Warehouse location selection with TOPSIS group decision-making under different expert priority allocations

Due to the finite number of location alternatives, usually pre-defined in the decision problem, to be evaluated under multiple, even conflicting criteria, the warehouse location selection could be appropriately framed as a multi-criterion decision-making (MCDM) problem (Özcan et al., 2011). As ill-defined formulations, MCDM problems often contain a criterion or criteria, from the set of criteria, which are subjective, with non-sharp information and limited measurement systems (Ocampo & Clark, 2015). The presence of both quantitative and qualitative factors (i.e., criteria) in the warehouse location selection process (Demirel et al., 2010) increases the complexity of the decision problem. As such, a decision regarding the location of a warehouse is generally one of the most critical and strategic decisions in logistics management and supply chain planning, mainly that such decision involves substantial capital investments and impacts future long-term capacity and inventory decisions (Demirel et al., 2010).

In a recent review by Yap et al. (2019) on the application of MCDM methods in site selection problems, to which warehouse location selection belongs, the analytic hierarchy process (AHP) of Saaty (1980) emerges as the widely used approach. In fact, one of the earliest works on solving warehouse location selection problem via MCDM methods was presented by Korpela and Tuominen (1996), with the AHP as their approach. Since then, the domain literature on this topic has flourished, and an increasing number of works that implemented MCDM methods and their hybrid, including their extensions via the use of fuzzy set theory, has been reported for the last decade. Some MCDM methods which were adopted in addressing the warehouse selection problem and closely related problems include approaches (i.e., pure or hybrid) based on the AHP (Alberto, 2000; García et al., 2014; Boltürk et al., 2016; Raut et al., 2017; Kabak & Keskin, 2018; Hakim & Kusumastuti, 2018; Singh et al., 2018; Franek & Kashi, 2017; Nevima & Kiszová, 2017), analytic network process — the generalisation of the AHP (Cheng et al., 2005), simple additive weighting (Chou et al., 2008; Dey et al., 2013), PROMETHEE II (Athawale et al., 2012), ELECTRE-II (He et al., 2017), the Choquet integral (Demirel et al., 2010), TOPSIS (Chu, 2002), VIKOR (Kutlu Gündoğdu & Kahraman, 2019), and cloud-based design optimisation (Temur, 2016), among others. Some works on this domain purposely combined two or more MCDM methods to overcome and complement the limitations of each technique and come up with a more powerful hybrid selection tool. In most cases, a different approach is adopted to address the prioritisation (or weighting) of the criteria and another tool for the ranking of alternative warehouse locations. These works include the integration of fuzzy TOPSIS-SAW-MOORA (Dey et al., 2016), fuzzy AHP-TOPSIS (Roh et al., 2018), and stochastic AHP and fuzzy VIKOR (Emeç & Akkaya, 2018). Note that this list is not intended to be comprehensive.

