Sustainable construction supply chain management with the spotlight of inventory optimization under uncertainty

Abstract

In this research, a supply chain network has been designed for inventory management using not only the project site storage facility but also an ancillary warehouse to keep materials. In order to make decision about the appropriate place for building the warehouse, multi-criteria decision-making techniques have been applied. Since the transportation sector, as the most important energy-consuming part, plays a significant role in global warming after power stations and the delivery of materials will have environmental impacts, this research tried to minimize the external cost of global warming caused by transportation. In this study, a mathematical formulation is presented to solve the problem of ordering the required amount to project site, while taking into account an ancillary warehouse. To quell the discussion, a numerical example has been demonstrated. The findings show that uncertainty considerations fortify the strict decision making and can increase the confidence level.

Appendices

Appendix 1

1.1 Best–worst method

BWM is a pairwise comparison-based multi-criteria decision-making method, which has been utilized in various fields such as green innovation (Gupta and Barua 2018), technology evaluation and selection (Ren 2018), logistics performance evaluation (Rezaei et al. 2018), research and development performance evaluation (Salimi and Rezaei 2018) and supply chain management.

The concept and purpose of this method will be delineated in following paragraphs.

Suppose we have n criteria and we want to make a pairwise comparison matrix, so as to begin the process of obtaining weights of each of the criteria. As it is shown below, in this matrix every of the elements is indicating the relative preference of criteria to each other; moreover, the pairwise comparison matrix will be filled by decision maker (DM) using 1–9 scale. For instance, a_ij is the relative preference of criterion I to the criterion j. If a_ij = 1, it shows that criterion I and criterion j have the same importance, and if a_ij > 1, it shows that i is regarded as much more important one. If a_ij = 9, it is an indication of extreme importance of I to j (Rezaei 2015).

$$\left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {a_{11} } & {a_{12} } \\ {a_{21} } & {a_{22} } \\ \end{array} } & \cdots & {\begin{array}{*{20}c} {a_{1n} } \\ a \\ \end{array}_{2n} } \\ \vdots & \ddots & \vdots \\ {\begin{array}{*{20}c} {a_{n1} } & {a_{n2} } \\ \end{array} } & \cdots & {a_{nn} } \\ \end{array} } \right)$$

(49)

Considering the reciprocal property of matrix, to complete the above-mentioned matrix, (n − 1)/2 pairwise comparisons will be done. The consistency of matrix is of high. According to what is mentioned in the literature, a pairwise comparison matrix is consistent if:

$$a_{ik} \times a_{kj} = a_{ij} ,\,\,\,\,\forall i,j$$

(50)

The above-mentioned issues were the basics of pairwise comparison matrix, but still a question will be left and that one is related to the assurance and reliability of this matrix. In order to use experts’ ideas, we need to be sure that those ideas are not biased and eventually can express the strength of criterion I to criterion j appropriately. The best–worst method is introduced by Rezaei (2015) in order to obtain weights of each criterion through comparing others criterion with the best one and also comparing them with the worst, so with applying this method DM can express his preferences more easily. At last, the comparing process will be facilitated.

In this section, the steps of BWM will be explained to derive the weights of the criteria (Rezaei 2015).

Step 1. In this step, the decision criteria will be determined. Criteria $\{ c_{1} , c_{2} ,c_{3} , \ldots ,c_{n} \}$ have been considered for decision making.

Step 2. The best and the worst criteria must be found. The most important or most desirable and also the least important criteria will be identified by DM.

Step 3. A number between 1 and 9 will be assigned to each criteria to highlight the preference of the best one over other criteria. This will at last make best-to-others vector

$$A_{B} = (a_{B1} ,a_{B2} , \ldots , a_{B2} )$$

(51)

Step 4. A number between 1 and 9 will be assigned to each criteria to highlight the preference of the worst one over other criteria. This will at last make others-to-worst vector.

$$A_{w} = \left( {a_{w1} ,a_{w2} , \ldots , a_{wn} } \right)^{\text{T}}$$

(52)

Step 5. The optimal weights $(w_{1} , w_{2} ,w_{3} , \ldots ,w_{n} )$ will be found in the last step. The optimal weight for each criteria is the one for which $\frac{{w_{B} }}{{W_{J} }} = a_{ij}$ and also $\frac{{w_{j} }}{{W_{W} }} = a_{jW}$.

