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Abstract

Due to the considerations of the carbon footprint of the product, low-carbon product family design has attracted more and more attention. The existing low-carbon product family design methods do not adequately take into account the component reliability and warranty period. At present, providing warranty for key components of the product is an important marketing tool, and it is widely adopted by manufacturers. In fact, when component reliability and warranty period are considered in a low-carbon product family design, they affect not only the profit of the product family, but also the GHG emission. To this end, the article proposes an optimization model that optimizes product family design, supplier selection and component warranty period in parallel. The objective of the model is to maximize the profit of the product family and minimize GHG emission while satisfying various constraints. The developed model is solved using a genetic algorithm. The method’s effectiveness is verified through a case study, and a sensitivity analysis is performed. The method can assist manufacturers in simultaneously determining the product family configuration, supplier selection and the component warranty period based on their goal preferences.

Keywords

Low-carbon design GHG emission product family design product configuration component warranty joint decision-making

Introduction

Over the last half century, the impacts of global warming have been felt most strongly in the world. The Greenhouse gas (GHG) emission (particularly carbon dioxide) is a significant cause of global climate change.¹ Related studies show that around 84% of energy-related CO₂ emission and 90% of the energy consumption can be attributed to various products.² Worldwide protocols (e.g. Kyoto Protocol, Copenhagen Protocol) have recommended that industries take action to reduce GHG emission from a product.

The GHG emission of a product is determined to a large extent during the design stage of the product. Low-carbon design is a design process that considers GHG emission in the life cycle of a product. The result of low-carbon design is the creation of products that reduce GHG emission. In recent years, the low-carbon design has been receiving increasing attention from academia and industry, especially low-carbon product family design. For instance, Wang et al.³ proposed a decision model for optimizing product family design considering the objective of GHG emission. Xiao et al.⁴ presented an approach for product family configuration that considers manufacturing methods. Since the choice of supplier affects both the production costs and the GHG emission of the product, Wang et al.⁵ investigated a parallel optimization approach to low-carbon product family design and supplier selection. Although some studies have focused on the low-carbon product family design, the component warranty and the component reliability are not fully considered. At present, product warranty is an important part of new product marketing and sales due to the fact that it can attract more consumers. Component reliability has a severe impact on the warranty servicing cost as well as the production cost. In addition to impact cost, component reliability is also related to GHG emission of production. Therefore, it is meaningful that the component warranty and the component reliability are included in low-carbon product family design. To fill the gap, the paper proposes a method for parallel optimization of product family design, supplier selection and component warranty period. In the method, the customer demand model that considers component warranty period is developed, and the component reliability is also considered in supplier selection. Moreover, the genetic algorithm was developed for solving the optimization problem. This method can assist manufacturers in simultaneously determining the product family configuration, supplier selection and the component warranty period based on their goal preferences.

To enhance the understanding of the article, the remainder of the paper is organized as follows: Section ‘Related works’ reviews existing relevant research, including low-carbon product design and product warranty. In Section ‘Problem definition’ the problem is described. An optimization model is established in Section ‘Development of optimization model’. Section ‘Optimization model solving method’ develops the corresponding algorithms for solving the model. An application case is used in Section ‘Case study’ to demonstrate the validity of the proposed model. Finally, Section ‘Conclusions’ concludes.

Related works

Low-carbon product design

In recent years, many methods have been put forward for low-carbon design. Song and Lee⁶ developed a design assistance system for designing low-carbon products using a bill of materials. Qi and Wu⁷ incorporated low-carbon technologies into the modular design of products. Su et al.⁸ presented a method for assessing GHG emission and costs in product design stage. For low-carbon product design, Zhang et al.⁹ proposed a method for identifying high carbon emission components. Kuo et al.¹⁰ investigated an approach for optimizing low-carbon product design with consideration of cost, supplier capability and transportation. Xu et al.¹¹ established a multi-objective optimization model for low-carbon product design to address the conflict between business, consumer and government. He et al.¹² studied on low-carbon product design for product life cycle. Chiang and Che¹³ gave a method to develop electronic products considering GHG emission. He et al.¹⁴ proposed a conceptual framework for low-carbon product design. Regarding material, structural layout and structural parameters, Zhang et al.¹⁵ presented a decision model to optimize material selection, structural layout and structural parameters for designing low-carbon component. Ai et al.¹⁶ proposed an low-carbon product conceptual design from the perspectives of technical systems and human use. All of the above methods are specific to a single product. For the past few years, the low-carbon product family design have started to be studied. For example, Tang et al.¹⁷ proposed an optimization model for low-carbon product configuration. Kim and Moon¹⁸ reported a method to design sustainable platforms based on sustainability value, risk value and commonality. Wang et al.¹⁹ presented a model for optimizing low-carbon product family design considering remanufactured products.

In summary, although a number of studies have been presented for low-carbon product family design, component reliability has not been given sufficient attention. The component reliability affects not only the warranty cost of a product family but also the GHG emission of the product family. Furthermore, due to differences in reliability of components from different suppliers, it is meaningful that supplier selection for low carbon-product family design should consider component reliability.

Product warranty

Product warranty is considered an important competitive strategy, which has been the subject of research since the 20th century.²⁰ Glickman and Berger²¹ first proposed a demand model for price and warranty length based on a general Cobb-Douglass type function to optimize price and warranty length, which was widely used in many studies. The ‘g.b .’ function was applied to profit maximization models and analysed by classical optimization methods. Menezes and Currim²² proposed a function containing a large number of independent variables related to the product, the firm, related products and rival firms to calculate the optimal warranty length. Huang et al.²³ developed the G.B. model to optimize the price and warranty length for goods. Manna²⁴ evaluated the optimization of price and warranty length for a two-dimensional warranty policy based on the g.b. function. Ketzenberg and Zuidwijk²⁵ investigated the optimal pricing, ordering and return policies for consumer products. Liu et al.²⁶ conducted a study on customizing the warranty duration of multiple products. The problem was solved by applying Markov chains in a cost-minimization model. Wu et al.²⁷ presented an optimization method to determine the price, warranty length and production rate of a product. Liu et al.²⁸ evaluated the cost of warranty length for multi-component products with renewal policies where the components were not independent. A stochastic process approach was applied to obtain mathematical equations for the mean and variance of the number of product warranty services under different configurations. Liu et al.²⁹ identified conditions for the optimal pricing and production strategy of a monopoly manufacturer under a free replacement warranty that is not renewed for the duration of a two-period plan.

