2 , ..., wn )T , we need to consider all LSPs hi (i = 1, 2, ..., m). Thus, we construct the weight matrix W =

(1) w1

j=1

j=1

n m



Solving Model (M-2) yields an optimal criteria weight vec-

⎛

ϕ

wj {pij [(v(˜aij )) + 1 − exp(−((v(˜aij )) − (v(˜a− )) ))]} j

L(w, ) = −

0<+ ≤1

m×n

ϕ

To solve the model, we construct the Lagrangian

j=1 ⎪ ⎪ ⎪ ⎪ w j ≥ 0, j = 1, 2, ..., m ⎪ ⎪ ⎩

(i)

i=1

ϕ

wj {pij [(v(˜aij )) + 1 − exp(−(((v(˜aij )) − (v(˜a+ )) ))]}+ j

⎧ n ⎪ ⎨ w 2 = 1

j

(wj )

n m



(n)

ω1

(n)

.. .

ω2 )(

.. .

)

(1) (2) (n) ωn wm wm · · · wm (1) (2) = ω1 w + ω2 w + ... + ωn w(n)

Yielding the weight vector w = (w1 , w2 , ..., wm )T of the attribute ej (j = 1, 2, ..., m). If the attribute weights are completely unknown, we can construct the simplicity model (M-3), following [42].

 s+ =

(˜a+ , a˜ + , ..., a˜ + ), 1 2 j

a˜ + j

s−

(˜a− , a˜ − , ..., a˜ − ). 1 2 j

a˜ − ij

=

=

i=1

4.3. Decision steps We now develop a novel method considering the regret/rejoice psychology of a decision maker to solve the MCDM problem, in which the expected service level of the omni-channel retailer and actual service level of the LSPs are considered. The method involves the following steps: Step 1: Determine the system and design ranges. From the actual conditions, the expert panel can identify the design range, and give the assessment value using linguistic terms. Step 2: Transform the linguistic terms into TFNs. Step 3: Compute the probability pij of achieving design range FRj for LSP hi , according to Eq. (3.4). If pij = 0, then hi is removed. Step 4: Identify PIS a˜ + and NIP a˜ − for each criterion, which are j j defined as follows:

(max a˜ ijl , max a˜ ijm , max a˜ iju ), i = 1, 2, ..., n. If j is a benefit attribute (min a˜ ijl , min a˜ ijm , min a˜ iju ), i = 1, 2, ..., n. If j is a cost attribute

 =

j=1

not change, as it is irrelevant to the parameter pair (, ) if  = .

(min a˜ ijl , min a˜ ijm , min a˜ iju ), i = 1, 2, ..., n. If j is a benefit attribute (max a˜ ijl , max a˜ ijm , max a˜ iju ), i = 1, 2, ..., n. If j is a cost attribute

.

W. Chen et al. / Applied Soft Computing 71 (2018) 353–363

359

Table 1 Expert panel assessments for the criteria of LSPs U

e1

e2

e3

e4

e5

e6

e7

e8

h1 h2 h3 h4 h5 h6

M MG MG VG G VG

G MG MG G MG MG

MG MG VG MG M MG

MG M G MG MG G

VG MG MG G G MG

MP P M P P MP

M M M MG MG MG

P MP MP MP P P

Table 2 Converted TFNs for Table 1. U

e1

e2

e3

e4

e5

e6

e7

e8

h1 h2 h3 h4 h5 h6

(3, 5, 7) (5, 7, 9) (5, 7, 9) (9, 10, 10) (7, 9, 10) (9, 10, 10)

(7, 9, 10) (5, 7, 9) (5, 7, 9) (7, 9, 10) (5, 7, 9) (5, 7, 9)

(5, 7, 9) (5, 7, 9) (9, 10, 10) (5, 7, 9) (3, 5, 7) (5, 7, 9)

(5, 7, 9) (3, 5, 7) (7, 9, 10) (5, 7, 9) (5, 7, 9) (7, 9, 10)

(9, 10, 10) (5, 7, 9) (5, 7, 9) (7, 9, 10) (7, 9, 10) (5, 7, 9)

(1, 3, 5) (0, 1, 3) (3, 5, 7) (0, 1, 3) (0, 1, 3) (1, 3, 5)

(3, 5, 7) (3, 5, 7) (3, 5, 7) (3, 5, 7) (5, 7, 9) (5, 7, 9)

(0, 1, 3) (1, 3, 5) (1, 3, 5) (1, 3, 5) (0, 1, 3) (0, 1, 3)

Fig. 5. Functional requirements for defined criteria. Fig. 4. TFNs for intangible factors.

