# Product Selection and Components Replenishment Model of ATO Manufacturer under Heterogeneous Demand

Authors: Yuyu Li; Bo Huang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

How to satisfy customers’ heterogeneous demands by limited models of product is the key problem in an ATO (Assemble-to-Order) manufacturer’s operations management. Based on the condition that customers’ demands depend on their types which follow uniform distribution, we develop an inventory model to study how an ATO manufacturer selects the products from the candidate products in a same
product family and determines components replenishment quantities, in order to maximize its profit. We get the optimal set of products which an ATO manufacturer should assemble, as well as the components replenishment quantities through theoretical, numerical, and case analyses. Furthermore, we
propose an algorithm to obtain the optimal solution on the set of products and the components replenishment quantities. We found that an ATO manufacturer should make the performance and qualities of products over a certain level, and determine the optimal selling duration of products according to
their selling prices and qualities, as well as the purchase prices and holding costs of their components. This paper offers consultation for ATO manufacturers to make policies on product selection and components replenishment.


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## Body

## 1. Introduction

Due to the heterogeneity of customers and their demands, which means different customers have different income, price sensitivity, and preference on product usage; manufacturers in reality have to assemble a series of products with different usages and qualities (i.e., product family) to meet heterogeneous demands [1–3]. Take PC manufacturers as an example, they provide PC of low configurations (e.g., low speed processor and integrated graphics video card) at a low price to price sensitive customers while they provide PC of high configurations (e.g., high speed processor and discrete graphics video card) at a high price to customers with demand for high performance. Meanwhile, in order to reduce the costs due to the diversity of products, many manufacturers have adopted ATO (Assemble-to-Order), that is, only after receiving customers’ order, do they assemble products according to it [4, 5].Different with make-to-stock [6], there is no product inventory under ATO model, whose core is components replenishment accordingly [7]. Many researches on components replenishment of multiproduct ATO system have been made [8, 9]. Lu and Song studied a multiproduct ATO system in which customers may order different but possibly overlapping subsets of items, and determined the right base-stock level [10]. Betts and Johnston used a deterministic batch sizing model assuming that ATO inventory is finite to analyze the JIT replenishment or component substitution decisions [11], and further treated the stochastic version of the same problem and developed a tractable solution method for ATO decision problem [12]. DeCroix et al. identified several ways in which returns complicate the behavior of ATO system, and demonstrated how to handle these additional complexities when calculating or approximating key order-based performance metrics [13]. Zhao derived exact expressions for key performance metrics of a multiproduct and multi-component ATO system where demand follows compound Poisson processes, and develop an efficient sampling method to estimate these metrics [14]. However, these researches did not consider the heterogeneity of customers and their demands, and did not study the product selection problem.While the study of product selection forms a growing part of the literature in operation research [15–17], most of them are either about how to select and design product line according to product selection in order to enhance its performance, reduce costs, and raise profits [18], or about the product selection during the competition between traditional retailer or cross-channel competition [19]. To the best of our knowledge, there is little research on how ATO manufacturers select the optimal products and determine the components replenishment quantities to meet the heterogeneous customer demands.The remainder of the paper is organized as follows. Section2 is dedicated to the model development. The model and its optimal solution are analyzed in Section 3. We propose an algorithm for the optimal solution in Section 4. A numerical example is used to illuminate the model in Section 5. Section 6 is a case study. Finally, conclusions are drawn in Section 7.