Aside from MCDM techniques, different methods have been explored in addressing warehouse selection decisions. In general, these techniques are associated with mathematical programming, with various solution techniques such as search algorithms and heuristics (Tyagi & Das, 1995; Rosenwein, 1996). An early work of Lee et al. (1980) first proposed an integer goal programming formulation for a multi-criterion warehouse location problem. Since then, various extensions have been developed, including mixed-integer linear programming (Kratica et al., 2014; Brunaud et al., 2018; Vanichchinchai & Apirakkhit, 2018; You et al., 2019), non-linear programming (Monthatipkul, 2016), multi-objective optimisation model (Xifeng et al., 2013), and second-order cone programming (Wagner et al., 2009), among others. Due to the complexity of the formulation, and entrapment to the local optima as a direct consequence, various techniques were developed, such as the Lagrangian relaxation approach (Ozsen et al., 2008; Nezhad et al., 2013), approximation algorithms (Huang & Li, 2008), local search algorithm (Cura, 2010), branch-and-price algorithm (Klose & Görtz, 2007), hybrid firefly-genetic algorithm (Rahmani & MirHassani, 2014), evolutionary multi-objective optimisation (Rakas et al., 2004), two-stage robust models and algorithms (An et al., 2014), and weighted Dantzig-Wolfe decomposition and the path-relinking combined method (Li et al., 2014). Search heuristics were also proposed, including hybrid multi-start heuristic (Resende & Werneck, 2006), a discrete variant of unconscious search (Ardjmand et al., 2014 and Kratica et al., 2014), math heuristic (Rath & Gutjahr, 2014), greedy heuristic and fix-and-optimise heuristic (Ghaderi & Jabalameli, 2013), kernel search heuristic (Guastaroba & Speranza, 2014), iterated tabu search heuristic (Ho, 2015), modified Clarke and Wright savings heuristic (Li et al., 2015), and swarm intelligence based on sample average approximation (Aydin & Murat, 2013). Interpretive structural modelling was also adopted to solve a warehouse selection problem that incorporates the sustainability agenda (Jha et al., 2018). When formal mathematical programs are used to address warehouse location decisions, the factors are expected to be quantitative and measurable in such a way that they can be expressed as formal mathematical equations or inequalities. However, in most real-life cases, some factors relevant in the decision domain (e.g., quality of life, social and cultural, security) are qualitative and subjective, which could not be expressed as mathematical statements. Thus, MCDM methods are considered a holistic approach in addressing both quantitative and qualitative factors in the selection of a warehouse location.

In the context of MCDM applications for the selection of a warehouse location, the AHP, ELECTRE, and TOPSIS can be highlighted as the primary methods (Özcan et al., 2011). Özcan et al. (2011) made a comparative assessment of these methods, and the result is presented in Appendix 1. The assessment provides an insight into the performance of these methods in several areas. It is noteworthy that the TOPSIS has three main leverages: (1) the number outranking relations is one, which implies efficiency in judgment elicitations, (2) it is able to handle a large number of alternatives and criteria with objective and quantitative data, and (3) it generates a global, net order. These characteristics are appropriate in addressing a warehouse location selection, particularly when the number of criteria and alternatives is large, and the efficiency in generating the results from judgment elicitations is given a priority.

Initially proposed by Hwang and Yoon (1981), the foundation of TOPSIS lies at the notion of the distance function where the best alternative is chosen on the basis of maximising the distance from the negative ideal solution and minimising the distance from the positive ideal solution. Aside from location decision problems, TOPSIS has been used in a broad application domain, such as performance evaluation with the use of financial investment decisions (Kim et al., 1997), and financial ratios (Deng et al., 2000), personnel selection (Kelemenis & Askounis, 2010), strategy formulation (Ocampo, 2019), among others. Note that this list is not intended to be comprehensive. Reviews on the applications of TOPSIS have been reported by Behzadian et al. (2012), Shukla et al. (2017), and Yadav et al. (2018). TOPSIS leverages its advantages on simplicity and the tractability of the notion of distance based on ranking a set of alternatives (Chou et al. 2008; Özcanet al., 2011). It has efficient computational requirements due to its more straightforward evaluation techniques (Chou et al., 2008; Özcan et al., 2011; Roszkowska, 2011; Vavrek et al., 2017; Stankevièienë & Nikanorova, 2020).

The computational steps of the TOPSIS approach are provided below.

Step 1: Establish a decision matrix to evaluate the alternatives (e.g., supplier selection attributes) under different criteria. The structure of the evaluation can be expressed as follows: (1)Dk=(f11kf12k⋯f1nkf21kf22k⋯f2nk⋮⋮⋱⋮fm1kfm2k⋯fmnk){D^k} = \left({\matrix{{f_{11}^k} & {f_{12}^k} &\cdots& {f_{1n}^k}\cr{f_{21}^k} & {f_{22}^k} &\cdots& {f_{2n}^k}\cr \vdots&\vdots&\ddots&\vdots \cr{f_{m1}^k} & {f_{m2}^k} &\cdots& {f_{mn}^k}\cr}} \right) where fijkf_{ij}^k , i = 1, …, m, j = 1, …, n, k = 1, …, M, represents the evaluation score on the performance (or relevance) of the ith alternative on the jth criterion, elicited by the kth decision-maker.

