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

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## Abstract

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## 1. Introduction

**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.

## 2. The VSPLCC Procurement Approaches

#### 2.1. Linear Programming Technique

**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.

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

#### VSPLCC Procurement Problem

- (i)
- One item is purchased from each vendor.
- (ii)
- Quantity discounts are not considered.
- (iii)
- No shortage of the item is allowed for any of the vendors.
- (iv)
- The lead time and demand for the item are constant and known with certainty.

#### 2.3. VSPLCC Procurement Model

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

^{1}, A

^{2}, …, A

^{n}. The jth row of matrix A

^{j}is the same as the jth row of the initial matrix A, where the supplementary matrix ${{\displaystyle ({{\displaystyle A}}^{j})}}^{{{\displaystyle T}}^{}*}=\left[{{\displaystyle a}}_{1}^{j},{{\displaystyle a}}_{2}^{j},\dots ,{{\displaystyle a}}_{n}^{j}\right]$ and each row of the matrix A

^{j}is computed as follows (${{\displaystyle T}}^{*}$: Transpose): ${{\displaystyle a}}_{j}^{j}={{\displaystyle a}}_{j},$ ${{\displaystyle a}}_{1}^{j}={{\displaystyle ({{\displaystyle a}}_{j1})}}^{-1}{{\displaystyle a}}_{j}^{j},$ ${{\displaystyle a}}_{2}^{j}={{\displaystyle ({{\displaystyle a}}_{j2})}}^{-1}{{\displaystyle a}}_{j}^{j},\dots ,{{\displaystyle a}}_{n}^{j}={{\displaystyle ({{\displaystyle a}}_{jn})}}^{-1}{{\displaystyle a}}_{j}^{.j}.$ Next, we construct the supertransitive approximation, ${A}^{s}=\Vert {{\displaystyle a}}_{ij}^{s}\Vert ,$ i, j = 1, 2, …, n, by taking the geometric mean of the corresponding elements from the supplementary matrices A

^{1}, A

^{2},…, A

^{n}. More formally, ${{\displaystyle a}}_{ij}^{s}={{\displaystyle ({{\displaystyle a}}_{ij}^{1}\times {{\displaystyle a}}_{ij}^{2}\times ......\times {{\displaystyle a}}_{ij}^{n})}}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}$. Then we obtain the largest value of A

^{s}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, ${{\displaystyle \alpha}}_{jt}$ is the weighting coefficient that shows the relative importance at the four stages of the PLC.

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

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

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

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

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

_{it}$\in $ {0, 1} is a binary variable attached to $\left|{{\displaystyle f}}_{it}(X)-{{\displaystyle y}}_{it}\right|,$ which can be either achieved or released in Equation (25). In terms of real conditions, b

_{it}is subject to some appropriate constraints according to real needs.

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

_{it}, ${{\displaystyle b}}_{i+1,t}$ and ${{\displaystyle b}}_{i+2,t}$ are binary variables. As a result, ${{\displaystyle b}}_{i+1,t}$ or ${{\displaystyle b}}_{i+2,t}$ must equal 1 if b

_{it}= 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

- Step 1:
- Construct the model for VSPLCC procurement.
- Step 2:
- Step 3:
- Calculate the criteria of weighted geometric mean for solving VSPLCC procurement problem.
- Step 4:
- Repeat the process individually for each of the remaining objectives. It determines the lower and upper bounds of the optimal values for each objective corresponding to the set of constraints.
- Step 5:
- Use these limited values as the lower and upper bounds for the crisp formulation of the VSPLCC procurement problem.
- Step 6:
- Based on Steps 4–5 we can find the lower and upper bounds corresponding to the set of solutions for each objective. Let ${{\displaystyle Z}}_{jt}^{-}$ and ${{\displaystyle Z}}_{jt}^{+}$ denote the lower and upper bound, respectively, for the jt th objective (Z
_{jt}) (Amid, Ghodsypour; O’Brien, 2011) [35]. - Step 7:
- Using the weighted geometric mean with a supertransitive approximation to solve Model 1 by following Equations (11)–(17).
- Step 8:
- Formulate and solve the equivalent crisp model of the weighted geometric mean max-min for the VSPLCC procurement problem to solve Model 2 by following Equations (18)–(24).
- Step 9:
- Use the weighted geometric mean and the no-PW (penalty weights) formulation of the fuzzy optimization problem to solve Model 3 by following Equations (25)–(28).
- Step 10:
- Formulate Model 4 using the weighted geometric mean and the PW formulation of the fuzzy optimization problem by following Equations (29)–(32). Assume that the purchasing company manager sets a PW of five for a 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 (Chang, 2008) [28].
- Step 11:
- The four stages of the PLC cost matrix are given as follows (Demirtas; Ustun, 2009) [37]:$$\left(\begin{array}{cccc}1.92& 1.52& 1.23& 1.82\\ 1.04& 0.92& 0.86& 1.00\\ 3.94& 3.52& 3.05& 3.56\end{array}\right)$$
- Step 12:
- Assume that the four stages of the PLC budget matrix are given as follows:$$\left(\begin{array}{cccc}\mathrm{25,000}& \mathrm{26,500}& \mathrm{27,400}& \mathrm{26,000}\\ \mathrm{100,000}& \mathrm{120,000}& \mathrm{125,000}& \mathrm{110,000}\\ \mathrm{35,000}& \mathrm{36,000}& \mathrm{37,500}& \mathrm{35,200}\end{array}\right)$$
- Step 13:
- Solve the MOLP and MCGP models for the fuzzy optimization problem.
- Step 14:
- Analyze the PLCCs and capacity limitations for the four stages. The procedure of the VSPLCC procurement problem-solving model is illustrated through a numerical example. Figure 2 shows the use of the AHP with a WGM and supertransitive approximation with a WGM technique to the MOLP and MCGP approach models to solve VSPLCC procurement problems.