## 2. Model Development

### 2.1. Problem Description and Assumption

A product family ATO manufacture is able to use common components and special components to assemblen models of seasonal products belonging to the same family. These products with different performances or qualities and selling price are candidate ones for the ATO manufacturer’s final optimal set of products, which maximize its profit. In reality, a product consists of multiple common components and special components. However, the parameters of these components, such as purchase price and holding cost per unit per time, are fixed constants. Therefore, for simplifying the model and analysis, we can use a common component c to represent all common components of product family, and special component i for all special components of product i, i=1,2,…,n. So, a product i consists of one unit each of component c and i [20, 21]. In additional, we number all products in order of their values or qualities, that is, the values or qualities of product i follows V1<V2<⋯<Vn. The structure of the assemble-to-order system is shown in Figure 1.Figure 1
The assemble-to-order system.The potential demand of whole product family isD and the market segmentation or actual demand of product i ismi, i=1,2,…,n. The ATO manufacture only selects the products with positive market segmentation to assemble from the whole family according to the heterogeneous preference and demand of customers, as well as the quality Vi and selling price Si of product i. In practice, the price of many ATO products, PC, for example, is decided by market because of the substitution among different types of product and a number of competitors. Therefore, we consider the selling price, Si, is exogenous and follows S1<S2<⋯<Sn.Customers are heterogeneous and customer type, which can be consider as a customer’s importance weight for producti, which follows a uniform distribution of[0,1]. Therefore, a customer of type θ gets a value θVi from buying a unit of product i, and the highest price that he is willing to pay for a unit of product i is θVi [22, 23]. A customer only buys a unit of one product, which provides the highest positive customer surplus (the margin between what he is willing to pay and the selling price, θVi-Si). If the highest customer surplus is less than zero, he will not buy any products. The quality and selling price of products i, Vi, and Si, as well as the distribution of customer type θ are common knowledge, that is, the information is known to all the players, including the ATO manufacturer and all customers, and each player knows that all the players know it, each player knows that all the players know that all the players know it, and so forth ad infinitum.The process of the ATO manufacturer makes its product selection and components replenishment policies as follows. AsVi, Si, and θ are all common knowledge, the manufacturer can shortlist the preliminary set of products with positive market segmentation, mi, for final selection. Then, it deters the final set of products with positive optimal selling duration, Ti, according to the market segmentation of products, the demand substitution among the preliminary set of products, as well as the purchase price, Pi and Pc, and unit holding cost of components, Hi and Hc. Subsequently, it deters the purchase quantity of every kind of components, Qi and Qc, and replenish it according to its optimal selling duration, and will make no replenishment during the whole selling season. Finally, it assembles and sales product after receiving customer order, therefore, it has no product inventory but components inventory.The process of customers making purchase decision is as follows. When a potential customer arrives, he makes the decision of whether and which product should he buy according to products’ qualities, selling price, and his heterogeneous preference on products’ qualities with the goal of maximizing his surplus. If he decides to buy, the potential demand becomes actual demand. If the product he decides to buy is in short supply, he makes a new decision of whether and which substitution should he buy among the spot products to maximize his surplus. Denote the substitution product for producti, i=1,2,…,n, as product j, where j=1,2,…,n, j≠i, as all the other products except product i may be the substitution product. If no product provides a positive customer surplus, he gives up purchase.


### 2.2. Notation

The other notations in this paper are as follows, wherei,j=1,2,…,n, j≠i:ρji
: substitution parameter of producti, namely, when product i is in short supply, the proportion of customer who wants to buy it chooses to buy product j, ∑ρji≤1;Ti: selling duration of product i, decision variable;Qi: selling quantity of product i, and replenishment quantity of special component i, decision variable;Pi: purchase price of special component i;Hi: holding cost per unit per time of special component i;Qc: replenishment quantity of common component c, Qc=∑Qi, decision variable;Pc: purchase price of common component c;Hc: holding cost per unit per time of common component c.


### 2.3. The ATO Inventory Model

Without losing generalization, renumbering all products in order of their selling duration, we getTI-1<TI, I=1,2,…,n. When product I is sold out, the percentage of buying the substitutions is ∑j=I+1n‍ρjI, and then percentage of giving up purchase is 1-∑j=I+1n‍ρjI. Letting T0=0, we can get the demand per time of special component I at t time of kth period (t∈[Tk-1,Tk]) as
(1)DIk={D(mI+∑j=1k-1‍ρijmj),k≤I;0,k>I.
And the demand per time of common component c at t time of kth duration as
(2)Dck=D∑I=kn‍[mI+∑j=1k-1‍(ρIjmj)],k=1,2,…,n.
Now, we can get the total holding cost of all components as
(3)Hc∑k=1n‍{∑j=kn‍Dck(Tk-Tk-1)20000000+(Tk-Tk-1)∑j=kn‍[Dcj+1(Tj+1-Tj)]}+∑I=1n‍∑k=1I‍{∑j=kI-1‍HIDIk(Tk-Tk-1)2200000000000+HI(Tk-Tk-1)∑j=kI-1‍[DIj+1(Tj+1-Tj)]},
the sales income of products as
(4)∑I=1n‍(SIQI),
the purchase cost of all components as
(5)Pc∑I=1n‍QI+∑I=1n‍(PIQI),
and the profit of ATO manufacturer as
(6)π=∑i=In‍(SIQI)-Hc∑k=1n‍{∑j=kn‍Dck(Tk-Tk-1)22000000000+(Tk-Tk-1)∑j=kn‍[Dcj+1(Tj+1-Tj)]}-∑I=1n‍∑k=1I{∑j=kI-1‍HIDIk(Tk-Tk-1)22‍000000000+HI(Tk-Tk-1)∑j=kI-1‍[DIj+1(Tj+1-Tj)]}-Pc∑I=1n‍QI-∑I=1n‍(PIQI).