Step 2: Aggregate the individual decision matrices using an aggregation function. One of the highly adopted aggregation functions is the arithmetic mean. Thus, the aggregate score f_ij can be obtained as fij=∑k=1Mfijk{f_{ij}} = \sum\nolimits_{k = 1}^M {f_{ij}^k} . The resulting aggregate decision matrix is shown in Equation (2). (2)D=(f11f12⋯f1nf21f22⋯f2n⋮⋮⋱⋮fm1fm2⋯fmn)D = \left({\matrix{{{f_{11}}} & {{f_{12}}} &\cdots& {{f_{1n}}}\cr{{f_{21}}} & {{f_{22}}} &\cdots& {{f_{2n}}}\cr \vdots&\vdots&\ddots&\vdots \cr{{f_{m1}}} & {{f_{m2}}} &\cdots& {{f_{mn}}}\cr}} \right)

Step 3: Obtain the priority weights of the criteria. The priority weight of a criterion j is expressed as w_j. Any prioritisation technique generates this. Note that the TOPSIS approach provides no specific method for obtaining the priority weights of the criteria.

Step 4: Calculate the normalised decision matrix R = (r_ij)_m×n. The normalised value r_ij is obtained as: (3)rij=fij(Σi=1mfij2)12 ∀i=1,…,m, j=1,…,n{r_{ij}} = {{{f_{ij}}} \over {{{(\Sigma _{i = 1}^mf_{ij}^2)}^{{1 \over 2}}}}}\,\,\,\,\forall i = 1, \ldots,m,\,\,\,j = 1, \ldots,n

Step 5: Calculate the weighted normalised decision matrix V = (v_ij)_m×n. Each element denoted as v_ij, is obtained by (4)vij=rij×wj∀i=1,…,m, j=1,…,n\matrix{{{v_{ij}} = {r_{ij}} \times {w_j}} & {\forall i = 1, \ldots,m,\,\,\,j = 1, \ldots,n}\cr}

Step 6: Determine the positive ideal solution (PIS), denoted by V⁺, and the negative ideal solution (NIS), denoted by V⁻: (5)V+={v1+,...,vn+}={(maxi vij:j∈J1),(mini vij:j∈J2)}\matrix{{{V^ +} = \{v_1^ +,...,v_n^ + \}= \{(\mathop {max}\limits_i \,{v_{ij}}:j \in {J_1}),}\cr{(\mathop {min}\limits_i \,\,{v_{ij}}:j \in {J_2})\}}\cr}(6)V-={v1-,...,vn-}={(mini vij:j∈J1),(maxi vij:j∈J2)}\matrix{{{V^ -} = \{v_1^ -,...,v_n^ - \}= \{(\mathop {min}\limits_i \,{v_{ij}}:j \in {J_1}),}\cr{(\mathop {max}\limits_i \,\,{v_{ij}}:j \in {J_2})\}}\cr} where J₁ is associated with the benefit (i.e., maximising) criteria, and J₂ is associated with the cost (i.e., minimising) criteria.