To sum up, although there are many studies on marketing strategies and product warranty in particular, few studies have considered component warranty in low-carbon product family design. This study attempts to provide a joint optimization model for low-carbon product family design that considers component warranty period and supplier selection.

Problem definition

The optimization problem in this study can be described as follows: a product is developed into a modular architecture with multiple functional modules. Each functional module has a number of candidate module instances. To meet the needs of different customer groups, the company plans to launch a product family containing several product variants. By combining different module instances, various product variants can be configured. In this study, it is assumed that module instances are supplied from suppliers and the products are assembled by the leading manufacturer. Typically, several types of module instances are available from each supplier, and a single module instance can be supplied by different suppliers. Different module instances have different performance, reliability, production GHG emission, etc. Different suppliers offer various discount schemes. To encourage business, the leading manufacturer offers warranty policies on some key components when selling their products. The research question in this study is to determine module instances configuration of the product family, supplier selection and warranty length of the component to maximize the profitability and minimize the expected GHG emission. The decision variables in this research are as follows:

$\begin{matrix} R_{ij}^{(r)} = \\ {\begin{matrix} 1 & if j th instance of i th module (M_{ij}) selected for \\ the r th product variant \\ (i = 1, 2, \dots, I; j = 1, 2, . . ., J; r = 1, 2, \dots, R) \\ 0 & otherwise \end{matrix} \end{matrix}$ (1)

$\begin{matrix} x_{ij}^{(r) z} = \\ {\begin{matrix} 1 & if j th instance of i th module (M_{i, j}) for the \\ r th product variant is provided by z th supplier \\ (k = 1, 2, . . ., K; i = 1, 2, \dots, I; j = 1, 2 \dots, J; \\ r = 1, 2, \dots, R; z = 1, 2 \dots, Z) \\ 0 & otherwise \end{matrix} \end{matrix}$ (2)

$\begin{matrix} p_{v}^{r} = {\begin{matrix} 1 & if v th price is determined to the \\ r th product variant \\ (v = 1, 2, \dots, V; r = 1, 2, \dots, R) \\ 0 & otherwise \end{matrix} \end{matrix}$ (3)

$\begin{matrix} W_{ij}^{s (r) q} = \\ {\begin{matrix} 1 & if s th warrant length of j th instance of i th \\ module is selected for the r th product variant \\ in k th market (s = 1, 2, . . . ., S; i = 1, 2, \dots, I; \\ j = 1, 2, . . ., J; r = 1, 2, \dots, R; k = 1, 2, \dots, K) \\ 0 & otherwise \end{matrix} \end{matrix}$ (4)

Development of optimization model

Customer preference modelling considering the warranty length of component

The purpose of product family design is to meet the customer needs of different market segments. The product market needs to be first divided into multiple segments. It is often assumed that customers in the same market segment have similar buying preferences. The utility function has been shown to be an effective measure of customer preferences in the product design phase. Based on the utility function, a decision model is developed to evaluate the customer preference. According to the part-value model,³⁰ the utility of a product is equal to the sum of the utilities of all components. In addition to the utility of the components influencing the customer’s purchase decision, the length of the product’s warranty will undoubtedly affect the customer’s purchase decision. Therefore, the utility of the warranty length of the component needs to be considered in the customer preference model. With consideration of the warranty length of the component, the utility of the rth product variant in the qth market segment ( $U_{pro}^{r (q)}$ ) is calculated as follows:

$U_{pro}^{r (q)} = \sum_{r = 1}^{R} \sum_{s = 1}^{S} \sum_{q = 1}^{Q} \sum_{i = 1}^{I} \sum_{j = 1}^{J} (μ_{ij}^{(q)} + {wl}_{ij}^{s (q)} v_{ij}^{s (r)}) + η_{r}$ (5)

where $μ_{ij}^{(q)}$ is the utility of module instance M_ij in the qth segment. $η_{r}$ is constant. $μ_{ij}^{(q)}$ is given by conjoint analysis. ${wl}_{ij}^{s (q)}$ is the utility of sth warrant length of M_ij in the qth market.

When the product price is considered, the surplus utility for a product is as follows:

$S^{r (q)} = U_{pro}^{r (q)} - p_{r}$ (6)

where $S^{r (q)}$ represents the surplus utility of rth product variant in the qth market, $p_{r}$ is the selling price for rth product variant.

Product sales

For product, whether a consumer buys the product or not is largely affected by the residual utility of the product and the residual utility of competitors’ products. In this context, probabilistic choice rules are widely used to describe customers’ purchasing decision behaviour. In a probabilistic choice rule, utility is a random variable, and the customer’s choice to purchase a product follows a random utility maximization criterion. The multinomial logit choice (MNL) rule as a typical probabilistic choice rule is used in this paper.³¹ According to the MNL rule, the probability of a product selected in a market is expressed as:

$P_{r}^{(q)} = \frac{e^{μ s^{r (q)}}}{\sum_{r = 1}^{R} e^{μ s^{r (q)}} + \sum_{r = 1}^{G^{e}} e^{μ s_{e}^{r (q)}} + \sum_{r = 1}^{G^{c}} e^{μ s_{c}^{r (q)}}}$ (7)

where $P_{r}^{(q)}$ is choice probability of the rth product variant chosen in the qth market, $μ$ is the scaling parameter in the MNL rule, $G^{e}$ and $G^{c}$ represent the number of product categories of competing companies and the number of product categories launched in the market by this company, respectively. $S_{e}^{r (q)}$ and $S_{c}^{r (q)}$ are the surplus utility of existing and competitive products, respectively.

The demand function of rth product variant in qth market ( $Q_{r}^{(q)}$ ) is represented as follows:

$Q_{r}^{(q)} = n_{q} P_{r}^{(q)}$ (8)

where n_q is the total number of product demand in the qth market.

