*Article* **Considering Product Life Cycle Cost Purchasing Strategy for Solving Vendor Selection Problems**

**Chien-Wen Shen 1, Yen-Ting Peng <sup>1</sup> and Chang-Shu Tu 2,3,4,\***


Received: 30 May 2019; Accepted: 2 July 2019; Published: 9 July 2019

**Abstract:** The framework of product life cycle (PLC) cost analysis is one of the most important evaluation tools for a contemporary high-tech company in an increasingly competitive market environment. The PLC-purchasing strategy provides the framework for a procurement plan and examines the sourcing strategy of a firm. The marketing literature emphasizes that ongoing technological change and shortened life cycles are important elements in commercial organizations. From a strategic viewpoint, the vendor has an important position between supplier, buyer and manufacturer. The buyer seeks to procure the products from a set of vendors to take advantage of economies of scale and to exploit opportunities for strategic relationships. However, previous studies have seldom considered vendor selection (VS) based on PLC cost (VSPLCC) analysis. The purpose of this paper is to solve the VSPLCC problems considering the situation of a single buyer–multiple supplier. For this issue, a new VSPLCC procurement model and solution procedure are derived in this paper to minimize net cost, rejection rate, late delivery and PLC cost subject to vendor capacities and budget constraints. Moreover, a real case in Taiwan is provided to show how to solve the VSPLCC procurement problem.

**Keywords:** vendor selection; product life cycle; multi-objective linear programming; multi-choice goal programming

#### **1. Introduction**

Modern businesses face an increasingly competitive market environment, in which companies need to shorten product life cycle (PLC) to bring their good products to market quickly, and thereby increase their competitive advantages. In particular, the PLC of electronic products has become shorter to support the timing of marketing [1]. A significant challenge faced by the vendor–buyer supply chain (SC) is how to deal with the arrangement of the vendor's uncertain lead time and the buyer's random demand over the selling season [2]. Accurately determining timing for purchasing is an important issue for procurement plans. The PLC-purchasing strategy (PS) offers a framework for procurement plans and examines the sourcing strategy of a firm [3,4]. PLC is a descriptive framework that classifies the development of product-markets into four stages: Introduction, growth, maturity, and decline. In the introduction stage, there are few competitors in the market. This provides innovators with a chance to use a price-skimming strategy to recoup their product development costs and encourage knowledge of the new product. In the growth stage, overall market sales increase radically, attracting many new market entrants. The decline stage is entered when overall market sales begin to fall. During this stage products are withdrawn from the market and firms reduce their marketing expenditures to

cut costs [5]. It can be seen that using the framework of PLC can act as a guideline to aid purchasing managers in fitting the performance of their ever-expanding duties and tasks for the optimal profit of the company. Purchasing planners have known that they want to achieve this desired elasticity by fitting procurement actions to each PLC phase. The emphasis on this procurement planning is on the timing of the changes in purchasing activities to create the best utilization of company resources [6]. Schematically, the PLC can be approximated by a bell-shaped curve that is divided into several stages. The PLC is typically depicted as a unit sales curve of a product category over time [7–16]. Another important issue faced by firms is the vendor selection (**VS**) problem. Supply professionals must balance their firm's quality and delivery policies with the cost saving and flexibility profit offered by vendors, so a vendor's product manufacturing skills are attractive early on the relationship but efficiency dictates in later stages [7]. The purchasing firm's preferences or weights associated with various vendor attributes may vary during different stages of the PLC. The concept of PLC cost (**PLCC**) originates from the US Department of Defense and is focused on a product's entire value chain from a cost perspective since the development phase of a product's life, through design, manufacturing, marketing/distribution and finally customer services [8]. Elmark and Anatoly (2006) indicated that the PLCC is the total cost of acquiring and utilizing a system over its complete life span [10]. Vasconcellos and Yoshimura (1999) proposed a breakdown structure to identify the main activities for the active life cycle of automated systems [11]. Spickova and Myskova (2015) proposed activity based costing, target costing and PLC techniques for optimal costs management [12]. Sheikhalishahi and Torabi (2014) proposed a VS model considering PLCC analysis for manufacturers to deal with different vendors offering replaceable/spare parts [13]. Narasimhan and Mahapatra (2006) developed a multi-objective decision model that incorporates a buyer's PLC-oriented relative preferences regarding multiple procurement criteria for a portfolio of products [3]. Life cycle costing is concerned with optimizing the total costs in the long run, which consider the trade-offs between different cost elements during the life stages of a product [17]. In brief, the PLCC methodology aims to assist the producer to forecast and manage costs of a product during its life cycle. PLCC is a good technique used to assess the performance of a PLC. It can evaluate the total cost incurred in a PLC and assist managers in making decisions in all stages [9]. Their research aims to obtain a comprehensive estimation of the total costs of alternative products or activities in the long run. It is usually possible to affect the future costs beforehand by either planning the use of an asset or by improving the product or asset itself [18]. Previous studies, however, have seldom examined the VSPLCC procurement problem in the situation of single buyer–multiple supplier. The contribution of the study is to consider a VSPLCC problem with a single-buyer multiple-supplier procurement problem. We integrate VS and PLCC (VSPLCC) procurement planning into a model for enterprise to reduce their purchasing cost. Based on the literature reviews and discussions with experts in this field, we obtained important criteria, including price, transportation cost, quality, quality certification, lead time, necessary buffer stock, goodwill, PLC cost, vendor reliability, and vendor-area-specific experience in the VSPLCC problem of real case example. In addition, we would like to maximize the benefit of the procurement process and must continue to reduce purchasing costs as well as aim to achieve minimal costs to obtain the maximum benefit. To help purchasing managers effectively perform and coordinate these responsibilities with their jobs, we need to reconceptualize their role for procurement [14,15]. A new VSPLCC procurement model is then proposed to solve the problem of real case example procurement problem and is presented based on the modified dataset of the auto parts manufacturers' example, and a numerical example is adopted from a light-emitting diode company in Taiwan. Our study considers the following goals: For more realistic applications, net cost minimization, rejection rate minimization, and late delivery minimization, minimization of PLCC, and vendor capacities and budget constraints. Moreover, multi-objective linear programming (MOLP) and multi-choice goal programming (MCGP) approaches are integrated to solve this VSPLCC procurement problem.