Step 5: Defuzzification. To simplify the calculation, we use the signed distance to obtain the defuzzified values of the fuzzy criteria. Let R = (rij )m×n indicate the defuzzified value of using the signed distance. Here rij =

a˜ ijl + 2 · a˜ ijm + a˜ iju 4

, i ∈ m, j ∈ n.

Step 6: Determine the weights of the criteria. Build the optimization model (M-1) based on the obtained probability pij . If the weight information is incomplete, convert model (M-1) to model (M-2). Solving Model (M-2) yields the weight vector w = (w1 , w2 , ..., wn )T . If the weight information is completely unknown, use Eq. (4.6) to get the weight vector. Step 7: Find the perceived utility values and select the best LSP hk , with zk = max zi . i=1,2,..,n

5. Case study 5.1. Case We provide an illustrative example to present the application of the proposed method for the LSP selection problem in an omnichannel environment. Suppose a retailer wants to select an LSP for the omni-channel. After preliminary screening, six potential LSPs h1 , h2 , h3 , h4 , h5 , h6 are shortlisted for further evaluation. Assume that the experts provide their preference information on the candidate LSPs with regard to criteria by using linguistic terms, as depicted in Fig. 4. Step 1: Determine system and design ranges In this case, the LSP selection decision is made on the basis of eight main criteria of flexibility and reliability e1 , service quality e2 , reputation e3 , financial record e4 , information system strength e5 , expected cost e6 , business growth potential e7 , and impact on

the environment e8 . The six LSPs are evaluated using the evaluation criteria. The decision information is expressed in linguistics terms (see Table 1). From the actual conditions of the candidate firms, the expert panel will discuss and identify the design range as follows: (each criterion is measured on a scale from 1 to 10) FRe1 =Flexibility and reliability must be over 5.5: (5.5, 10, 10). FRe2 =Service quality must be over 7: (7, 10, 10). FRe3 = Reputation must be over7.5: (7.5, 10, 10). FRe4 = Financial record must be over 6: (6, 10, 10). FRe5 = Information system strength must be over 7: (7, 10, 10). FRe6 = Expected cost must be less than 4: (0, 0, 4). FRe7 = Business growth potential must be over 5: (5, 10, 10). FRe8 = Impact on environment must be less than 4: (0, 0, 4). The FRs defined above for each criterion are given in Fig. 5. Step 2: Transform the linguistic terms into TFNs using Fig. 4, as shown in Table 2. Step 3: From Eq. (3.4) and the design and system ranges, the probability pij of achieving the functional requirement FRej for LSP hi can be obtained, as shown in Table 3. From Table 3, the probability of success of LSP h5 with respect to the criteria reputation is 0. So, LSP h5 is eliminated. Step 4: Identify PIS and NIS. In this example, we obtain PIS and NIS as shown in Table 4. Step 5: Using Definition 2, we obtain the defuzzified values of a˜ ij , as shown in Table 5. Calculate the defuzzified value of PIS and NIS, respectively. PIS= (9.75, 8.75, 9.75, 8.75, 9.75, 1.25, 7, 1.25); NIS = (5, 7, 5, 5, 7, 3, 5, 3). Step 6: Determine the weights of the criteria Following [15,39], we set ϕ = 0.88 and  = 0.3 to construct Model (M-1). If the information about attribute weights is incomplete, assume that the information about the attribute weights, given by expert

360

W. Chen et al. / Applied Soft Computing 71 (2018) 353–363

Table 3 Probability of success pij . U

e1

e2

e3

e4

e5

e6

e7

e8

h1 h2 h3 h4 h5 h6

0.0865 0.4712 0.4712 1.0000 0.9769 1.0000

0.7500 0.2000 0.2000 0.7500 0.2000 0.2000

0.1250 0.1250 1.0000 0.1250 0.0000 0.1250

0.3750 0.0417 0.9000 0.3750 0.3750 0.9000

1.0000 0.2000 0.2000 0.7500 0.7500 0.2000

0.3750 0.9000 0.0417 0.9000 0.9000 0.3750

0.1429 0.1429 0.1429 0.1429 0.5714 0.5714

0.9000 0.3750 0.3750 0.3750 0.9000 0.9000

e1

e2

e3

e4

e5

e6

e7

e8

(9, 10, 10) (3, 5, 7)

(7, 9, 10) (5, 7, 9)

(9, 10, 10) (3, 5, 7)

(7, 9, 10) (3, 5, 7)

(9, 10, 10) (5, 7, 9)

(0, 1, 3) (3, 5, 7)

(5, 7, 9) (3, 5, 7)

(0, 1, 3) (1, 3, 5)

Table 4 Ideal solution with TFNs.