When a product family has been developed, the total expected revenue T_rev is calculated as follows:

$T_{rev} = \sum_{q = 1}^{Q} \sum_{r = 1}^{R} Q_{r}^{(q)} p_{r}$ (9)

Cost calculation for product family

The cost C can be divided into production cost and expected warranty cost. C is expressed as follows:

$C = C^{\Pr o} + C^{War}$ (10)

where $C^{\Pr o}$ is the production cost, $C^{War}$ is the warranty cost.

Production cost

The production cost includes intra-firm production cost and purchasing cost, and can be presented as follows:

$C^{\Pr o} = C^{intra} + C^{pur}$ (11)

$C^{intra}$ denotes the cost incurred in the production processes of products inside the firm. It can be further divided into two parts, including fixed part ( $C^{intra (fix)}$ ) and variable part ( $C^{intra (var)}$ ). The fixed part is the cost for product development, management cost, etc. The variable part mainly refers to the assembly cost, packaging cost, etc.

The fixed part of $C^{intra}$ is related to the developed number of product variant (W_pr), and it can be described as follows:

$C^{intra (fix)} = φ W_{pr}$ (12)

where $φ$ represents the fixed part of the intra-firm production cost for developing one product variant.

$C^{intra (var)} can be expressed as :$

$C^{intra (var)} = \sum_{q = 1}^{Q} \sum_{r = 1}^{R} \sum_{i = 1}^{I} \sum_{j = 1}^{J} Q_{r}^{(q)} c_{ij}^{intra (var)} R_{ij}^{(r)}$ (13)

where $c_{ij}^{intra (var)}$ is the unit variable production cost for M_ij.

$C^{\Pr o}$ also consists of the fixed part ( $C^{pur (fix)}$ ) and variable part ( $C^{pur (var)}$ ). Fixed part is related to the number of used suppliers, and typical costs for this section include negotiation costs, contracting costs, etc. $C^{pur (fix)}$ is expressed as follows:

$C^{pur (fix)} = \sum_{t = 1}^{T} S F_{t} L_{t}$ (14)

where SF_t is the fixed cost of adopting the tth supplier, L_t is equal to 0 if the tth supplier is not adopted; otherwise 1.

$C^{pur (var)}$ indicates the purchasing cost and the transportation cost. With consideration of discounts offered by suppliers, $C^{pur (var)}$ is calculated as follows:

$\begin{matrix} C^{pur (var)} = \sum_{z = 1}^{Z} [\sum_{i = 1}^{I} \sum_{j = 1}^{J} \sum_{q = 1}^{Q} \sum_{r = 1}^{R} Q_{r}^{(q)} R_{ij}^{(r)} x_{ij}^{(r) z} f_{ij}^{z}] (1 - τ_{z}) \\ + \sum_{z = 1}^{Z} [\sum_{i = 1}^{I} \sum_{j = 1}^{J} \sum_{q = 1}^{Q} \sum_{r = 1}^{R} Q_{r}^{(q)} R_{ij}^{(r)} x_{ij}^{(r) z} w_{ij}] D^{z} C_{Tr} \end{matrix}$ (15)

where $f_{ij}^{z}$ is the price of the module instance (M_ij) provided by the zth supplier, $τ_{z}$ denotes the discount rate provided by the zth supplier, $w_{ij}$ is the weight of M_ij, D_z is the distance from the supplier z to the leading company, $C_{Tr}$ denotes the unit transportation cost.

Warranty cost of component

It is assumed that the product component under consideration is repairable and that any faults can be corrected. The repair action is considered as a minimum repair that brings the system to essentially the same state as it was before the failure occurred. The parameter $θ$ is used as the design variable to represent the reliability of a component. The smaller the value $θ$ , the higher the component reliability. The failure distribution can be characterized according to the design variable $θ$ (e.g. $θ$ is the failure rate which is exponentially distributed and the scale parameter of the Weibull distribution). $θ_{max}$ represents the upper limit of achievable reliability subject to technical and budgetary constraints, and $θ_{m in}$ represents the lower limit of reliability required for satisfactory operation of the component.

The incidence of faults can be expressed as a fault density function and a fault distribution function as follows:

$r_{ij} (t, θ_{ij}) = \frac{f (t, θ_{ij})}{1 - F (t, θ_{ij})}$ (16)

where $θ_{ij}$ is the reliability variable of the module instance $M_{ij}$ , $r_{ij} (t, θ_{ij})$ is rate of occurrence of failures for $M_{ij}$ , $f (t, θ_{ij})$ indicates the product time to failure density function, $F (t, θ_{ij})$ represents the product time to failure distribution function with no preventive maintenance.

Let $S_{ij} (0, W)$ be the expected number of failures of module instance $M_{ij}$ during the warranty period W, and it can be calculated as follows:

$S_{ij} (0, W) = \int_{0}^{W} r_{ij} (t, θ_{ij}) dt$ (17)

When a component is returned for repair in warranty period, the repair cost will incurred, including shipping costs, material costs and so on. The expected warranty cost ( $ω_{ij}^{W}$ ) for unit module instance $M_{ij}$ during the warranty period W is given by

$ω_{ij}^{W} = λ_{ij} \int_{0}^{W} r_{ij} (t, θ_{ij}) dt$ (18)

where $λ_{ij}$ is the average cost per repair for module instance $M_{ij}$ .

The total expected warranty cost for product family of production is as follows:

$war r^{tol} = \sum_{i = 1}^{I} \sum_{j = 1}^{J} \sum_{q = 1}^{Q} \sum_{r = 1}^{R} Q_{r}^{(q)} R_{ij}^{(r)} ω_{ij}^{W}$ (19)

GHG emission model of product family

In this research, the GHG emission of a product family is mainly generated from component manufacturing ( $E_{com}^{T}$ ), transportation ( $E_{tra}^{T}$ ), production ( $E_{pro}^{T}$ ), supplier selection ( $E_{sup}^{sel}$ ) and component maintenance within the warranty period ( $E_{rep}^{T}$ ), and it is expressed as:

$E^{T} = E_{com}^{T} + E_{tra}^{T} + E_{pro}^{T} + E_{sup}^{T} + E_{rep}^{T}$ (20)

$E_{com}^{T} can be calculated as follows :$

$E_{com}^{T} = \sum_{i = 1}^{I} \sum_{j = 1}^{J} \sum_{q = 1}^{Q} \sum_{r = 1}^{R} Q_{r}^{(q)} R_{ij}^{k} e_{ij}$ (21)

where $e_{ij}$ is the GHG emission for $M_{ij}$ .