The paper is organized as follows. We review the literature regarding the quantitative methods for the VS decision in Section 1. Section 2 presents the formulations and solutions to the VSPLCC procurement problem using both MOLP and MCGP approaches. In Section 3, the solution procedures of the two approaches for VSPLCC procurement problem are presented based on the modified dataset of the auto parts manufacturers' example and a numerical example is adopted from a light-emitting diode company in Taiwan [19]. In Section 4 Solution results of MOLP and MCGP are provided. Conclusions regarding the managerial implications and limitations solving the VSPLCC procurement problem in the four stages of the PLC with MOLP and MCGP approaches are addressed in Section 5.

#### **2. The VSPLCC Procurement Approaches**

#### *2.1. Linear Programming Technique*

Linear programming (LP) is a powerful mathematical technique which can be used to solve PLC problem. Azapagic and Clift (1998) applied LP to assess the environmental performance of a product system [20]. Dowlatshahi (2001) developed a conceptual framework to tactically consider PLC costs [21]. Zimmermann (1978) showed that a problem with fuzzy goals and constraints can be reformulated as conventional LP problem [22]. Ghodsypour and O'Brien (1998) utilized AHP and LP to develop a decision support system for solving VS problems [23]. Kumar et al. (2004) used fuzzy GP to address the effects of information uncertainty on the VS problems [24]. Amid et al. (2006) developed a fuzzy multi-objective LP model to overcome the VS problem with vague information [25]. In addition, Kagnicioglu (2006) first compared two fuzzy multi-objective methods for VS problems [26]. Chang (2007, 2008) proposed the MCGP method which allows one goal mapping multiple aspiration levels to find the best achievement levels for multiple objective decision making (**MODM**) problems [27,28]. Accordingly, in order to improve the quality of decision making for solving the VSPLCC procurement problem, we integrate AHP and MCGP methods, wherein both qualitative and quantitative issues are considered for more realistic VSPLCC applications. The AHP-MCGP method is also used to aid decision makers (DMs) in obtaining appropriate weights and solutions for the VSPLCC problem. The proposed VSPLCC procurement model can be easily used to select an appropriate vendor from a number of potential alternatives. The framework adopted for this study is shown in Figure 1.

**Figure 1.** Framework of the study.

The formulation of the VSPLCC procurement model requires the following assumptions, indices, decision variables and parameters.

#### *2.2. Fuzzy Multi-Objective Models for the VSPLCC Procurement*

VSPLCC Procurement Problem


The sets of indices, parameters, and decision variables for the VSPLCC model are listed in Table 1.


**Table 1.** Nomenclature [fuzzy parameters are shown with a tilde (~)].

#### *2.3. VSPLCC Procurement Model*

The multi-objective VSPLCC procurement problem with four fuzzy objectives and some constraints are as follows:

$$\text{Min } Z\_{1t} \stackrel{\pi}{=} \sum\_{i=1}^{n} \sum\_{t=4}^{4} P\_{it} X\_{it} \text{ the total net cost} \tag{1}$$

$$\text{Min } Z\_{2t} \stackrel{\Rightarrow}{=} \sum\_{i=1}^{n} \sum\_{t=1}^{4} Q\_{it} X\_{it} \text{ the reject items for vendor } i \tag{2}$$

$$\text{Min } Z\_{\mathfrak{H}} \equiv \sum\_{i=1}^{n} \sum\_{t=1}^{4} L\_{it} X\_{it} \text{ the late delivered items for vendor } i \tag{3}$$

$$\text{Min } Z\_{4l} \cong \sum\_{i=1}^{n} \sum\_{t=1}^{4} C\_{it} X\_{it} \text{ the product life cycle } \cos t \text{ for window } i \tag{4}$$