## 2.1. Problem Description and Assumption

A product family ATO manufacture is able to use common components and special components to assemblen models of seasonal products belonging to the same family. These products with different performances or qualities and selling price are candidate ones for the ATO manufacturer’s final optimal set of products, which maximize its profit. In reality, a product consists of multiple common components and special components. However, the parameters of these components, such as purchase price and holding cost per unit per time, are fixed constants. Therefore, for simplifying the model and analysis, we can use a common component c to represent all common components of product family, and special component i for all special components of product i, i=1,2,…,n. So, a product i consists of one unit each of component c and i [20, 21]. In additional, we number all products in order of their values or qualities, that is, the values or qualities of product i follows V1<V2<⋯<Vn. The structure of the assemble-to-order system is shown in Figure 1.Figure 1
The assemble-to-order system.The potential demand of whole product family isD and the market segmentation or actual demand of product i ismi, i=1,2,…,n. The ATO manufacture only selects the products with positive market segmentation to assemble from the whole family according to the heterogeneous preference and demand of customers, as well as the quality Vi and selling price Si of product i. In practice, the price of many ATO products, PC, for example, is decided by market because of the substitution among different types of product and a number of competitors. Therefore, we consider the selling price, Si, is exogenous and follows S1<S2<⋯<Sn.Customers are heterogeneous and customer type, which can be consider as a customer’s importance weight for producti, which follows a uniform distribution of[0,1]. Therefore, a customer of type θ gets a value θVi from buying a unit of product i, and the highest price that he is willing to pay for a unit of product i is θVi [22, 23]. A customer only buys a unit of one product, which provides the highest positive customer surplus (the margin between what he is willing to pay and the selling price, θVi-Si). If the highest customer surplus is less than zero, he will not buy any products. The quality and selling price of products i, Vi, and Si, as well as the distribution of customer type θ are common knowledge, that is, the information is known to all the players, including the ATO manufacturer and all customers, and each player knows that all the players know it, each player knows that all the players know that all the players know it, and so forth ad infinitum.The process of the ATO manufacturer makes its product selection and components replenishment policies as follows. AsVi, Si, and θ are all common knowledge, the manufacturer can shortlist the preliminary set of products with positive market segmentation, mi, for final selection. Then, it deters the final set of products with positive optimal selling duration, Ti, according to the market segmentation of products, the demand substitution among the preliminary set of products, as well as the purchase price, Pi and Pc, and unit holding cost of components, Hi and Hc. Subsequently, it deters the purchase quantity of every kind of components, Qi and Qc, and replenish it according to its optimal selling duration, and will make no replenishment during the whole selling season. Finally, it assembles and sales product after receiving customer order, therefore, it has no product inventory but components inventory.The process of customers making purchase decision is as follows. When a potential customer arrives, he makes the decision of whether and which product should he buy according to products’ qualities, selling price, and his heterogeneous preference on products’ qualities with the goal of maximizing his surplus. If he decides to buy, the potential demand becomes actual demand. If the product he decides to buy is in short supply, he makes a new decision of whether and which substitution should he buy among the spot products to maximize his surplus. Denote the substitution product for producti, i=1,2,…,n, as product j, where j=1,2,…,n, j≠i, as all the other products except product i may be the substitution product. If no product provides a positive customer surplus, he gives up purchase.


## 2.2. Notation

The other notations in this paper are as follows, wherei,j=1,2,…,n, j≠i:ρji
: substitution parameter of producti, namely, when product i is in short supply, the proportion of customer who wants to buy it chooses to buy product j, ∑ρji≤1;Ti: selling duration of product i, decision variable;Qi: selling quantity of product i, and replenishment quantity of special component i, decision variable;Pi: purchase price of special component i;Hi: holding cost per unit per time of special component i;Qc: replenishment quantity of common component c, Qc=∑Qi, decision variable;Pc: purchase price of common component c;Hc: holding cost per unit per time of common component c.