## 3. Numerical Example

_{2}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 CO

_{2}annually.

_{i}in $), the percentage of rejections (R

_{i}), the percentage of late deliveries (L

_{i}), the PLCC (C

_{i}), the PLC of the vendors’ capacities (Ui), the vendors’ quota flexibility (F

_{i}, on a scale from 0 to1), the vendors’ ratings (R

_{i}, on a scale from 0 to 1), and the budget allocations for the vendors (B

_{i}) were also considered.

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

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

_{1}= 0.2958, w

_{2}= 0.0579, w

_{3}= 0.0863, w

_{4}= 0.0365, w

_{5}= 0.1291, w

_{6}= 0.1254, w

_{7}= 0.0392, w

_{8}= 0.0199, w

_{9}= 0.0151, and w

_{10}= 0.1949 (see Section 4.1: Using the AHP process with a geometric mean).

_{1}= 0.3020, w

_{2}= 0.0611, w

_{3}= 0.0810, w

_{4}= 0.0272, w

_{5}= 0.1226, w

_{6}= 0.1294, w

_{7}= 0.0376, w

_{8}= 0.01936, w

_{9}= 0.0142, and w

_{10}= 0.2057 and its corresponding eigenvalue ${{\displaystyle \lambda}}_{\mathrm{max}}$ 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).

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

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

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

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

_{11}= 5000 (due to the absence of PW constraints), b

_{11}= 1 and b

_{51}= 1. The forced bound order quantity of vendor 1 was 5000 (i.e., for model 3 at the first stage (Introduction), x

_{11}= 5000) (see Table 7, Table 8, Table 9, Table 10 and Table 11). With regards to the other approaches (i.e., the MCGP approach with the geometric mean and the PW approach), b

_{12}= 1 and b

_{62}= 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

_{22}= 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

_{12}= 1 and b

_{62}= 1, then forces b

_{it}equal to zero used to adjust the purchasing quantity) (see Table 8, Table 9, Table 10, Table 11 and Table 12). We found the MCGP model to be stable with regard to the PLCC in all of the stages (see Table 13, Table 14 and Table 15).

#### 4.1. Analysis of Results

_{4t}(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.

## 5. Conclusions and Managerial Implications

#### 5.1. Conclusions

#### 5.2. Managerial Implications

#### 5.3. Limitations

#### 5.4. Future Directions

## Author Contributions

## Funding

## Conflicts of Interest

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

i | Index for vendor, for all i = 1, 2, ..., n |

j | Index for objectives, for all j = 1, 2, ..., J |

k | Index for constraints, for all k = 1, 2, ..., K |

t | index objectives and constraints for all at four PLC stages t = 1, 2, 3, 4 |

Decision Variable | |

X_{it} | Ordered quantity given to the vendor i, t = 1, 2, 3, 4 index for all at four PLC stages |

Parameters | |

${{\displaystyle \tilde{D}}}_{t}$ | Aggregate demand for the item over a fixed planning period, t = 1, 2, 3, 4 index for all at four PLC stages |

n | Number of vendors competing for selection |

p_{it} | Price of a unit item of ordered quantity x_{i} for vendor i, t = 1, 2, 3, 4 index for all at four PLC stages |

Q_{it} | Percentage of the rejected units delivered for vendor i, t = 1, 2, 3, 4 index for all at four PLC stages |

L_{it} | Percentage of the units delivered late for vendor i, t = 1, 2, 3, 4 index for all at four PLC stages |

C_{it} | Product life cycle cost of ordered for vendor i, t = 1, 2, 3, 4 index for all at four PLC stages |

${{\displaystyle \tilde{U}}}_{it}$ | Upper limit of the quantity available for vendor i, t = 1, 2, 3, 4 index for all at four PLC stages |

r_{it} | Vendor rating value for vendor i, t = 1, 2, 3, 4 index for all at four PLC stages |

P_{it} | The total purchasing value that a vendor can have, t = 1, 2, 3, 4 index for all at four PLC stages |

f _{it} | Vendor quota flexibility for vendor i, t = 1, 2, 3, 4 index for all at four PLC stages |

F_{it} | The value of flexibility in supply quota that a vendor should have, t = 1, 2, 3, 4 index for all at four PLC stages |

B_{it} | Budget constraints allocated to each vendor, t = 1, 2, 3, 4 index for all at four PLC stages |

Vendor No. | P_{i} ($) | R_{i} (%) | L_{i} (%) | C_{i} ($) | U_{i} (Units) | r_{i} | F_{i} | B_{i} ($) |
---|---|---|---|---|---|---|---|---|

1 | 3 | 0.05 | 0.04 | 1.92 | 5000 | 0.88 | 0.02 | 25,000 |

2 | 2 | 0.03 | 0.02 | 1.04 | 15,000 | 0.91 | 0.01 | 100,000 |

3 | 6 | 0.01 | 0.08 | 3.94 | 6000 | 0.97 | 0.06 | 35,000 |

**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).

$(\mathbf{min}.)\text{}\mathit{\mu}=1$ | $(\mathbf{max}.)\text{}\mathit{\mu}=0$ | |
---|---|---|

Main Goals | ||

(G_{l}) Net cost objective | 57,000 | 71,833 |

(G_{2}) Rejection objective | 413 | 521 |

(G_{3}) Late deliveries objective | 604 | 816 |

(G_{4}) PLC cost objective | 10,000 | 90,000 |

(G_{5}) Vendor 1 | 5000 | 5500 |

(G_{6}) Vendor 2 | 15,000 | 16,500 |

(G_{7}) Vendor 3 | 6000 | 6600 |

Budget constraints |