The following constraints are given for the VSPLCC procurement problem:

$$\sum\_{i=1}^{n} \sum\_{t=1}^{4} X\_{it} \ge \widetilde{D}\_t \text{ (aggregate demand constraint)}\tag{5}$$

$$X\_{\text{il}} \le \overline{l} \overline{l}\_{\text{il}} \text{ : } = 1, \text{ } 2, \dots, n \text{ : } = 1, \text{ } 2, \text{ } 3, \text{ } 4, \text{ (capacity constraint)}\tag{6}$$

$$\sum\_{i=1}^{n} \sum\_{t=1}^{4} r\_{it}(X\_{it}) \ge p\_{it}; \ t = 1, \ 2, \ 3, \ 4, \ \text{(total items purchased constraint)}\tag{7}$$

$$\sum\_{i=1}^{n} \sum\_{t=1}^{4} f\_{\text{it}}(X\_{\text{it}}) \le F\_{\text{it}}; \ t = 1, \ 2, \ 3, \ 4, \ (\text{quota constraint}) \tag{8}$$

$$P\_{\rm il}X\_{\rm il} \le B\_{\rm il}; \; i = 1, \; 2, \ldots, n, \; t = 1, \; 2, \; 3, \; 4, \; \text{(tudget constraint)}\tag{9}$$

$$X\_{\text{if}} \ge 0, \text{ i } = 1, \text{ } 2, \dots, n, \text{ } t = 1, \text{ } 2, \text{ } 3, \text{ } 4. \text{ (non-negative constraint)}\tag{10}$$

Equation (5) presents the aggregate demand constraint larger than quantity of items supplied over a fixed planning period. Equation (6) presents the vendor product capacity constraint based on the uncertain aggregate demand. Equation (7) presents the incorporate total item purchasing value constraint. Equation (8) presents the flexibility of the vendors' quota. Equation (9) presents the budgetary constraint where no vendor can exceed the budgeted allocated to vendors. Finally, Equation (10) presents the non-negativity constraint prohibiting negative orders. Generally, the tilde sign (~) indicates that the environment objectives function and constraints are fuzzy [29,30]. The fuzzy decision can be either symmetric or asymmetric depending on whether the objectives and constraints have equal or unequal weights [26,30,31]. These weights can be derived using techniques such as the AHP with a geometric mean (see details of the process in Chakraborty, Majumder; Sarkar, 2005) [32].

#### *2.4. The Solution of the VSPLCC Procurement Problem Using the Weight Additive Approach*

In this section, we present the general multi-objective model for solving the VS problem. To specify the weights of the goals and constraints in a fuzzy environment, we can use a fuzzy approach, instead of having the DM subjectively assign values to these weights. To obtain the supertransitive approximation of the previous comparison matrix, we construct supplementary matrices *A*1, *A*2, ... , *An*. The *j*th row of matrix *A<sup>j</sup>* is the same as the *j*th row of the initial matrix *A*, where the supplementary matrix (*A<sup>j</sup>* ) *<sup>T</sup>*<sup>∗</sup> = *a j* 1, *a j* <sup>2</sup>, ... , *a j n* and each row of the matrix *A<sup>j</sup>* is computed as follows (*T*<sup>∗</sup> : Transpose): *a j <sup>j</sup>* = *aj*, *a j* <sup>1</sup> = (*aj*1) −1 *a j j* , *a j* <sup>2</sup> = (*aj*2) −1 *a j j* , ... , *a j <sup>n</sup>* = (*ajn*) −1 *a* .*j <sup>j</sup>* . Next, we construct the supertransitive approximation, *As* = *a<sup>s</sup> ij* , *i*, *j* = 1, 2, ... , *n*, by taking the geometric mean of the corresponding elements from the supplementary matrices *A*1, *A*2, ... , *An*. More formally, *as ij* = (*a*<sup>1</sup> *ij* <sup>×</sup> *<sup>a</sup>*<sup>2</sup> *ij* <sup>×</sup> ......×*a<sup>n</sup> ij*) 1 *<sup>n</sup>* . Then we obtain the largest value of *As* with an eigenvector method. The corresponding eigenvector is the optimal weight for the criteria [26,33]. In the solution to the VSPLCC problem model, the AHP with weighted geometric mean (WGM) is calculated using a supertransitive approximation. Thus, these weights are assigned separately. In these equations, α*jt* is the weighting coefficient that shows the relative importance at the four stages of the PLC.

The following crisp simplex objective programming function used to solve VSPLCC procurement problem.