## 2.3. The ATO Inventory Model

Without losing generalization, renumbering all products in order of their selling duration, we getTI-1<TI, I=1,2,…,n. When product I is sold out, the percentage of buying the substitutions is ∑j=I+1n‍ρjI, and then percentage of giving up purchase is 1-∑j=I+1n‍ρjI. Letting T0=0, we can get the demand per time of special component I at t time of kth period (t∈[Tk-1,Tk]) as
(1)DIk={D(mI+∑j=1k-1‍ρijmj),k≤I;0,k>I.
And the demand per time of common component c at t time of kth duration as
(2)Dck=D∑I=kn‍[mI+∑j=1k-1‍(ρIjmj)],k=1,2,…,n.
Now, we can get the total holding cost of all components as
(3)Hc∑k=1n‍{∑j=kn‍Dck(Tk-Tk-1)20000000+(Tk-Tk-1)∑j=kn‍[Dcj+1(Tj+1-Tj)]}+∑I=1n‍∑k=1I‍{∑j=kI-1‍HIDIk(Tk-Tk-1)2200000000000+HI(Tk-Tk-1)∑j=kI-1‍[DIj+1(Tj+1-Tj)]},
the sales income of products as
(4)∑I=1n‍(SIQI),
the purchase cost of all components as
(5)Pc∑I=1n‍QI+∑I=1n‍(PIQI),
and the profit of ATO manufacturer as
(6)π=∑i=In‍(SIQI)-Hc∑k=1n‍{∑j=kn‍Dck(Tk-Tk-1)22000000000+(Tk-Tk-1)∑j=kn‍[Dcj+1(Tj+1-Tj)]}-∑I=1n‍∑k=1I{∑j=kI-1‍HIDIk(Tk-Tk-1)22‍000000000+HI(Tk-Tk-1)∑j=kI-1‍[DIj+1(Tj+1-Tj)]}-Pc∑I=1n‍QI-∑I=1n‍(PIQI).