$E_{tra}^{T}$ is the GHG emission generated from the transportation of component and is described as follows:

$E_{tra}^{T} = \sum_{z = 1}^{Z} \sum_{i = 1}^{I} \sum_{j = 1}^{J} \sum_{q = 1}^{Q} \sum_{r = 1}^{R} Q_{r}^{(q)} R_{ij}^{(r)} x_{ij}^{(r) z} w_{ij} D^{z} E_{Tr}$ (22)

where $E_{Tr}$ is the unit transportation GHG emission.

$E_{pro}^{T}$ is the GHG emission from the production processes of products inside the firm. It can be further divided into two parts, fixed part ( $E_{pro}^{fix}$ ) and variable part ( $E_{pro}^{var}$ ).

$E_{pro}^{fix}$ mainly includes the GHG emission from product development, management and so on. $E_{pro}^{fix}$ is related to the number of product variant (W_pr), and it is formulated as follows:

$E_{pro}^{fix} = E_{V} W_{pr}$ (23)

where $E_{V}$ is the fixed GHG emission for developing one product variant.

$E_{pro}^{var}$ mainly includes the GHG emission from product assembly, and it can be formulated as follows:

$E_{pro}^{var} = \sum_{q = 1}^{Q} \sum_{r = 1}^{R} \sum_{i = 1}^{I} \sum_{j = 1}^{J} Q_{r}^{(q)} e_{ij}^{(ass)} R_{ij}^{(r)}$ (24)

where $e_{ij}^{(ass)}$ is the GHG emission for $M_{ij}$ .

$E_{sup}^{sel}$ , including GHG emission from negotiation communication, relationship maintaining between firm and supplier and so on, represents the GHG emission of adopting supplier. It is formulated as follows:

$E_{sup}^{sel} = \sum_{z = 1}^{Z} F_{z} G_{z}$ (25)

where F_z is GHG emission of adopting the zth supplier, G_z is equal to 0 if the zth supplier is not adopted; otherwise 1.

$E_{rep}^{T}$ is the repair GHG emission during the warranty period. The expected GHG emission ( $E_{ij}^{W}$ ) for unit module instance $M_{ij}$ during the warranty period W is given by

$E_{ij}^{W} = E_{ij} \int_{0}^{W} r_{ij} (t, θ_{ij}) dt$ (26)

where $E_{ij}$ is the average GHG emission per repair for module instance $M_{ij}$ .

The total expected GHG emission in warranty period W for product family of production is as follows:

$E_{rep}^{T} = \sum_{z = 1}^{Z} [\sum_{i = 1}^{I} \sum_{j = 1}^{J} \sum_{q = 1}^{Q} \sum_{r = 1}^{R} Q_{r}^{(q)} R_{ij}^{(r)} E_{ij}^{W}]$ (27)

Objective function

Maximizing total profit is the first objective of optimization in this research. The total profit equals the expected revenue (T_rev) minus total cost (C), and it can be calculated as follows:

$\begin{matrix} Max Δ = \sum_{q = 1}^{Q} \sum_{r = 1}^{R} Q_{r}^{(q)} p_{r} - φ W_{pr} \\ - \sum_{q = 1}^{Q} \sum_{r = 1}^{R} \sum_{i = 1}^{I} \sum_{j = 1}^{J} Q_{r}^{(q)} c_{ij}^{intra (var)} R_{ij}^{(r)} - \sum_{t = 1}^{T} S F_{t} L_{t} \\ - \sum_{z = 1}^{Z} [\sum_{i = 1}^{I} \sum_{j = 1}^{J} \sum_{q = 1}^{Q} \sum_{r = 1}^{R} Q_{r}^{(q)} R_{ij}^{(r)} x_{ij}^{(r) z} f_{ij}^{z}] (1 - τ_{z}) \\ - \sum_{z = 1}^{Z} [\sum_{i = 1}^{I} \sum_{j = 1}^{J} \sum_{q = 1}^{Q} \sum_{r = 1}^{R} Q_{r}^{(q)} R_{ij}^{(r)} x_{ij}^{(r) z} w_{ij}] D^{z} C \\ + \sum_{i = 1}^{I} \sum_{j = 1}^{J} \sum_{q = 1}^{Q} \sum_{r = 1}^{R} Q_{r}^{(q)} R_{ij}^{(r)} ω_{ij}^{W} \end{matrix}$ (28)

Minimizing GHG emission is the second optimization objective, and it is formulated as follows:

$\begin{matrix} min E^{T} = \sum_{i = 1}^{I} \sum_{j = 1}^{J} \sum_{q = 1}^{Q} \sum_{r = 1}^{R} Q_{r}^{(q)} R_{ij}^{(r)} e_{ij} \\ + \sum_{z = 1}^{Z} \sum_{i = 1}^{I} \sum_{j = 1}^{J} \sum_{q = 1}^{Q} \sum_{r = 1}^{R} Q_{r}^{(q)} R_{ij}^{(r)} x_{ij}^{(r) z} w_{ij} D^{z} E_{Tr} \\ + E_{V} W_{pr} + \sum_{q = 1}^{Q} \sum_{r = 1}^{R} \sum_{i = 1}^{I} \sum_{j = 1}^{J} Q_{r}^{(q)} e_{ij}^{(ass)} R_{ij}^{(r)} \\ + \sum_{z = 1}^{Z} F_{z} G_{z} + \sum_{z = 1}^{Z} [\sum_{i = 1}^{I} \sum_{j = 1}^{J} \sum_{q = 1}^{Q} \sum_{r = 1}^{R} Q_{r}^{(q)} R_{ij}^{(r)} E_{ij}^{W}] \end{matrix}$ (29)

Optimization constraints

Instance selection constraint

In a product variant, only one instance can be selected for a function module. It is expressed as follows:

$\sum_{j = 1}^{J} R_{ij}^{(r)} = 1, r = 1, 2, \dots, R; i = 1, 2, \dots, I .$ (30)