**Model 1**: The weighted additive (WA) approach [34], which is formulated as follows:

$$\text{Max}\sum\_{j=1}^{s}\sum\_{t=1}^{4}\alpha\_{jt}\lambda\_{jt}^{\*}\tag{11}$$

$$\text{s.t.} \,\lambda\_{j\text{t}} \le \mu\_{z/\text{t}}(\text{x}), \; j = 1, 2, \dots, q, \; t = 1, 2, 3, 4,\tag{12}$$

γ*rt* ≤ μ*hrt*(*x*), *r* = 1, 2, ... , *h* = 1, 2, 3, 4, (13)

$$g\_{mt}(\mathbf{x}) \le b\_{mt}, \ m = \ 1, \ \dots, \ p, \ t = \ 1, \ 2, \ 3, \ 4, \tag{14}$$

$$
\lambda\_t \in \langle 0, 1 \rangle, \ t = 1, \ 2, \ 3, \ 4, \tag{15}
$$

$$\sum\_{j=1}^{s} \sum\_{t=1}^{4} \alpha\_{jt} = 1, \alpha\_{jt} \ge 0, \ t = 1, \ 2, \ 3, \ 4, \tag{16}$$

$$\mathbf{x}\_{\text{nt}} \ge \mathbf{0}, \; n = \; 1, \; 2, \; \dots, \; i = \; 1, \; 2, \; 3, \; 4. \tag{17}$$

See Amid et al. (2011) [35] for a more detail.

#### *2.5. The Solution of the VSPLCC Procurement Problem Based on Lin's Weighted Max-Min Approach*

Lin (2004) proved that a weighted max–min (WMM) approach could find an optimal solution such that the ratio of the achievement level approximates the ratio of the weight as closely as possible. He noted that the WA model gives heavier weights to objectives of higher achievement levels than do other models. However, the ratio of the achievement levels is not necessarily the same as that of the objectives' weights [35,36]. Thus, to obtain the solution of the VSPLCC procurement problem model, WMM model is used as follows:

**Model 2**: Lin's WMM approach (Lin, 2004) [36]:

$$\mathbf{Max} \,\,\lambda\_{\mathbf{f}\_{\mathbf{f}}} \tag{18}$$

s.t. *wjt*λ*jt* <sup>≤</sup> <sup>μ</sup>*zjt*(*x*), *<sup>j</sup>* = 1, 2, ... , *<sup>q</sup>*, *<sup>t</sup>* = 1, 2, 3, 4, (19)

$$\forall \gamma\_{\underline{r}} \le \mu\_{\underline{h}\underline{r}}(\mathbf{x}), \; r = 1, \; 2, \; \dots, \; h = 1, \; 2, \; 3, \; 4, \tag{20}$$

$$g\_{mt}(\mathbf{x}) \le b\_{mt}, \ m = \ 1, \ \dots, \ p, \ t = \ 1, \ 2, \ 3, \ 4, \tag{21}$$

$$
\lambda\_t \in \{0, 1\}, \ t = 1, \ 2, \ 3, \ 4, \tag{22}
$$

$$\sum\_{j=1}^{s} \sum\_{t=1}^{4} \alpha\_{jt} = 1, \; \alpha\_{jt} \ge 0, \; t = 1, \; 2, \; 3, \; 4, \tag{23}$$

$$\mathbf{x}\_{\text{n!}} \ge \mathbf{0}, \; n = \; 1, \; 2, \; \dots, \; i\_{\text{.}} = \; 1, \; 2, \; 3, \; 4. \tag{24}$$

#### *2.6. The Solution of the VSPLCC Procurement Problem Based on MCGP Approaches*

In real decision-making problems, goals are often interrelated in which DMs can set more aspiration levels using the idea of multi-choice aspiration level (MCAL) to find more appropriate resources so as to reach the higher aspiration level in the initial stage of the solution process (Chang, 2007) [27]. To address this issue, the MCGP AFM (achievement function model) models are developed below.

MCGP AFM (case I)( **Model 3**): The MCGP AFM (case I) is used in the case of "the more, the better" as follows. Minimize

$$\sum\_{i=1}^{n} \sum\_{t=1}^{4} \left[ w\_{it} (d\_{it}^{+} + d\_{it}^{-}) + \alpha\_{it} (e\_{it}^{+} + e\_{it}^{-}) \right]$$

$$= 1 - d\_{-}^{+} + d\_{-}^{-} = h\_{i} \,\mu\_{i}, \, i = 1, 2, \ldots, \, t = 1, 2, 3, \, 4. \tag{25}$$

$$\text{s.t.} \, f\_{\text{it}}(X) b\_{\text{it}} - d\_{\text{it}}^{+} + d\_{\text{it}}^{-} = b\_{\text{it}} y\_{\text{it}}, \text{ i } = \text{ 1, 2, \dots, \text{ }, \text{ t.} = \text{ 1, 2, 3, 4, \dots} \tag{25}$$

$$y\_{it} - e\_{it}^{+} + e\_{it}^{-} = g\_{it, \text{max}} \text{ } i = 1, 2, \text{ } \dots \text{, } n \text{ } t = 1, 2, 3, 4 \text{ } \tag{26}$$

$$g\_{it, \text{min}} \le y\_{it} \le g\_{it, \text{max}}, \ t = 1, \ 2, \ \dots, \ n \, t = 1, \ 2, \ 3, \ 4, \tag{27}$$

$$d\_{\rm it}^{+}, d\_{\rm it}^{-}, e\_{\rm it}^{+}, e\_{\rm it}^{-} \ge 0, \; i = 1, 2, \ldots, \; t = 1, 2, 3, 4. \tag{28}$$

*X* ∈ *F* where *F* is a feasible set and *X* is unrestricted in sign. Where *bit* ∈ {0, 1} is a binary variable attached to *fit*(*X*) <sup>−</sup> *yit* , which can be either achieved or released in Equation (25). In terms of real conditions, *bit* is subject to some appropriate constraints according to real needs.