## 3. Model Analysis

For a better analysis and understanding of the model, we introduce two fictitious products labeled as product 0 and productn+1. The qualities and selling prices of product 0 and n+1 are S0=V0=0,  Sn+1=Vn+1=+∝. Then, there are n+2 kinds of products for a customer’s choosing (i.e., j=1,2…,n, changes into j=0,1,…,n+1), but nobody will choose these two fictitious products as the quality of product 0 is V0=0 while the selling price of product n+1 is S++∝. Therefore, introducing these two fictitious products has no effect on the result of the model.Lettingγji=(Si-Sj)/(Vi-Vj), i=1,2,…,n, j=0,1,…,n+1, and j≠i, we can get Lemma 1 as follows.Lemma 1.
A customer of typeθi∈[Max[γji,j=0,1,…,i-1],Min[γji,j=i+1,i+2,…,n+1]]is willing to buy producti,i=1,2,…,n.Proof.
Only if customer surplus gotten from buying producti,  i=1,2,…,n, is nonnegative;that is, θiVi-Si≥0, and no less than that from product j, j=0,1,…,n+1 and j≠i; that is, θiVi-Si≥θiVj-Sj, will a customer of type θi buy product i.
FormθiVi-Si≥0, we get θi≥(Si/Vi), so a customer of type θi∈[Si/Vi,1]is willing to buy product i if there is no other products. From S0=V0=0 and Sn+1=Vn+1=+∝, we get γ0i=((Si-S0)/(Vi-V0))=(Si/Vi), γn+1i=((Si-Sn+1)/(Vi-Vn+1))=1, so θi∈[Si/Vi,1] can be denoted as θi[γ0i,γn+1i].
FromθiVi-Si≥θiVj-Sj and V1<V2<⋯<Vn, we get θi≥(Si-Sj)/(Vi-Vj) when j<i and θi≤(Si-Sj)/(Vi-Vj) when j>i.
Therefore, a customer of typeθi∈[Max[γji,j=0,1,…,i-1],Min[γji,j=i+1,i+2,…,n+1]] is willing to buy product i.LettingγLi=Max[γji,j=0,1,…,i-1], L∈[0,i-1], γUi=Min[γji,j=i+1,i+2,…,n+1], and  U∈[i+1,n+1], we can get Proposition 2 as follows.Proposition 2.
The feasible set of products that an ATO manufacturer should assemble consists of products with qualitiesVi>((Si-SL)+(SU-Si)VL)/(SU-SL), i=1,2,…,n.Proof.
From Lemma1, we know that a customer of type θi∈[γLi,γUi] is willing to buy product i, i=1,2,…,n. Therefore, only when γLi<γUi, is there realistic demand on product i. Solving γLi=(Si-SL)/(Vi-VL)<(Si-SL)/(Vi-VL)=γUi, we can get Vi>((Si-SL)+(SU-Si)VL)/(SU-SL), i=1,2,…,n. Therefore, the feasible set of products that an ATO manufacturer should assemble consists of products with qualities Vi>((Si-SL)+(SU-Si)VL)/(SU-SL), i=1,2,…,n.Among then kinds of products, there are N kinds of products with realistic demand, where N=1 when there is only one kind of product with realistic demand, N=n when all the n kinds of products with realistic demand. Therefore, we denote the number of product with realistic demand by N, N=1,2,…,n.As some of the products with realistic demand may be less profitable than other products of the same product family, or the holding cost per unit per time of their special component are too high, these products should not be produced or their selling duration should be shorter than other products. Therefore, after getting the feasible set of products with realistic demand, an ATO manufacturer will determine the final set of products which consists of products with positive selling duration in order to maximize its profit.We rearrange theseN kinds of products with realistic demand in order of their qualities as V1<V2<⋯<VN, and denote these products by product i, i=1,2,…,N. Meanwhile, denote these N kinds of products in order of selling duration by product I. Now, we can get the new profit function of an ATO manufacturer as follows
(7)π=∑I=1N‍(SIQI)-Hc∑k=1n‍{∑j=kn‍Dck(Tk-Tk-1)22000000000+(Tk-Tk-1)∑j=kn‍[Dcj+1(Tj+1-Tj)]}-∑I=1n‍∑k=1I‍{∑j=kn‍HIDIk(Tk-Tk-1)22000000000+HI(Tk-Tk-1)∑j=kI-1‍[DIj+1(Tj+1-Tj)]}-Pc∑I=1N‍QI-∑I=1N‍(PIQI).Before analyzing the manufacturer’s optimal product selection and components replenishment policies, we determine the market segmentation and substitution parameter of producti,  mi and ρji,  i,j=1,2,…,N,  j≠i.Lemma 3.