Supplier selection constraint

In a product variant, the instance of a function module only can be supplied by one supplier. The constraint can be expressed as follows:

$\sum_{z = 1}^{Z} x_{ij}^{(r) z} = 1, i = 1, 2, \dots, I; j = 1, 2 \dots, J; r = 1, 2, \dots, R .$ (31)

Warranty length constraint

Regardless of the sales volume of a product variant in a market segment, only a warranty length can be selected for a function module of a product variant in a market segment. This relationship can be formulated as follows:

$\begin{matrix} \sum_{q = 1}^{Q} W_{ij}^{s (r) q} = 1, s = 1, 2, . . . ., S; i = 1, 2, \dots, I; \\ j = 1, 2, . . ., J; r = 1, 2, \dots, R . \end{matrix}$ (32)

Optimization model representation

Based on the above analysis, the optimization model can be formulated as a constrained bi-objective programming problem, which is shown as follows:

$\begin{matrix} Objective 1 : Eq . (28) \\ Objective 2 : Eq . (29) \\ s . t . Eqs . (30 - 32) \end{matrix}$ (33)

Optimization model solving method

The proposed model is a large-scale hybrid non-linear programming problem. Because the solution space rapidly expands as the number of function modules and candidate instances, among others, increase, it is very difficult to obtain the solution of the optimization model using traditional methods. A meta-heuristic algorithm is considered to achieve the near-optimal solutions for the optimization problem. Due to its simple computation and robust search abilities,^32,33 the genetic algorithm (GA) is adopted in this paper.

Chromosome representation

As shown in Figure 1, the integer-coding method is used in GA. A chromosome includes four sections: a product configuration section, a supplier selection section, a warranty length selection section and a price selection section. In product configuration section, the value in the gene indicates which module instance has been selected. For example, the second gene has a value of ‘2’, which indicates that M_2,2 was selected for the second function module of product variant 1. Supplier selection is indicated by the supplier selection section. For instance, a value of ‘1’ in the second gene of the supplier selection section indicates that the first supplier (P¹) is selected to provide the module instance (M_2,2) for the configuration of product variant 1. The price of the product is pre-discrete. The price selected for the product variant is indicated by the gene value in the price selection section. Similarly, the warranty length of module instance is indicated by the gene value in warranty length selection section.

Figure 1.

An example of chromosome representation.

Fitness function

To solve the optimization model, the weighting method³⁴ to assign positive weights X₁ and X₂ (X₁ + X₂ = 1) to the two objectives is applied. The first measures profit (y₁), while the second measures GHG emission (y₂). As profit and GHG emission have different magnitudes and units, each objective must be normalized. Weighted additive utility function with normalization is as follows:

$Y^{'} (k) = X_{1} y_{1}' (k) + X_{2} y_{2}' (k)$ (34)

where $y_{i}^{'}$ is normalized values, each normalized objectiveis defined as:

$y_{i}' (k) = \frac{y_{i} (k) - y_{i, min}}{y_{i, max} - y_{i, min}}$ (35)

where $y_{i, min}$ and $y_{i, max}$ represent the minimum and maximum values of the objective function $y_{i} (k)$ , respectively.

Genetic operators

(1) Crossover. In the present study, the uniform crossover method was used.³⁵ First, a random crossover mask with a value of 0 or 1 was generated for the genes of the chromosome. Secondly, if the corresponding crossover mask value is 1, the biparental genes are exchanged. An example of a homogeneous crossover is shown in Figure 2, where two offspring are generated in the light of the crossover mask.

(2) Mutation. The mutation operation used in this study is based on the idea of neighbourhood. A neighbourhood of a gene is defined as an increasing or a decreasing change in the value of an integer contained in the gene. Individuals in a population mutate with a given probability. The mutation operation consists of first selecting a number of genes in an individual at random and then changing the values of these genes to their neighbourhoods.

(3) Selection mechanism. The roulette wheel selection method was used in this study for the selection operation.³⁶ In the roulette wheel selection method, the greater the fitness of an individual the greater the probability that it will be selected as a parent and produce offspring in the next generation.

Figure 2.

An example of crossover.

Case study

Case introduction and calculation

For meeting the diversified customer demand, electronic dictionary manufacturers want to develop a product family. The product architecture includes six functional modules: product housing module (M₁), storage module (M₂), voice module (M₃), key module (M₄), control module (M₅) and display module (M₆). For each function module, a set of candidate module instance can be selected to configure the product. By analysing market information, there are three market segments, and the segment market and competitive product information are shown in Table 1. Tables 2 to 4 list information about module instances, candidate vendors, warranty and so on. Table 5 provides the discount rates for different suppliers. After analysing the highest utility and cost, the range of possible product prices can be limited to [49.5, 78.1], and in the case study, the product prices are discretized into a set of integer prices from $60 to $80. The component failure distribution of the product is considered to be an exponential distribution with parameter $θ$ , therefore $F (t, θ) = 1 - e^{- θ t}$ .

Table 1.

The surpluses utility ($) of the competitive products and market size.

Market	Segment 1	Segment 2	Segment 3
Demand (PCS)	230,000	320,000	75,000
Utility of competitor product 1	63.81	60.15	65.22
Utility of competitor product 2	57.31	62.05	60.00
Utility of existing product	56.23	59.11	61.03

Table 2.

Related information of module instance.