PIS NIS

Table 5 Defuzzified values of TFNs. U

e1

e2

e3

e4

e5

e6

e7

e8

h1 h2 h3 h4 h5 h6

5 7 7 9.75 8.75 9.75

8.75 7 7 8.75 7 7

7 7 9.75 7 5 7

7 5 8.75 7 7 8.75

9.75 7 7 8.75 8.75 7

3 1.25 5 1.25 1.25 3

5 5 5 5 7 7

1.25 3 3 3 1.25 1.25

For simplicity, we only enumerate some combinations for  +  = 1, as shown in Table 6. H = {w1 ≤ 0.3, 0.1 ≤ w2 ≤ 0.2, w3 − w1 ≤ 0.1, w5 − w2 ≥ w4 With  +  = / 1, we list only three indicative cases and the ranking results as shown in Table 7. − w8 , 0.1 ≤ w5 ≤ 0.3, 0.2 ≤ w6 ≤ 0.3, w7 ≥ w8 , w8 ≥ 0.05}. From Tables 6 and 7, in the case when the decision maker has a Here we assume that the decision maker is neutral to regret and regret degree less than the rejoice degree, though the weights of the rejoice, that is  = 0.5,  = 0.5. Solving Model (M-2) yields the criteria are changed, the rank orders are still Z4 Z6 Z1 Z3 Z2 . weight vector: This means that as long as the expert panel is more likely to rejoice, the best LSP is h4 . When the decision maker is neutral to regret w = (0.2305, 0.1000, 0.0562, 0.0708, 0.2345, 0.2000, 0.0500, 0.0500).and rejoice, the optimal LSP is h . When the decision maker tends 3 to regret more than rejoice, then choosing either LSP h3 or h2 is Step 7: Calculate the perceived utility values: optimal. panel, is shown as follows, respectively:

Z1 = 0.0706

,,,,

Rank all the LSPs based on the perceived utility: Z3 Z6 Z4

Z1 Z2 , and thus the most qualified LSP is h3 . Hence, LSP h3 is chosen. If the information about the attribute weights is completely unknown (with  = 0.5,  = 0.5), we utilize Eq. (4.6) to get the criteria weights vector:

(2) If the information about the attribute weights is completely unknown

Similar to the case of  +  = 1, we list 6 combinations, the results are shown in Table 8. With  +  = / 1, similarly, we list only three representative cases and the ranking results as shown in Table 9. w = (0.2373, 0.0695, 0.1146, 0.1571, 0.1243, 0.1184, 0.0488, 0.1301)T . From Tables 8 and 9, we note that in cases when the decision maker has a regret degree less than the rejoice degree, though the Calculate theperceived utility values: weights of criteria are changed, the rank order is still Z4 Z6 Z3

Z 1 Z2 . This means that if the decision maker prefers to rejoice, Z1 = 0.0462 ,,,, the best LSP is h4 . When the decision maker is neutral to regret and Rank all the LSPs based on the perceived utility: Z3 Z4 Z6

rejoice, the optimal LSP is h3 . When the decision maker has more Z2 Z1 , again the most qualified LSP is h3 , i.e. LSP h3 is chosen. regret than rejoice, the rank order is Z2 Z1 Z3 Z6 Z4 and LSP h2 is selected. From the above analysis, we can infer that as long as the decision 5.2. Sensitivity analysis maker prefers to rejoice, the optimal LSP is h4 . When the decision maker is indifferent to regret and rejoice, then LSP h3 is the As a decision maker can regret and rejoice at the same time, the best. If the decision maker tends to regret, the best LSP is h2 or h3 , bounded rational decision making model with (, ) level of regret depending on the weights of the attributes. and rejoice could also be applied to this problem. In order to reflect the influence of parameter (, ) on the results, we use different tuples (, ) and assess the obtained weights of the criteria and ranking of the LSPs.