To meet the requirements mentioned for obtaining criteria’s weights, the maximum absolute differences ${\kern 1pt} \left| {\frac{{w_{B} }}{{W_{J} }} - a_{ij} } \right|$, $\left| { \frac{{w_{j} }}{{W_{W} }} - a_{jW} } \right|$ for all j need to be minimized. Hence, the model is shown in Eq. (53):

$$\hbox{min} \mathop {\hbox{max} }\limits_{j} \left\{ {\left| {\frac{{W_{B} }}{{W_{j} }} - a_{Bj} } \right|,\left| {\frac{{W_{j} }}{{W_{w} }} - a_{jw} } \right|} \right\}$$

(53)

s.t.

$$\begin{aligned} & \mathop \sum \limits_{j} W_{j} = 1 \\ & W_{j} \ge 0,\,\,\,{\text{for}}\, \,{\text{all}}\, j \\ \end{aligned}$$

The linear form is also obtained as:

$$\hbox{min} \xi$$

(54)

s.t.

$$\begin{aligned} \left| {\frac{{W_{B} }}{{W_{j} }} - a_{Bj} } \right|W_{j} & \ge 0,\,\,\,{\text{for}}\,{\text{all}}\,\,j \le \xi \,\,\,{\text{for }}\,{\text{all}}\,j \\ \left| {\frac{{W_{j} }}{{W_{w} }} - a_{jW} } \right| & \le \xi \,\,\,{\text{for }}\,{\text{all}}\,\, j \\ W_{j} & \ge 0,\,\,\,{\text{for }}\,{\text{all}}\,\, j \\ \end{aligned}$$

1.2 TOPSIS method

The technique for order preference by similarity to ideal solution (TOPSIS) was first introduced by Hwang and Yoon in 1981. The goal of this method is to rank the alternatives by calculating the distance of each alternative from the positive ideal solution and the negative ideal solution for problems in decision making, thus to determine the optimum alternative.

The steps of TOPSIS method presented by Chen (2019) are as followings.

Step 1. The decision matrix R = {r_ij}s, in which r_ij (i = 1, 2, …, m; j = 1, 2, …, n) is the value of the jth attribute in the ith alternative will be identified in this step.

Step 2. The difference of attributes and order of magnitude needs to be considered, and then, the decision matrix R will be normalized and the normalized matrix will be transformed to R′ = {r_ij′}.

Step 3. The weighted normalized decision matrixes will be found:v_ij = W_jr_ij′.

Step 4. The $D_{\text{IS}}$ and D_NIS will be identified by the following equations:

$$\begin{aligned} S_{i}^{ + } & = \sqrt {\sum\nolimits_{j = 1}^{n} {\left( {v_{ij} - v_{j}^{ + } } \right)^{2} } } \\ S_{i}^{ - } & = \sqrt {\sum\nolimits_{j = 1}^{n} {\left( {v_{ij} - v_{j}^{ - } } \right)^{2} } } \\ \end{aligned}$$

(55)

Step 5. The relative closeness of each alternative will be calculated in this step by the following equation:

$$RC_{i} = \frac{{S_{i}^{ - } }}{{S_{i}^{ + } + S_{i}^{ - } }}$$

(56)

The value of relative closeness reflects the relative superiority of the alternatives. Larger $RC _{i}$ indicates that the alternative i is relatively better, whereas smaller $RC _{i}$indicates this alternative is relatively poorer.

Appendix 2

Table 14 shows the amount of demand for materials at different periods during the planning horizon. Table 15 lists the parameters for each supplier. Table 16 shows the estimated purchase prices in fuzzy numbers.

Table 14 Material demand in planning horizon

Table 15 Parameters used in the mathematical formulation

Table 16 Material price in planning horizon

Appendix 3

The following questionnaire examines the importance of 11 criteria for selecting the best location for ancillary warehouse construction in a construction project. The name of each criterion and its exact explanation are given in front of it, please give a score to each criterion according to the set spectrum.