Step 7: Calculate the separation measures, using the m-dimensional Euclidean distance. The separation measure Di+D_i^ + of each alternative i from the PIS is given as (7)Di+={Σj=1n(vij-vj+)2}2¯ ∀i=1,…,mD_i^ += {\left\{{\Sigma _{j = 1}^n{{\left({{v_{ij}} - v_j^ +} \right)}^2}} \right\}^{\bar 2}}\,\,\,\forall i = 1, \ldots,m Similarly, the separation measures Di-D_i^ - of each alternative from the NIS is as follows: (8)Di-={Σj=1n(vij-vj-)2}2¯ ∀i=1,…,mD_i^ -= {\left\{{\Sigma _{j = 1}^n{{\left({{v_{ij}} - v_j^ -} \right)}^2}} \right\}^{\bar 2}}\,\,\,\forall i = 1, \ldots,m

Step 8: Calculate the relative closeness to the ideal solution and rank the alternatives in descending order. The relative closeness coefficient of the alternative j with respect to PIS V⁺ can be expressed as: (9)Ci=Di-Di-+Di+ ∀i=1,…,m{C_i} = {{D_i^ -} \over {D_i^ -+ D_i^ +}}\,\,\forall i = 1, \ldots,m

ABC-G Enterprises is a product distributor of one of the largest brewing companies in the Philippines. It is a local distributor which is located in Cebu, an island in the central Philippines. With an increasing trend for the demand for their products, the company is in the process of finding a new location where they can build their second warehouse. The new warehouse is intended to stock a significant volume of their products to respond to an expected increase in customer demand. Two possible location alternatives were identified by ABC-G Enterprises. One possible location is at a 10-kilometre distance from their current headquarters, with an area of around 380 square meters. The second alternative has an area of approx. 300 square meters and is located within a 9-kilometre distance. Aside from the available area and the distance of the possible location, the company is also considering other salient criteria. For brevity, we refer to Talamban warehouse and Compostela warehouse for the first and second alternative, respectively. In determining the best location, the final decision lies with the administration team, which is composed of the President, Administration Manager, Senior Manager, and Assistant Manager, who are usually involved in making the crucial decisions of the company. Thus, there is a need for ABC-G Enterprises to carry out an analytic multi-criterion group decision-making process to identify the best location for the warehouse.

The proposed TOPSIS group decision-making process in this work consists of the following steps:

Step 1: Set up the decision warehouse location selection problem.

The decision problem is shown in Fig. 1. It shows the evaluation of two alternative warehouses (i.e., Talamban Warehouse and Compostela Warehouse) under 22 selection criteria. Current literature offers a number of selection criteria for warehouse location selection. Appendix 2 presents the majority of these criteria. These criteria are generic to some extent but may not be applicable in some cases, depending on the decision problem under consideration. The two warehouse location alternatives are evaluated with this set of criteria. Table 1 presents these criteria, corresponding codes for the brevity of presentation, and a brief description.

Fig. 1

Step 2: Assign the importance weights of the expert decision-makers.

For this work, the administration team, which is composed of four members, becomes the expert group tasked to elicit judgments within the TOPSIS approach. Thus, the proposed approach becomes a TOPSIS group decision-making problem. However, in this case, the members of the expert group are non-homogeneous in their expertise of the decision problem, as well as their power in decision-making within the company. Thus, the aggregation described in Step 2 must be revised. Instead, expert decision-makers are assigned corresponding priority weights. These weights represent the importance of their inputs to the group decision. One crucial point that must be addressed in the process of generating these weights. To address this, three possible scenarios were explored; that is, three sets of priority weights of decision-makers were generated. Priority weights were obtained based on (S1) the level of their expertise in warehouse location selection, (S2) their power in making decisions, (S3) equal weights. Table 2 provides the rating scale that was used to generate the importance weights of the decision-makers. The weights of the decision-makers based on the three scenarios (i.e., S1, S2, S3) are shown in Table 3.

Based on S1, the Assistant Manager got the highest priority since it has a better knowledge of the warehouse operations. The S2 scenario yields the President's judgments with the highest weight as it has the most significant power in making decisions for the company. Finally, Table 3 shows with equal priority weights, that the level of influence of the four decision-makers on the warehouse location selection is equal.