Customers of typeθi∈[γi-1i,γi+1i] buy product i, so the market segmentation of product i is mi=[γi-1i-γi+1i], i=1,2,…,N.Proof.
Firstly, we apply reduction to absurdity to proveγi-1i=Max[γji,j=1,2,…,i-1],  i=1,2,…,N.
Assuming thatγi-ki=Max[γji,j=1,2,…,i-1], where, k=2,3,…,i-1; that is, γi-1i<γi-ki, we get that the surplus of customers of type θi∈[γi-1i,γi-ki] obtained from product i is more than that from product i-1 while less than that from product i-k, so they will buy product i-k. Form the fact that the upper limit of customer type θi-1 is γUi-1=Min[γji-1,j=1,2,…,n+1]≤γii-1=γi-1i, we get θi-1≤θi-k, while from the fact that θi-1≤γi+1i, θi+1≥γii+1 and γi+1i=(Si-Si+1)/(Vi-Vi-1)=(Si+1-Si)/(Vi+1-Vi)=γii+1, we get θi≤θi+1. Therefore, the assumption that γi-ki=Max[γji,j=1,2,…,j-1] is wrong, which means that γi-1i=Max[γji,j=1,2,…,i-1], i=1,2,…,N.
Now, we prove thatγi-1i=Max[γ0i,γi-1i], i=1,2,…,N.
Wheni=1, γ0i=γi-1i, so γi-1i=Max[γ0i,γi-1i]. When i=2,3,…,N, the surplus of customers of type θ≥γi-1i obtained from buying product i is positive and bigger than that obtained from product i-1, or else product i should not be assembled, so γi-1i>γ0i; that is, γi-1i=Max[γ0i,γi-1i]. Now, we can make the conclusion that γi-1i=Max[γ0i,γi-1i], i=1,2,…,N; that is, the lower limit of θi is γi-1i.
Similarly as above, we can prove that if the upper limit ofθi was not γi+1i but γi+ki,  k=2,3,…,n+1-i, the customers of type θ∈[γi+1i,γi+ki] should buy product i+k; that is, θi+k≤θi+1, which contradicts θi≤θi+1. Therefore, the upper limit of θi is γi+1i.
From what was discussed above, we can make the conclusion that customers of typeθ∈[γi-1i,γi+1i] are willing to buy product i, whose market segmentation is mi=(γi+1i-γi-1i)/(1-0)=γi+1i-γi-1i,  i=1,2,…,N.Lemma 4.
When producti, i=1,2,…,N, is in short supply, and products from i to i-k and from i to i+l,  k=1,2,…,i-1, l=0,1,…,N-i, are all in short supply too, the percentage of customers who were willing to buy product i, that is, customers of type θi∈[γi-1i,γi+1i] change to buy product j, j=1,2,…,i-k-1,i+l+1,…,N, as follows:
(8)ρji={γi+1i-γi+l+1i-l-1mi,j=i+l+1,j≠N+1;γi+l+1i-l-1-γi+1imi,j=i-k-1,j≠0;0,other.Proof.
As the lower and upper limit of customer typeθi is individually γi-1i and γi+1i, i=1,2,…,N, when product i is in short supply, customers who were willing to buy it will just buy product i-1 or i+1. As a matter of course, if product i+1 is in short supply too, they will buy product i-1 or i+2, generally, if products from i to i-k and from i to i+l, k=0,1,…,i-1, l=0,1,…,N-i, are all in short supply too, they will buy product i-k-1 or i+l+1. Therefore, ρji=0 when j≠i-k-1, i+l+1.
When products fromi-k to i+l are all in short supply, the upper limit of customer type θi-k-1 and the lower limit of customer type θi+l+1 change into γi+l+1i-k-1, therefore, among the customers of type θi∈[γi-1i,γi+1i], those of type θi∈[γi+l+1i-k-1,γi+1i] will buy product i+l+1, and those of type θi∈[γi-1i,γi+l+1i-k-1] will buy product i-l-1, in other words, ρji=(γi+1i-γi+l+1i-k-l)/mi, where j=i+l+1, and ρji=(γi+l+1i-k-l-γi-1i)/mi, wherej=i-l-1.
As product 0 andN+1 are fictitious, we can get that ρ(N+1)i=ρ0i=0.There areN! kinds of permutations that products in order of qualities are ranged in order of selling duration, from [product 1, product 2,…, product N] to [product N, product N-1,…, product 1]. Therefore, we should analyze the ATO manufacturer’s profits under these N! kinds of permutations to obtain the optimal profit and selling duration of all products.We take the first permutation, [product 1, product 2,…, productN], as an example to show the process of obtaining the optimal solution.Under the first permutation, producti is product I, i,I=1,2,…,N; therefore, from (1), we can get the selling quantity of product i and the replenishment quantity of special component i as