MCGP AFM (case II) (**Model 4**): The MCGP AFM (case II) is used in the case of "the less, the better" as follows. Minimize

$$\sum\_{i=1}^{n} \sum\_{t=1}^{4} \left[ w\_{it} (d\_{it}^{+} + d\_{it}^{-}) + a\_{it} (e\_{it}^{+} + e\_{it}^{-}) \right]$$
 
$$\text{s.t. } f\_{it}(X) b\_{it} - d\_{it}^{+} + d\_{it}^{-} = b\_{it} y\_{it}, \text{ i } = 1, \text{ 2, ..., } n, \text{ } t = 1, \text{ 2, 3, 4, } \tag{29}$$

$$y\_{it} - e\_{it}^{+} + e\_{it}^{-} = g\_{it, \text{max}} \text{ } i = 1, 2, \text{ } \dots \text{, } n \text{ } t = 1, 2, 3, 4 \text{ } \tag{30}$$

$$\text{g}\_{\text{it,min}} \le y\_{\text{it}} \le g\_{\text{it,max}}, \text{ i = 1, 2, \dots, n = 1, 2, 3, 4,\tag{31}$$

$$d\_{\rm it}^{+}, d\_{\rm it}^{-}e\_{\rm it}^{+}, e\_{\rm it}^{-} \ge 0, \; i = 1, 2, \ldots, \; t = 1, 2, 3, 4. \tag{32}$$

where all variables are defined as in model 3. The mixed-integer terms Equations (29) and (32) can easily be linearized using the linearization method (Chang, 2008) [28]. As seen in Equations (25), (29)–(31), there are no selection restrictions for a single goal, but some dependent relationships exist among the goals. For instance, we can add the auxiliary constraint *bit* ≤ *bi*+1,*<sup>t</sup>* + *bi*+2,*<sup>t</sup>* to the MCGP AFM, where *bit*, *bi*+1,*<sup>t</sup>* and *bi*+2,*<sup>t</sup>* are binary variables. As a result, *bi*+1,*<sup>t</sup>* or *bi*+2,*<sup>t</sup>* must equal 1 if *bit* = 1. This means that if goal 1 has been achieved, then either goal 2 or goal 3 has also been achieved.

#### *2.7. The Solution Procedure of VSPLCC Procurement Problem*

In order to solve the VSPLCC procurement problem, the following procedure is then proposed.



Step 12: Assume that the four stages of the PLC budget matrix are given as follows:



**Figure 2.** Using AHP and supertransitive approximation with a WGM algorithm with the MOLP and MCGP approach models to solve VSPLCC problems.

#### **3. Numerical Example**

As global warming intensifies, carbon dioxide emissions is an important issue in the warming caused by greenhouse gases. Reducing the greenhouse effect and protecting the Earth's environment are important goals associated with the use of white light-emitting diodes (LED) since they consume substantially less electrical power than other light sources. White-light LED power can reduce the amount of crude oil used in power plants and substantially reduce the generation of CO2 emissions, which helps to significantly reduce contributions to the greenhouse effect. Thus, according to the estimate from the optoelectronics industry development association (OIDA), using white LED lighting technology could reduce emissions worldwide by 2.5 billion tons of CO2 annually.

We used the VSPLCC procurement model to solve a real case in the distribution department for the Everlight Company (the leading LED manufacturer in Taiwan), which is part of a multi-national group in the LED research and development (R and D) sector. External purchases account for more than 75% of the total annual costs, and the firm works on a make-to-order basis. The company's management aimed to improve the efficiency of the purchasing process and reconsider the company's sourcing strategies. A manager felt that the company must evaluate and certify the company's vendors to ensure reductions in product inventory and time to market. The company sought to develop longer-term, trust-based relationships with a smaller group of vendors, and the company manager appointed a team to recommend three or four suitable vendors. This team consisted of several managers from various departments, including purchasing, marketing, quality control, production, engineering and R& D. The members of the team organized several meetings to create profiles for the competing vendors and constructed an initial set of three vendors for evaluation purposes. A VSPLCC procurement model was then developed to select the appropriate vendors and to determine their quota allocations in uncertain environments.