Step 3: Generate the priority weights of the criteria. Decision-makers were asked to rate the importance of each criterion on a scale from 1 to 10, where 1 and 10 representing the lowest and highest importance, respectively. With S1, S2, and S3, three sets of priority weights were also generated for the criteria set. Table 4 presents the criteria weights on the three different scenarios.

Table 4 shows the weights of each criterion for all scenarios, i.e., S1, S2, and S3. It must be noted that the weights of the criteria presented in Table 4 are aggregate weights concerning the corresponding importance of the decision-makers in each scenario, as shown in Table 3. For instance, the weight of C1 for S1, written as w1S1w_1^{S1} , is computed using the following: (10)w1S1=Σkx1kS1ωkS1ΣjΣkxjkS1ωkS1w_1^{S1} = {{{\Sigma _k}x_{1k}^{S1}\omega _k^{S1}} \over {{\Sigma _j}{\Sigma _k}x_{jk}^{S1}\omega _k^{S1}}} where w1S1w_1^{S1} denotes the weight of criterion 1 in S1, x1kS1x_{1k}^{S1} is the score of criterion 1 under S1 as elicited by decision-maker k, and ωkS1\omega _k^{S1} represents the weight of decision-maker k under S1. For instance, w1S1=(10×0.2258)+(10×0.2258)+(10×0.2581)+(10×0.2903)131.8710=0.0758w_1^{S1} = {{(10 \times 0.2258) + (10 \times 0.2258) + (10 \times 0.2581) + (10 \times 0.2903)} \over {131.8710}} = 0.0758

Step 4: Decision-makers elicit judgments on the decision matrix. Using the rating scale for cost criteria (i.e., Table 5), and benefit criteria (i.e., Table 6), decision-makers elicit judgments on the performance of the jth alternative (i.e., warehouse) on the ith criterion.

The decision matrix is shown in Table 7.

Step 5: Generate aggregate decision matrices. Three aggregate decision matrices were generated, which corresponded to S1, S2, and S3 scenarios. Since the decision-makers had different importance weights at different scenarios, the aggregation function described in Step 2 of Section 2.2 had to be revised. To incorporate the importance weights of the decision-makers, the aggregation function is developed in Equation (11). (11)fijσ=14Σkfijk,σωkσf_{ij}^\sigma= {1 \over 4}{\Sigma _k}f_{ij}^{k,\sigma}\omega _k^\sigma where fijσf_{ij}^\sigma represents the aggregate performance score of the ith alternative (i.e., warehouse) with respect to jth criterion, under scenario σ (i.e., S1, S2, S3), fijk,σf_{ij}^{k,\sigma} is the score of the ith alternative with respect to jth criterion, under scenario σ evaluated by the kth decision-maker, and ωkσ\omega _k^\sigma is the importance weight of decision-maker k under scenario σ. The aggregate decision matrices are shown in Table 8.

For instance, using Equation (11), the value of f11S1f_{11}^{S1} can be obtained: f11S1=(4×0.2258)+(5×0.2258)+(5×0.2580)+(4×0.2932)4=1.1210f_{11}^{S1} = {{(4 \times 0.2258) + (5 \times 0.2258) + (5 \times 0.2580) + (4 \times 0.2932)} \over 4} = 1.1210

Step 6: Calculate the normalised decision matrices. Using Equation (3), the normalised decision matrices are obtained.

Step 7: Calculate the weighted normalised decision matrices. Using Equation (4) with inputs from the priority weights of the criteria under the different scenarios, the weighted normalised decision matrices are obtained. Table 10 presents these matrices.

Step 8: Determine V⁺ and V⁻ for S1, S2, and S3.

The PIS (V⁺) and the NIS (V⁻) are obtained using Equation (5) and Equation (6), respectively. Table 11 describes these values.

Step 9: Generate the separation measures Di+D_i^ + and Di-D_i^ - . The separation measures Di+D_i^ + and Di-D_i^ - are obtained using Equation (7) and Equation (8), respectively. The results are shown in Table 12.