(9)Qi=∑k=1i‍[Dik(Tk-Tk-1)],i=1,2,…,N.Substituting (9) into (7), we can get the ATO manufacturer’s profit under the first permutation as
(10)π1=∑i=1N‍{(Si-Pc-Pi)∑k=1i‍[Dik(Tk-Tk-1)]}-Hc∑k=1N‍{(Tk-Tk-1)∑j=kN‍[Dcj+1(Tj+1-Tj)]2000000000+Dck(Tk-Tk-1)22∑j=kN‍}-∑i=1N‍∑k=1i‍{∑j=ki-1‍HiDik(Tk-Tk-1)22000000000+Hi(Tk-Tk-1)∑j=ki-1‍[Dij+1(Tj+1-Tj)]}.The optimal selling duration of producti, Ti*, i=1,2,…,N, is the solution of first order derivative of the ATO manufacturer’s profit, π1, with respect to the selling duration, Ti, equaling 0, that is, the solution of the following equation:
(11)∂π1∂Ti=∑j=iN‍[(Sj-Pj-Pc)Dji]-∑j=i+1N‍[(Sj-Pj-Pc)Dji+1]-Hc(Dci-Dci+1)Ti-Ti∑j=i+1N‍[Hj(Dji-Dji+1)]-HiDiiTi=0,i=1,2,…,N.Solving (11), we can get
(12)Ti*=(∑j=iN‍[(Sj-Pj-Pc)Dji]000-∑j=i+1N‍[(Sj-Pj-Pc)Dji+1])×(∑j=i+1N‍Hc(Dci-Dci+1)+HiDii+∑j=i+1N‍[Hj(Dji-Dji+1)])-1,0000000000000000i=1,2,…,N.SubstitutingTi* into (9), we get the optimal purchase quantity of product i as
(13)Qi*=∑k=1i‍[Dik(Tk*-Tk-1*)],i=1,2,…,N.Similarly, we can get the function of the selling quantity of producti and special component i, Qi, i=1,2,…,N, as well as the ATO manufacturer’s profit, πj, under the jth permutation, j=1,2,…,N!. Subsequently, solving ∂πj/∂TI=0, we can get the optimal selling duration of product I, TI*, I=1,2,…,N, as well as the optimal selling duration and replenishment quantity of product i, Ti* and Qi*, according to the corresponding relationship between product i and product I under the jth permutation.Proposition 5.
The optimal replenishment quantity of special componenti, Qi*, i=1,2,…,N, increases with the extending of the optimal selling duration of product i, Ti*, while it is unrelated to the optimal selling duration of product j, Ti*, j=1,2,…,N and j≠i if product i is in short supply before product j or decreases with the extending of  Tj* if products from j to i-1 is in short supply before product i.Proof.
Solving the first order derivative ofQi* with respect to Ti*, we can get ∂Qi*/∂Ti*=Dii>0, i=1,2,…,N. Therefore, Qi*increases with the extending of Ti*.
Let producti and product j, j=1,2,…,N and j≠i, individually be the k~th and k~th product of being in short supply. If k^>k~, we can get that Qi* is unrelated to form Qi*=∑k=1k~‍[Dik(Tk*-Tk-1*)]. From Lemma 4, we can get that ρij>0 if products from j to i-1 are in short supply before product i. Therefore, we can get that Qi* decreases with the extending of Tj* from ∂Qi*/∂Ti*=Dik^-Dik^+1<0.Proposition 6.
The optimal selling duration of producti, Ti*, i=1,2,…,N, is not related to the potential total demand per time of whole product family, D, but related to the products’ selling price, the purchase price, and the holding cost per unit of components.Proof.
SubstitutingQI=∑k=1I‍[DIk(Tk-Tk-1)]into (7), we can get
(14)π=∑i=IN‍{(SI-PI-Pc)∑k=1I‍[DIk(Tk-Tk-1)]}-Hc∑k=1N‍{(Tk-Tk-1)∑j=kN‍[Dcj+1(Tj+1-Tj)]000000000+Dck(Tk-Tk-1)22∑j=kN‍}-HI∑I=1N‍∑k=1I‍{∑j=kI-1‍DIk(Tk-Tk-1)22000000000000+(Tk-Tk-1)∑j=kI-1‍[DIj+1(Tj+1-Tj)]}.
From (14), we can find that there are the demand of special component in the kth duration, DIk, k≤I, or the demand of common component in the kth duration Dck, k≤N, in every formula of the function of π, as well as the first order derivative of π with respect to TI,  ∂π/∂TI. Furthermore, from (1) and (2), we know that both DIk and Dck are equal to the potential total demand per unit of whole product family, D, multiplied by a certain constant. Therefore, we can eliminate D from equation ∂π/∂TI=0, and get the optimal selling duration, Ti*,which is unrelated to D but related to the products’ selling price, SI, the purchase price and holding cost per unit of components, PI, Pc and HI, Hc.