The team considered some objective functions and constraints as follows: Minimizing the net cost, minimizing the net rejections, minimizing the net late deliveries, minimizing the PLC cost, vendor capacity limitations, vendor budget allocations. The other considerations were: Price quoted (*Pi* in \$), the percentage of rejections (*Ri*), the percentage of late deliveries (*Li*), the PLCC (*Ci*), the PLC of the vendors' capacities (*Ui*), the vendors' quota flexibility (*Fi*, on a scale from 0 to1), the vendors' ratings (*Ri*, on a scale from 0 to 1), and the budget allocations for the vendors (*Bi*) were also considered.

The least amount of flexibility in the vendors' quotas is calculated as *Q* = *F* × *D*, and the smallest total purchase value is calculated as *P* = *R* × *D*. If the overall flexibility (*F*) is 0.03 on a scale of 0–1; if the overall vendor rating (*R*) is 0.92 on a scale of 0–1; and if the aggregate demand (*D*) is 20,000; then the least amount of flexibility in the vendors' quotas (*F*) and the smallest total purchase value of the supplied items (*P*) are 600 and 18,400, respectively. The three vendor profiles are shown in Table 2.


**Table 2.** Vendor source data for the problem.

In this case, the linear membership function is used to fuzzify the right-hand side of the constraints in the VSPLCC problem. The values of the uncertainty levels for all of the fuzzy parameters were taken as 10% of the corresponding values of the deterministic model. The datasets for the values at the lowest and highest aspiration levels of the membership functions are given in Table 3.

**Table 3.** Limiting values in the membership function for net cost, rejections, late deliveries, PLC cost, vendor capacities and budget information. (Data for all four stages: Introduction, growth, maturity, decline).


#### *3.1. Application of the WA Approach to the Numerical Example*

We obtained the solution using the WA approach of Tiwari et al. (1987), and in the next section we show the procedure by using the WGM AHP to construct a WGM supertransitive approximation to obtain the binary comparison matrixes.

Using the WGM AHP with WGM Supertransitive Approximation to Solve the VSPLCC Procurement Problem

Before determining the solution, we determined the weights of the AHP with the geometric mean process (see Chakraborty et al. 2005 [32]). Evaluating and selecting vendors is a typical MCDM problem involving multiple criteria that can be formulated by both qualitative and quantitative [38]. The VS problem involves tangible and intangible criteria, which may vary depending on the type of product being considered and may include many judgmental factors [24,39,40].

These criteria are shown in Figure 3. The VSPLCC procurement problem addresses how optimally performing vendors can be selected given the desired criteria. The AHP is one of the most widely used MCDM methods; it can be used to handle multiple criteria. The criteria for the VS problem are shown in Table 4. Based on the ratings obtained using the questionnaire, the average matrix is shown in Table 5. The maximum value of the eigenvector for the above matrix λmax is 10.77 [32]. The consistency index C.I. is given by (λmax − *n*)/(*n* − 1) = 0.09. The random index for the matrix of order 10 [41,42]. R.I. is 1.49. The consistency ratio C.R. is given by C.I./R.I. = 0.06, which is not greater than 0.1 (<0.1 acceptable).


**Figure 3.** Criteria for the VSPLCC problem.


**Table 4.** VSPLCC criteria and abbreviations (adopted and modified from Kumar et al., 2009).


**Table 5.** The geometric mean matrix for the criteria of the VSPLCC problems.

The AHP process with a geometric mean was applied to this comparison matrix, and the following weights were obtained [33]: *w*<sup>1</sup> = 0.2958, *w*<sup>2</sup> = 0.0579, *w*<sup>3</sup> = 0.0863, *w*<sup>4</sup> = 0.0365, *w*<sup>5</sup> = 0.1291, *w*<sup>6</sup> = 0.1254, *w*<sup>7</sup> = 0.0392, *w*<sup>8</sup> = 0.0199, *w*<sup>9</sup> = 0.0151, and *w*<sup>10</sup> = 0.1949 (see Section 4.1: Using the AHP process with a geometric mean).

The supertransitive approximation method is only used with the WA approach with a geometric mean to matrix *A*. Supertransitive approximation matrix *A* is constructed using the following algorithm described in Section 3.2: Solution to the VSPLCC procurement problem via the WA approach. The ten supplementary matrices corresponding to *A* are:


The supertransitive approximation method was applied to this comparison matrix, and the ollowing weights were obtained: *w*<sup>1</sup> = 0.3020, *w*<sup>2</sup> = 0.0611, *w*<sup>3</sup> = 0.0810, *w*<sup>4</sup> = 0.0272, *w*<sup>5</sup> = 0.1226, *w*<sup>6</sup> = 0.1294, *w*<sup>7</sup> = 0.0376, *w*<sup>8</sup> = 0.01936, *w*<sup>9</sup> = 0.0142, and *w*<sup>10</sup> = 0.2057 and its corresponding eigenvalue λmax is 9.94 [33]. Table 6 shows the AHP method weight with geometric mean and the supertransitive approximation with the geometric mean. For this VSPLCC procurement problem, we obtained the optimal quota allocations (i.e., the purchasing order), vendor product capacity limitations, and the budget constraints of the different vendors by using the WA approach model (i.e., Model 1) in accordance with Equations (11)–(17).