Module	Instance	Utility in segment 1	Utility in segment 2	Utility in segment 3	Variable unit cost ($)	Variable unit emission (g)	Weight (g)	GHGemission (g)
M₁	M_1,1	16.8	15.4	15.2	0.8	0.3	120	36.9
	M_1,2	17	15.6	15.2	0.8	0.5	118	34.2
	M_1,3	17.2	16	16	1	0.3	119	35.4
	M_1,4	17.2	16.7	14.9	1	0.3	122	37.3
M₂	M_2,1	27.2	27.6	26.9	0.7	0.1	78	195
	M_2,2	27.3	28.6	26.9	0.8	0.3	81	213
	M_2,3	27.3	27.5	27.5	0.8	0.1	84	220
	M_2,4	27.3	27.5	28.7	0.9	0.2	80	207
M₃	M_3,1	13.3	15	13.9	0.7	0.5	92	268
	M_3,2	13.2	15.2	14.3	0.7	0.5	90	254
	M_3,3	13.3	14.8	15	0.8	0.3	93	282
M₄	M_4,1	11.3	11.3	11.1	0.6	0.3	89	301
	M_4,2	11.3	11.5	11.1	0.6	0.1	92	358
	M_4,3	11.5	11.5	11.5	0.8	0.1	95	362
M₅	M_5,1	22.9	22.4	18.8	0.7	0.2	101	431
	M_5,2	23	22.8	20.2	0.8	0.3	103	443
	M_5,3	23.1	22.5	20.7	0.8	0.4	104	464
M₆	M_6,1	19.2	17.2	17.2	0.6	0.1	83	143
	M_6,2	19.3	17.2	17.2	06	0.2	81	121
	M_6,3	19.3	17.1	17.3	0.6	0.3	84	157

Table 3.

Information about utility of module warranty and maintenance.

Module	Warranty length (year)	Module warranty utility in segment 1	Module warranty utility in segment 2	Module warranty utility in segment 3	Average maintenance cost/GHG emission per time
M₁	0.5	2.8	3.1	2.7	3.8/21
	1	3.5	3.4	3.2
	1.5	4.1	3.8	3.9
	2	4.9	4.2	4.0
M₂	0.5	6.5	5.8	6.6	7.2/162
	1	6.8	6.4	6.8
	1.5	6.9	6.6	7.0
	2	7.1	6.9	7.1
M₃	0.5	3.2	3.5	3.1	4.3/195
	1	3.6	3.8	3.5
	1.5	3.9	4.0	3.8
	2	4.1	4.1	4.0
M₄	0.5	2.6	2.4	2.7	3.5/200
	1	2.9	2.8	3.0
	1.5	3.1	3.0	3.1
	2	3.4	3.2	3.2
M₅	0.5	4.8	4.4	4.3	6.2/308
	1	5.0	4.5	4.7
	1.5	5.4	4.8	5.2
	2	6.0	5.2	6.0
M₆	0.5	3.5	3.7	3.5	6.8/74
	1	3.7	3.8	3.8
	1.5	4.1	4.0	3.9
	2	4.3	4.2	4.2

Table 4.

Information about candidate suppliers and the reliability of component.

Supplier	Instances (purchase price $/reliability parameter)	Distance(km)
P ¹	M_1,1(4.90/0.29), M_1,2(5.10/0.32), M_1,3(5.20/0.39), M_1,4(5.40/0.25), M_3,1(6.40/0.38), M_3,2(6.50/0.35), M_3,3(6.70/0.34)	681
P ²	M_2,1(12.00/0.38), M_2,2(11.50/0.36), M_2,3(11.60/0.31), M_4,1(5.40/0.3), M_4,2(5.40/0.4), M_4,3(5.30/0.38), M_6,1(9.50/0.38), M_6,2(9.60/0.38), M_6,3(9.60/0.36)	1009
P ³	M_1,1(4.80/0.39), M_1,2(5.20/0.33), M_1,3(5.29/0.4), M_1,4(5.38,0.4), M_3,1(6.50/0.37), M_3,2(6.49/0.38), M_3,3(6.38/0.35), M_5,1(11.58/0.37), M_5,2(11.69/0.36), M_5,3(11.82/0.34)	896
P ⁴	M_2,1(11.19/0.36), M_2,2(11.28/0.36), M_2,3(11.37/0.30), M_2,4(11.58/0.30), M_4,1(5.19/0.29), M_4,2(5.27/0.4), M_4,3(5.29/0.39), M_6,1(9.79/0.37), M_6,2(9.87/0.36), M_6,3(9.85/0.34)	987
P ⁵	M_1,1(4.81/0.28), M_1,2(5.19/0.30), M_1,3(5.28/0.37), M_3,1(6.49/0.38), M_3,2(6.46/0.34), M_3,3(6.38/0.32)	1100
P ⁶	M_2,1(11.17/0.36), M_2,2(11.37/0.35), M_2,3(11.45/0.30), M_2,4(11.58/0.29), M_5,1(11.37/0.36), M_5,2(11.48/0.35), M_5,3(11.69/0.33)	756
P ⁷	M_2,1(11.28/0.36), M_2,3(11.49/0.35), M_2,4(11.68/0.33), M_3,1(6.47/0.37), M_3,2(6.58/0.38), M_3,3(6.5/0.39)	653
P ⁸	M_1,1(4.78/0.29), M_1,2(5.09/0.33), M_1,3(5.28/0.38), M_1,4(5.32/0.4), M_3,1(6.47/0.38), M_3,2(6.48/0.38), M_3,3(6.59/0.39), M_5,1(10.83/0.36), M_5,2(11.18/0.38), M_5,3(11.49/0.39), M_6,1(9.86/0.38), M_6,2(9.89/0.37), M_6,3(9.85/0.37)	782
P ⁹	M_4,1(5.39/0.31), M_4,2(5.47/0.33), M_4,3(5.59/0.35), M_5,1(11.68/0.37), M_5,2(11.69/0.36), M_5,3(11.8/0.38)	914
P ¹⁰	M_1,1(4.99/0.30), M_1,3(5.18/0.39), M_1,4(5.28/0.39), M_3,1(6.39/0.37), M_3,2(6.48/0.38), M_3,3(6.32/0.37), M_6,1(10.05/0.38), M_6,2(10.08/0.39), M_6,3(10.11/0.39)	1045
P ¹¹	M_2,1(11.13/0.36), M_2,2(11.24/0.36), M_2,3(11.38/0.35), M_2,4(11.49/0.35), M_4,1(5.48/0.32), M_4,2(5.46/0.32), M_4,3(5.49/0.33), M_5,1(11.53/0.36), M_5,2(11.5/0.34), M_5,3(11.59/0.34)	995
P ¹²	M_3,1(6.29/0.37), M_3,2(6.38/0.36), M_3,3(6.48/0.36), M_4,1(5.29/0.32), M_4,2(5.38/0.31), M_4,3(5.29/0.32), M_5,1(11.47/0.36), M_5,2(11.58/0.36), M_5,3(11.59/0.37)	745
P ¹³	M_2,1(10.88/0.4), M_2,2(11.19/0.36), M_2,3(11.26/0.35), M_2,4(11.37/0.35), M_5,1(11.78/0.36), M_5,2(11.79/0.36), M_5,3(11.68/0.35), M_6,1(9.87/0.38), M_6,2(11/0.37), M_6,3(10/0.37)	925

Table 5.