## 4. The Algorithm

Step 1.
Determine the feasible set of products with realistic demand byVi>((Si-SL)VU+(SU-Si)VL)/(SU-SL), and arrange these products in order of their qualities, that is, V1<V2⋯<VN.Step 2.
Determine the market segment of producti, mi, and its substitution parameter, ρji, i,j=1,2,…,N, j≠i.Step 3.
Determine the function of selling quantity of producti and replenishment quantity of special component i from Lemmas 3 and 4, and the ATO manufacturer’s profit πj under the jth permutation, j=1,2,…,N!.Step 4.
Solving∂πj/∂TI=0 to get the optimal selling duration of product I, TI*, I=1,2,…,N, as well as the optimal selling duration and replenishment quantity of products i, Ti*, and Qi*, as well as the optimal profit, πj*, according to the corresponding relationship between product i and product I under the jth permutation.Step 5.
Comparing the optimal profit,πj*, under all N! kinds of permutation to get the ATO manufacturer’s optimal profit π*=Maxπj*, j=1,2,…,N!.Step 6.
Exportπ* and corresponding optimal selling duration of product i, Ti*, the optimal replenishment quantity of special component i and common component c, Qi*, and Qc*, i=1,2,…,N.


## 5. Numerical Analysis

In this section, we provide a numerical example to illuminate our model and propositions. We consider that an ATO manufacturer is planning to select product to assemble from product family consisting of three products, among which producti consists of one unit common component c and special component i, i=1,2,3. The parameters of the ATO manufacturer are as follows: D=10000, S1=1450, S2=1800, S3=2350, P1=550, P2=850, P3=1350, and Pc=600, H1=20, H2=40, H3=75, and Hc=35.Firstly, fromVi>((Si-SL)VU+(SU-Si)VL)/(SU-SL), we get the feasible product set consisting of product 1 and 3. Therefore, there are two kinds of permutation [product 1, product 3] and [product 3, product 1].Secondly, we get the market segmentation of product 1 and 3 asm1=0.055 and m3=0.182, as well as the substitution parameter ρ31=0.633 and ρ13=1.Thirdly, we can obtain the optimal profit under the first permutation asπ1*=1.53×106 and that under the second permutation as π2*=2.10×106.Finally, comparingπ1* with π2*, we get the ATO manufacturer’s optimal profit as π*=2.10×106, and the corresponding optimal solutions as T1*=5.45, T3*=1.82, Q1*=9613, Q3*=3306, and Qc*=12919.We further show how the selling price of product 1 impacts the optimal solutions, the impact of the selling price of products 2 and 3 is similar.From Table1, we can find that m1 and T3* decrease while T1*, m3, and π* increase with the increase of S1 because parts of customers who were planning to buy product 1 give up and parts buy product 3. Meanwhile, as the cost of product 1 remains unchanged, its marginal profit becomes greater and the ATO manufacturer prefers to sell more of product 1. From Lemma 4, we know that when product 3 is in short supply, all its customers change to buy product 1, therefore, the ATO manufacturer should reduce T3* while extending T1* to earn more profit, π*. With the further increase of S1, no customer buys it, parts of customers who were planning to buy it change to buy product 2, and parts buy product 3, so the product set consists of product 2 and 3, and m3 gets a further expanding.Table 1
The optimal solutions with respect toS1.
S
1
T
1
*
T
2
*
T
3
*
m
1
m
2
m
3
π
*
1350
3.64
—
3.64
0.199
—
0.091
1383210
1400
4.55
—
2.73
0.127
—
0.136
1774140
1450
5.45
—
1.82
0.055
—
0.182
2103090
1500
—
4.67
1.42
—
0.003
0.214
1851890


## 6. Case Analysis

In this section, we provide a real case of ChongQing JianShe Motorcycle Co., LTD. (shortened as JianShe) to illuminate our model and propositions. JianShe is a state-owned enterprise providing products including motorcycle manufacturing, distribution, related services, and automotive components. We take the series of JS110-B as the subject to study its product selection and components replenishment policies.The JS110-B series consists of 3 models, including LingYa, LingYing, and BF3, in other words,n=3. The features of JS110-B are as follows. Firstly, it is of fashionable and unique appearance with sporty style. Secondly, low fuel consumption, which is economical; small vibration, which enables you to enjoy the comfort like a car. Thirdly, different configurations are available, including cargo carrier, medium carriers, small rear carriers, burglar alarm, as well as wide and narrow tires, to meet the needs of different users.As the demand for BF3 model is rare, JianShe has stopped providing BF3, we should remove it from our feasible set of products with realistic demand, in other words,N=2. Now, there are 2 models of products with realistic demand, LingYa and LingYing, for which we should determine the optimal selling duration and components replenishment quantities.The special components of LingYing model (labeled as product 1) and LingYa model (labeled as product 2) are covers and an engine, the other components are common components. We simplify covers and engine as a special component, and label the one for LingYing model as special component 1 and the one for LingYa model as special component 2. Furthermore, we simplify all common components as a common component labeled as common componentc.As JianShe replenish the components for LingYing and LingYa every two months, we treat a replenishment circle as a selling season to analyze our model. Through a survey of market and JianShe’s operation management, we get that the parameters of LingYing model and LingYa model are as follows:D=3000 units per day, S1=4050 yuan per unit, S2=4250 yuan per unit, P1=1275 yuan per unit, P2=1450 yuan per unit, Pc=1800 yuan per unit, H1=7 yuan per unit per day, H2=7.5 yuan per unit per day, and Hc=10 yuan per unit per day.Following the algorithm put forward in this paper, we get the optimal replenishment policy for LingYing and LingYa as follows. The optimal selling duration and quantity of LingYing are individuallyT1*=57.5 days and Q1*=10566 units, and those of LingYa are individually T2*=50 days and Q2*=3659 units. Therefore, JianShe should replenish special component Q1*=10566 units for LingYing, Q2*=3659 units for LingYa, as well as component Qc*=14225 units for these two models.According to our survey, the average selling quantity per month in 2009 of LingYing is 5770 units and that of LingYa is 1769 units. Our theoretical result,Q1*/2=5283 units and Q2*/2=1830 units, is close to the reality.


## 7. Conclusions

We developed an inventory model, in which customers are heterogeneous and their type follows uniform distribution, to study how an ATO manufacturer makes its product selection and components replenishment policies. Furthermore, we proposed an algorithm to obtain the optimal solution, and use a numerical example and a real case to illuminate the model. We found that an ATO manufacturer should make the performance and qualities of products over a certain level, and determine the optimal selling duration of products according to their selling prices and qualities, as well as the purchase prices and holding costs of their components.



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*Source: 714715-2013-11-21.xml*