5.3. Comparison and discussion

(1) If the information about the attribute weights is incomplete

In this section, some further discussions are provided to demonstrate the characteristics of the proposed method. A comparative

W. Chen et al. / Applied Soft Computing 71 (2018) 353–363

361

Table 6 LSP assessment results under incomplete weight information ␰ + ␨ = 1. regret

rejoice

Weights





w1

w2

w3

w4

w5

w6

w7

w8

Ranking

0.0 0.1 0.4 0.5 0.7 1.0

1.0 0.9 0.6 0.5 0.3 0.0

0.2243 0.2246 0.2257 0.2385 0.0779 0.0796

0.1398 0.1181 0.1176 0.1000 0.1236 0.1234

0.0466 0.0464 0.0471 0.0562 0.0236 0.0234

0.0658 0.0656 0.0651 0.0708 0.0087 0.0087

0.2175 0.2393 0.2386 0.2345 0.1472 0.1469

0.2000 0.2000 0.2000 0.2000 0.2402 0.2396

0.0530 0.0530 0.0529 0.0500 0.3289 0.3283

0.0530 0.0530 0.0529 0.0500 0.0500 0.0500

Z4 Z4 Z4 Z3 Z3 Z3

Z6

Z6

Z6

Z6

Z2

Z2

Z1

Z1

Z1

Z4

Z1

Z1

Z3

Z3

Z3

Z1

Z6

Z6

Z2

Z2

Z2

Z2

Z4

Z4

Table 7 / 1. LSP assessment results under incomplete weight information ␰ + ␨ = regret

rejoice

Weights

Ranking





w1

w2

w3

w4

w5

w6

w7

w8

0.1 0.4 0.7

0.5 0.4 0.2

0.2247 0.2385 0.0787

0.1181 0.1000 0.1235

0.0464 0.0562 0.0235

0.0655 0.0708 0.0087

0.2393 0.2345 0.1470

0.2000 0.2000 0.2399

0.0530 0.0500 0.3286

0.0530 0.0500 0.0500

Z4 Z6 Z1 Z3 Z2 Z3 Z6 Z4 Z1 Z2 Z3 Z2 Z1 Z6 Z4

Table 8 LSP assessment results under completely unknown information ␰ + ␨ = 1. regret

rejoice

0.0 0.1 0.4 0.5 0.7 1.0

1.0 0.9 0.6 0.5 0.3 0.0

Weight

Ranking

w1

w2

w3

w4

w5

w6

w7

w8

0.2331 0.2347 0.2567 0.2373 0.2078 0.2193

0.1472 0.1472 0.1463 0.0695 0.1482 0.1478

0.1165 0.1172 0.1275 0.1146 0.1046 0.1100

0.1876 0.1885 0.2000 0.1571 0.1744 0.1804

0.1744 0.1749 0.1816 0.1243 0.1667 0.1702

0.0360 0.0342 0.0092 0.1184 0.0648 0.0518

0.0606 0.0609 0.0642 0.0488 0.0568 0.0585

0.0445 0.0424 0.0145 0.1301 0.0767 0.0621

Z4 Z4 Z4 Z3 Z2 Z2

Z6

Z6

Z6

Z4

Z1

Z1

Z3

Z3

Z3

Z6

Z3

Z3

Z1

Z1

Z1

Z2

Z6

Z6

Z2

Z2

Z2

Z1

Z4

Z4

Table 9 / 1. LSP assessment results under completely unknown information+ = regret

rejoice

Weight





w1

w2

w3

w4

w5

w6

w7

w8

0.1 0.4 0.7

0.5 0.4 0.2

0.2363 0.2373 0.2133

0.1471 0.0695 0.1480

0.1180 0.1146 0.1072

0.1893 0.1571 0.1773

0.1754 0.1243 0.1684

0.0324 0.1184 0.0586

0.0611 0.0488 0.0576

0.0404 0.1301 0.0697

Ranking

study is conducted to examine the results of the proposed method along with those from other approaches. (1) Compared with ignoring the probability of success for each candidate under each criterion For simplicity, we only enumerate three representative combinations for  +  = 1, the results are shown in Table 10. From Table 10, we can see that the comparison results are different. The main reason for the difference is that there are complementarities among attributes if the probability of success is not considered. Due to the complementary of attributes, the candidate h5 performs well even if the attribute e3 (reputation) does not well. If we do not exclude it, it is likely to cause a huge loss to the retailer due to the neglect of the reputation. This can be effectively avoided in our proposed approach. (2) Compare against the traditional regret and TOPSIS. The traditional regret method has solved some multiple attribute decision making problems with just one reference point (PID or NID); whereas there are two reference points: PID and NID in the extended regret method. It is suitable for cautions DMs, because the DMs might like to have a decision, which not only makes as