Information about supplier discounts.

	Sales volume (in thousand $)	Percentage discount (%)
P ¹	(0, 320), (320, 650), (650, 950), (950, +∞)	0, 2, 4, 7
P ²	(0, 200), (200, 400), (400, 800), (800, +∞)	0, 2, 6, 9
P ³	(0, 500), (500, 600), (600, 900), (900, +∞)	0, 5, 8, 10
P ⁴	(0, 150), (150, 400), (400, 1000), (1000, +∞)	0, 1, 5, 8
P ⁵	(0, 300), (300, 600), (600, 1100), (1100, +∞)	0, 3, 8, 10
P ⁶	(0, 200), (200, 500), (500, 1200), (1200, +∞)	0, 2, 5, 10
P ⁷	(0, 600), (600, 600), (600, 1500), (1500, +∞)	0, 5, 8, 11
P ⁸	(0, 250), (250, 500), (500, 800), (800, +∞)	0, 3, 5, 8
P ⁹	(0, 200), (200, 900), (900, 1600), (1600, +∞)	0, 1, 7, 10
P ¹⁰	(0, 500), (500, 800), (800, 1000), (1000, +∞)	0, 4, 7, 10
P ¹¹	(0, 300), (300, 600), (600, 900), (900, +∞)	0, 5, 6, 9
P ¹²	(0, 300), (300, 600), (600, 1000), (1000, +∞)	0, 3, 5, 10
P ¹³	(0, 150), (150, 750), (750, 900), (900, +∞)	0, 1, 5, 9

The company intends to develop a product family comprising of two product variants. Four weight combinations were considered, including case 1 (X₁ = 1, X₂ = 0), case 2 (X₁ = 0.8, X₂ = 0.2), case 3 (X₁ = 0.7, X₂ = 0.3) and case 4 (X₁ = 0.5, X₂ = 0.5). The developed GA was adopted for solving the optimization problem. The algorithm was implemented in MATLAB 2014b. The GA parameters are set as follows: the population size is 10000; the crossover rate is 0.75; the mutation rate is 0.25. The optimized decision variables for cases 1–4 are provided in Table 6. It can be seen that in these four cases, the product family configuration, component warranty period, etc., are different. The optimization results are shown in Figure 3. With X₂ increasing from 0 to 1, the amount of GHG emission is reduced, and the profit are decreasing. When profit reaches an optimal near-optimal value 1.109 × 10⁶, the GHG emission reaches the maximum value 7.853 × 10⁵. It shows that there is a compromise between the two goals in the case study. Therefore, in order to reduce the environmental impact of their products, manufacturers need to set reasonable weights for each objective.

Table 6.

Optimized decision variables for cases 1–4.

		Module
		M₁	M₂	M₃	M₄	M₅	M₆
Product variant 1	Instance selection Case 1/2/3/4	M_1,2/M_1,4/M_1,4/M_1,3	M_2,1/M_2,2/M_2,1/M_2,4	M_3,2/M_3,3/M_3,2/M_3,2	M_4,2/M_4,1/M_4,1/M_4,1	M_5,2/M_5,2/M_5,1/M_5,2	M_6,1/M_6,1/M_6,3/M_6,3
	Warranty length Case 1/2/3/4	0.5/1/1/2	0.5/1/1/2	0.5/1/1/2	0.5/1/1/2	0.5/1/1/2	0.5/1/1/2
	Supplier selection Case 1/2/3/4	P⁵/P¹/P¹/P¹	P²/P⁴/P²/P⁴	P⁵/P⁷/P⁷/P⁵	P¹¹/P⁹/P⁴/P⁴	P¹¹/P⁹/P¹³/P³	P⁸/P¹⁰/P¹³/P¹⁰
Product variant 2	Instance selection Case 1/2/3/4	M_1,4/M_1,4/M_1,2/M_1,2	M_2,4/M_2,4/M_2,2/M_2,3	M_3,1/M_3,1/M_3,2/M_3,3	M_4,1/M_4,2/M_4,3/M_4,3	M_5,2/M_5,1/M_5,2/M_5,3	M_6,1/M_6,2/M_6,1/M_6,3
	Warranty length Case 1/2/3/4	0.5/1/1/2	0.5/1/1/2	0.5/1/1/2	0.5/1/1/2	0.5/1/1/2	0.5/1/1/2
	Supplier selection Case 1/2/3/4	P⁸/P¹/P⁸/P¹	P⁴/P¹¹/P²/P⁷	P⁸/P⁷/P⁷/P⁷	P¹¹/P⁴/P⁴/P⁴	P¹¹/P⁶/P¹³/P³	P¹/P¹/P⁸/P¹³

Figure 3.

Relevant results of cases 1–4.

Impact of component warranty on optimization results

The purpose of the experiment is to observe the impact of component warranty on low-carbon product family design and supplier selection. Two experiments, including cases 5–6, were designed. In case 5, the product family design, supplier selection and the warranty period of the component are optimized in parallel. Compared to case 5, the reliability and the component warranty period are not considered in case 6. In cases 5–6, the weights are set as X₁ = 0.60, X₂ = 0.40, and two product variants are developed.

The optimized decision variables (e.g. module instance configuration of each product variant, supplier selection, etc.) for cases 5–6 are shown in Figure 4. It can be seen that the optimal product configuration is different between case 5 and case 6. For example, in case 6, M_4,1 is selected for configuring product variants, and it was not adopted in case 5. Except for the product configuration, the suppliers selected in case 5 and case 6 are also different. Figure 5 shows the relevant optimization results. Regarding the profit, case 5 is greater than that in case 6. This is because market demand is greater in case 5 than that in case 6. The main reason for this is that case 5 provides the component warranty, which can attract more consumers to buy. The same reason could explain why the GHG emission in case 5 is greater than in case 6. In terms of the total amount, the profit in case 5 is greater than that in case 6, and the GHG emission in case 5 is also greater than that in case 6. But the growth rates of profit and GHG emission are different. Compared to case 6, case 5 has a 18% increase in profit and only a 10% increase in GHG emission. The result of the experiment indicates that the integration optimization of product family design, supplier selection and component warranty can bring more profit and less GHG emission for company than that without considering component warranty. Hence, it is meaningful that component warranty length is considered simultaneously as the decision variable in low-carbon product family design. It also demonstrates the effectiveness of the proposed approach.