**Table 6.** AHP method weight and supertransitive approximation with geometric mean.


#### *3.2. Using Lin's WMM Approach to Solve the Numerical Example*

For this VSPLCC procurement problem illustrative example, we obtained the optimal quota allocations (i.e., the purchasing order) subject to vendor product capacity limitations and budget constraints among the different vendors with Lin's WMM [36].

#### 3.2.1. Using a MCGP AFM (Model 3: Case I) to Solve the Numerical Example

For this VSPLCC procurement problem, we obtained the optimal quota allocations (i.e., the purchasing order), supplier product capacity limitations and budget constraints among the different vendors by using the MCGP method and a no-PW approach (according to Equations (25)–(28)). This VSPLCC problem was then formulated as follows (using the first stage of the PLC for Model 1):

$$\begin{aligned} \text{Max} &= 0.2958\lambda\_1 + 0.0579\lambda\_2 + 0.0863\lambda\_3 + 0.0365\lambda\_4 + 0.1291\lambda\_5 + 01254\lambda\_6 + 0.0392\lambda\_7 \\ &+ 0.0199\lambda\_8 + 0.0151\lambda\_9 + 0.1949\lambda\_{10} \end{aligned}$$

Main Goals:

$$(\text{G}\_{11})\;3\mathbf{x}\_{11} + 2\mathbf{x}\_{21} + 6\mathbf{x}\_{31} = 57,000 \; (\text{G}11\_,\text{MIN}.) \; \text{or} \; 71,833 \; (\text{G}11\_,\text{MAX}.)\_{4}$$

$$(\text{G}\_{21})\ 0.05\text{x}\_{11} + 0.03\text{x}\_{21} + 0.01\text{x}\_{31} = \ 413\ (\text{G}\_{21}\ \text{MIN}\cdot)\ \text{or}\ 521\ (\text{G}\_{21}\ \text{MX}\cdot)\ \text{or}$$

(*G*31) 0.04*x*<sup>11</sup> + 0.02*x*<sup>21</sup> + 0.08*x*<sup>31</sup> = 604 (*G*31, MIN.) or 816 (*G*31, MAX.),

$$(\text{G4})\ 1.92 \text{x}\_{11} + 1.04 \text{x}\_{21} + 3.94 \text{x}\_{31} = 10,000 \text{ (G4}\_{\text{41\\_MIN}}) \text{ or } 90,000 \text{ (G4}\_{\text{41\\_MAX}}).$$

Capacity Constraints Goals:

(*G*51) *x*<sup>11</sup> = 5000 (*G*51, MIN.) or 5500 (*G*51, MAX.) (*X*11, Vendor 1 s product capacity),

$$(\text{G}\_{61}) \,\,\text{x}\_{21} = 15,000 \, (\text{G}\_{61\prime\text{ MIN.}}) \,\text{or} \,\, 165,000 \, (\text{G}\_{61\prime\text{ MAX.}}) \,\, (\text{X}\_{21}, \text{ Vendor } \,\text{2's product capacity}),$$

(*G*71) *x*<sup>31</sup> = 6000 (*G*71, MIN.) or 165, 000 (*G*71, MAX.) (*X*31, Vendor 3's product capacity),

*x*<sup>11</sup> + *x*<sup>21</sup> + *x*<sup>31</sup> = 20, 000 (Total demand constraint).

Budget Constraints Goals:

(*G*81) 3*x*<sup>11</sup> = 25, 000 (*G*81, MIN.) or 27, 500 (*G*81, MAX.) (*X*11, Vendor 1's budget constraint),

(*G*91) 2*x*<sup>21</sup> = 100, 000 (*G*91, MIN.) or 110, 000 (*G*91, MAX.) (*X*21, Vendor 2's budget constraint),

(*G*101) 6*x*<sup>31</sup> = 35, 000 (*G*101, MIN.) or 110, 000 (*G*101, MAX.) (*X*31, Vendor 3's budget constraint).

3.2.2. Using a MCGP AFM (Model 4: Case II) to Solve the Numerical Example

The subjectivity inherent to the determination of both the desired level of attainment for each goal and the penalty weights assigned to deviations from the goal may present a problem [19,36]. Suppose that the purchasing company's manager sets a penalty weight of five for the vendor missing the net cost goal, four for missing the rejection goal, three for missing the late deliveries goal, and two for exceeding the PLC cost goal [28]. For this VSPLCC procurement problem, we obtained the optimal quota allocations (i.e., the purchasing order), supplier product capacity limitations and budget constraints among the different vendors using the MCGP method and a PW approach in accordance with Equations (29)–(32). 4. Solution Results of the Two Types of MOLP and MCGP Model Approaches.