Z4 Z6 Z3 Z1 Z2 Z3 Z4 Z6 Z2 Z1 Z2 Z1 Z3 Z6 Z4

much profit as possible, but also avoid as much risk as possible. The traditional regret method assumes that the DM is complete rationality, whereas our paper assumes that the DM is bounded rationality. In the actual decision making process, the decision makers do not always fully regret or rejoice. Rather, they tend to experience partial regret and rejoice simultaneously. The extended regret method assumes that both gain and loss exist for decision making, which coincides with the intuition of human beings. The TOPSIS method does not consider the DM’s psychology, whereas the psychology of DM is taken into account in the extended regret method. The TOPSIS assumes that the DM is complete rationality, whereas the extended regret method assumes that the DM is bounded rationality. In the actual decision making process, the decision maker’s psychological behaviour has a great influence on the decision result. Different DMs have different attitudes towards risk. These similarities and differences among methodologies are shown in Table 11. In summary, there are some problems by using the existing methods in the process of decision making. From Table 11, we can see that the extended regret method can overcome the abovementioned drawbacks of some existing methods. Moreover, the proposed method is based on the new formulate for calculating the gain and loss for decision maker, which is more reasonable.

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W. Chen et al. / Applied Soft Computing 71 (2018) 353–363

Table 10 Comparison results for ␰ + ␨ = 1. regret

rejoice

Rank order





Consider the probability of success

The probability of success is not considered

0.0 0.5 1.0

1.0 0.5 0.0

Z4 Z6 Z1 Z3 Z2 Z3 Z6 Z4 Z1 Z2 Z3 Z2 Z1 Z6 Z4

Z3 Z4 Z6 Z1 Z5 Z2 Z1 Z2 Z5 Z3 Z6 Z4 Z2 Z5 Z4 Z1 Z6 Z3

Table 11 Comparison with the traditional regret and TOPSIS. Characteristic

Extended regret

Traditional regret

TOPSIS

Psychology Rationality Reference vector Ranking results

Consider rejoice and regret psychology Bounded rationality Two reference vectors (PIS and NIS) Ranking results changes with DM’s psychology

Only consider rejoice or regret psychology Complete optimistic or complete pessimistic Only one reference vector (PIS OR NIS) Unique

No consider the DM’s psychology Complete rationality Two reference vectors (PIS and NIS) Unique

6. Conclusion While many fuzzy MCDM methods have been applied to study the logistics provider selection problem, those methods seldom consider the decision maker’s psychological behavior and they pay little attention to the probability of success for all candidates under each criterion. Considering these deficiencies, from the perspective of regret aversion/rejoice preference, this paper proposes an MCDM model based on fuzzy axiomatic design and extended regret/rejoice theory to model the LSP selection problem under uncertain and incomplete information. Bounded rationality decision models determining the unknown weight vector are developed, leading us to make decisions under regret minimization and rejoice maximization. We have treated the case when the decision makers can regret and rejoice simultaneously. In short, we have developed a new evaluation system for the LSP selection problem in an omni-channel environment. Our proposed method has the following characteristics. First, the probability of achieving FRi (ej ) for the choice of logistics providers is obtained using the fuzzy axiomatic design method. It is a novel idea integrating the performance level expected of the LSP by the omni-channel retailer and the actual performance level of the LSP, through fuzzy axiomatic design to measure the probability of success of choosing the right logistics provider. Second, from the perspective of the regret aversion/rejoice preference, bounded rational decision models to find the weight vector are formed with either incomplete or completely unknown weight information. This study has some limitations. One of the limitations is that the design ranges are highly dependent on the knowledge of experts and the experts’ grasp of actual situation of the enterprise. If they lack the professional experience, this will be impact on the results obtained. One possible solution to address this issue would be to increase the number of experts. Another limitation is that the proposed work is lack of real data in our study. Future research could expand the scope of this paper by analyzing the multi-retailers and multi-logistics providers mutual evaluation and selection in omnichannel. Future studies could also test the success of LSP selection in omni-channel environment through conducting an experimental study of the application of the proposed model in a specific application domain.

Acknowledgements This work is supported by the Education Commission of Science and Technology plan projects of Chongqing (KJ1600317), the China Scholarship Council, the key project of National Social Science Foundation of China (No. 14AJL015), and the Humanities and Social

Sciences Foundation of the Ministry of Education of the People’s Republic of China (No. 13YJC630252).

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