Figure 4.

Optimized decision variables for cases 5–6: (a) Case 5 and (b) Case 6.

Figure 5.

Relevant results of cases 5–6.

Impact of warranty policy of same component on optimization result

Different product variants in a product family may be configured with some of the same components. There are two warranty polices considered for the same components. One warranty policy is to adopt the same warranty length. The other is that the same components can adopt different warranty lengths. The purpose of the experiment is to observe the influence of warranty period of same component on GHG emission and profit of the product family. Two experiments, including case 7 and case 8, were designed. In case 7, the same component adopts the same warranty length. In case 8, the same component can adopt the different warranty lengths. In case 7 and case 8, the weights are set to X₁ = 0.60, X₂ = 0.40, and two product variants are developed.

The optimal product configuration, supplier selection and component warranty length for cases 7–8 are shown in Figure 6. The relevant results of cases 7–8 are provided in Figure 7. The total profit in case 8 is greater than that in case 7, and the total GHG emission in case 8 is lower than that in case 7. It can be seen that both optimization objectives in case 8 are better than in case 7. This is because that the same component with same warranty length directly affects the warranty cost in case 7. In addition to affecting the warranty cost, it also indirectly affects the selection of module instances and supplier selection in product family configuration. The result of the experiment indicates that larger GHG emission and less profit may result if the same components are restricted using the same warranty length. Hence, in order to improve profitability and reduce GHG emission, it is better not to require the same warranty length for the same components.

Figure 6.

Optimized decision variables for cases 7–8: (a) Case 7 and (b) Case 8.

Figure 7.

Relevant results of cases 7–8.

The influence of warranty policies of product variants on optimization result

If all components in a product variant adopt the same warranty period, the warranty length is called the warranty length of the product variant in this research. A product family includes several product variants. These product variants can adopt the same warranty length or different warranty lengths. The experiment is performed to explore the effect of warranty policies of product variants on GHG emission and profit of the product family. The four experiments of the two comparative experiments, including case 9, case 10, case 11 and case 12, were designed. Cases 9–10 are a group of comparative experiments. Cases 11–12 are other set of comparative experiments. In cases 10 and 12, all product variants in a product family are set to adopt the same warranty length, whereas it is not required in case 9 and case 11. The number of developing product variants is different between the two comparative experiments. Two product variants are developed in cases 9–10, and three product variants are developed in cases 11–12. The weight for cases 9–12 is set as X₁ = 0.60, X₂ = 0.40.

The optimal product configuration, supplier selection and component warranty length for cases 9–10 are shown in Figure 8. The profit and GHG emission of the product family for cases 9–10 are shown in Figure 10. It can be seen that the profit of the product family in case 9 is greater than that in case 10. In GHG emission aspect, case 9 is lower than that in case 10. The experiment results show that it is better not required to adopt the same warranty length for different product variants. The same conclusions can be drawn from the other set of comparative experiments. The optimized results for cases 10–11 are shown in Figures 9 and 10. It indicates that the conclusion does not change with the number of developed product variants.

Figure 8.

Optimized decision variables for cases 9–10: (a) Case 9 and (b) Case 10.

Figure 9.

Optimized decision variables for cases 11–12: (a) Case 11 and (b) Case 12.

Figure 10.

Relevant results for cases 9–12.

Conclusions

Global warming has become the most important environmental issue in the world today. As an important means of reducing GHG emission, low-carbon product family design has attracted more and more attention. In previous studies, neither the component reliability nor component warranty length was not fully considered. To address this issue, this paper presents a decision model that deals with concurrent optimization of product family design, supplier selection and component warranty period with consideration of profit and GHG emission. Moreover, in order to effectively solve this optimization problem, a corresponding genetic algorithm has been developed. The effectiveness of the proposed method is verified through an example. The optimization results show that the integrated optimization of product family design, supplier selection and component warranty length is superior to not considering component warranty. If the same components in a product family were limited to adopting the same warranty length, this would increase GHG emission and decrease profits. It is also observed that it is better not to require the adoption of the same warranty period for different product variants than to require them to adopt the same warranty period.

In this research, it is assumed that all parameters in the model have been determined. However, in actual cases, some parameters are fuzzy or even random. Therefore, in the future, an uncertain optimization method considering parameter uncertainty for coordinating low-carbon product family design, supplier selection and component warranty length should be investigated to overcome the limitation of the proposed method. Additionally, identifying the suppliers’ demands (such as lean principles and sustainable activities) and considering them in the model is worth further research in the future.

Footnotes

The authors thank the anonymous reviewers for their critical and constructive review of the manuscript.

Handling Editor: Qing Wang

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: This work was supported by the Zhejiang Provincial Natural Science Foundation of China [grant number LQ21E050011];the Ningbo Natural Science Foundation [grant number 202003N4316 ];and the Science and Technology Innovation 2025 Major Project of Ningbo of China [grant numbers 2022Z008;2023Z022 ].

ORCID iD

Qi Wang

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An optimization model for low-carbon product family design considering component reliability and warranty period

Abstract

Keywords

Introduction

Related works

Low-carbon product design

Product warranty

Problem definition

Development of optimization model

Customer preference modelling considering the warranty length of component

Product sales

Cost calculation for product family

Production cost

Warranty cost of component

GHG emission model of product family

Objective function

Optimization constraints

Instance selection constraint

Supplier selection constraint

Warranty length constraint

Optimization model representation

Optimization model solving method

Chromosome representation

Fitness function

Genetic operators

Case study

Case introduction and calculation

Impact of component warranty on optimization results

Impact of warranty policy of same component on optimization result

The influence of warranty policies of product variants on optimization result

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References