#### **4. Solution Results of the Two Types of MOLP and MCGP Model Approaches**

After using the Lingo 11.0 software package solving the VSPLCC procurement problem, we found that Lin's (2004) [36] WMM approach and the MCGP method with the geometric mean and the PW approach have the same results in the first stage of the PLCs. With regards to the MCGP approaches with the geometric mean (no-PW restrictions), *x*<sup>11</sup> = 5000 (due to the absence of PW constraints), *b*<sup>11</sup> = 1 and *b*<sup>51</sup> = 1. The forced bound order quantity of vendor 1 was 5000 (i.e., for model 3 at the first stage (Introduction), *x*<sup>11</sup> = 5000) (see Tables 7–11). With regards to the other approaches (i.e., the MCGP approach with the geometric mean and the PW approach), *b*<sup>12</sup> = 1 and *b*<sup>62</sup> = 1. The forced bound order quantity of vendor 2 was greater than 15,000 (i.e., for model 4 at the second period (Growth), *x*<sup>22</sup> = 15,750). To guarantee the net cost goal, the rejection goal, or the late delivery goal, zero value should be achieved (e.g., if *b*<sup>12</sup> = 1 and *b*<sup>62</sup> = 1, then forces *bit* equal to zero used to adjust the purchasing quantity) (see Tables 8–12). We found the MCGP model to be stable with regard to the PLCC in all of the stages (see Tables 13–15).

**Table 7.** PLCC model during the first stage (Introduction).



**Table 8.** PLCC during the second stage (Growth).

**Table 9.** PLCC during the third stage (Maturity).


**Table 10.** PLCC during the fourth stage (Decline).



**Table 11.** PLCC during the first period (Introduction).

**Table 12.** PLCC during the second period (Growth).


**Table 13.** All of the models for order quantity of vendor *x*<sup>1</sup> in the fourth PLC stages.


**Table 14.** All of the models for order quantity of vendor *x*<sup>2</sup> in the four PLC stages.


**Table 15.** All of the models for order quantity of vendor *x*<sup>3</sup> in the four PLC stages.


#### *4.1. Analysis of Results*

Based on the solutions to the two type's goal-programming models, after using the Lingo 11.0 software package we summarized the results of the VSPLCC procurement problem in Tables 7–15. From *Z*4*<sup>t</sup>* (i.e., the PLC cost goal) of Figure 4, we can see that the maturity stage has the lowest PLC cost; in contrast, the growth and decline stages have similar costs, and the introduction stage has a high PLC cost. We found that the MCGP model demonstrated more stable control of the PLC cost over all of the stages.

**Figure 4.** *Z*4*t*: The results of the four VSPLCC models' solutions to the PLCC goal.

#### **5. Conclusions and Managerial Implications**

#### *5.1. Conclusions*

The results obtained using the MOLP and MCGP VSPLCC procurement problem models for determining vendor quotas in SCM if the capacity and budget constraints of each vendor are not known with certainty. The effectiveness of the VSPLCC procurement model was demonstrated with a real-world problem adopted from a leading LED company in Taiwan. Managers in high-tech companies can easily apply our VSPLCC procurement model to select their vendors in a fuzzy environment using the MOLP and MCGP approaches. We found in our study results that the weighted geometric mean with AHP and PW methods has good control conditions for constructing an MCGP AFM model (model 4) within four PLC stages.

#### *5.2. Managerial Implications*

Some managerial implications are found as follows: (i) doing so is practical because the no-PW and PW MCGP AFM model approaches (MCGP AFM models 3 and 4) do not require precise knowledge of all of the parameters, and they make the application of a fuzzy methodology more understandable [27,28,35]; (ii) the No-PW and PW MCGP models are demonstrated to be more stable over all of the PLC stages; (iii) company managers can easily use MOLP and MCGP model approaches to solve VSPLCC procurement problems; and (iv) this VSPLCC procurement model allows DMs to solve VSPLCC problems when considering their preferences.

#### *5.3. Limitations*

We integrate VS and PLCC procurement planning into a VSPLCC procurement model for enterprise to reduce their purchasing costs. In order to eliminate the MOLP and MCGP model approaches drawbacks and achieve the accurate results, we are comparing two GP with AHP supertransitive approximation with a WGM technique to verify the result of the VSPLCC procurement model. Otherwise, if DMs use a new AHP method and conjunction GP approach, it can be a different result in uncertain conditions.

#### *5.4. Future Directions*

In addition, integrating other mathematical models, such as the Pareto concept with AHP and ANP [37,43] with DEAHP [44], or AHP-QFD [45] with the MOGP [46] and MCGP [27,28,47] models to solve the VSPLCC procurement problems in a multi-item—multi-vendor environment that could be performed in conjunction with the various models [48].

**Author Contributions:** Conceptualization, Y.-T.P. and C.-W.S.; formal analysis, C.-S.T. and Y.-T.P.; writing—original draft preparation, C.-S.T. and Y.-T.P.; writing—review and editing, C.-S.T. and Y.-T.P.; supervision, C.-W.S.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **References**


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