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

How to satisfy customers’ heterogeneous demands by limitedmodels of product is the key problem in anATO (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 ATOmanufacturer 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.


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][2][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 basestock 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 multicomponent 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][16][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.Section 2 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.

Problem Description and Assumption.
A product family ATO manufacture is able to use common components and special components to assemble  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  to represent all common components of product family, and special component  for all special components of product ,  = 1, 2, . . ., .So, a product  consists of one unit each of component  and  [20,21].In additional, we number all products in order of their values or qualities, that is, the values or qualities of product  follows  1 <  2 < ⋅ ⋅ ⋅ <   .The structure of the assemble-to-order system is shown in Figure 1.
The potential demand of whole product family is  and the market segmentation or actual demand of product  is   ,  = 1, 2, . . ., .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   and selling price   of product .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,   , is exogenous and follows  1 <  2 < ⋅ ⋅ ⋅ <   .
Customers are heterogeneous and customer type, which can be consider as a customer's importance weight for product , which follows a uniform distribution of [0, 1].Therefore, a customer of type  gets a value   from buying a unit of product , and the highest price that he is willing to pay for a unit of product  is   [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,   −   ).If the highest customer surplus is less than zero, he will not buy any products.The quality and selling price of products ,   , and   , as well as the distribution of customer type  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.As   ,   , and  are all common knowledge, the manufacturer can shortlist the preliminary set of products with positive market segmentation,   , for final selection.Then, it deters the final set of products with positive optimal selling duration,   , according to the market segmentation of products, the demand substitution among the preliminary set of products, as well as the purchase price,   and   , and unit holding cost of components,   and   .Subsequently, it deters the purchase quantity of every kind of components,   and   , 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 product ,  = 1, 2, . . ., , as product , where  = 1, 2, . . ., ,  ̸ = , as all the other products except product  may be the substitution product.If no product provides a positive customer surplus, he gives up purchase.
And the demand per time of common component  at  time of th duration as Now, we can get the total holding cost of all components as the sales income of products as the purchase cost of all components as and the profit of ATO manufacturer as
Form     −   ≥ 0, we get   ≥ (  /  ), so a customer of type   ∈ [  /  , 1] is willing to buy product  if there is no other products.From  0 =  0 = 0 and  +1 =  +1 = + ∝, we Among the  kinds of products, there are  kinds of products with realistic demand, where  = 1 when there is only one kind of product with realistic demand,  =  when all the  kinds of products with realistic demand.Therefore, we denote the number of product with realistic demand by ,  = 1, 2, . . ., .
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 these  kinds of products with realistic demand in order of their qualities as  1 <  2 < ⋅ ⋅ ⋅ <   , and denote these products by product ,  = 1, 2, . . ., .Meanwhile, denote these  kinds of products in order of selling duration by product .Now, we can get the new profit function of an ATO manufacturer as follows Before analyzing the manufacturer's optimal product selection and components replenishment policies, we determine the market segmentation and substitution parameter of product ,   and   , ,  = 1, 2, . . ., ,  ̸ = .
Proof.Firstly, we apply reduction to absurdity to prove where,  = 2, 3, . . .,  − 1; that is,   −1 <   − , we get that the surplus of customers of type   ∈ [  −1 ,   − ] obtained from product  is more than that from product  − 1 while less than that from product −, so they will buy product −.Form the fact that the upper limit of customer type  −1 is   obtained from buying product  is positive and bigger than that obtained from product  − 1, or else product  should not be assembled, so Proof.As the lower and upper limit of customer type   is individually   −1 and   +1 ,  = 1, 2, . . ., , when product  is in short supply, customers who were willing to buy it will just buy product  − 1 or  + 1.As a matter of course, if product  + 1 is in short supply too, they will buy product  − 1 or  + 2, generally, if products from  to  −  and from  to  + ,  = 0, 1, . . ., −1,  = 0, 1, . . ., −, are all in short supply too, they will buy product  −  − 1 or  +  + 1.Therefore,   = 0 when  ̸ =  −  − 1,  +  + 1.When products from  −  to  +  are all in short supply, the upper limit of customer type  −−1 and the lower limit of customer type  ++1 change into  −−1 ++1 , therefore, among the customers of type ] will buy product  +  + 1, and those of type As product 0 and  + 1 are fictitious, we can get that  (+1) =  0 = 0.
There are ! kinds of permutations that products in order of qualities are ranged in order of selling duration, from [product 1, product 2,. .., product ] to [product , product  − 1,. .., product 1].Therefore, we should analyze the ATO manufacturer's profits under these ! kinds of permutations to obtain the optimal profit and selling duration of all products.
We take the first permutation, [product 1, product 2,. .., product ], as an example to show the process of obtaining the optimal solution.
Under the first permutation, product  is product , ,  = 1, 2, . . ., ; therefore, from (1), we can get the selling quantity of product  and the replenishment quantity of special component  as Substituting ( 9) into (7), we can get the ATO manufacturer's profit under the first permutation as The optimal selling duration of product ,  *  ,  = 1, 2, . . ., , is the solution of first order derivative of the ATO manufacturer's profit,  1 , with respect to the selling duration,   , equaling 0, that is, the solution of the following equation: Solving (11), we can get Substituting  *  into (9), we get the optimal purchase quantity of product  as Similarly, we can get the function of the selling quantity of product  and special component ,   ,  = 1, 2, . . ., , as well as the ATO manufacturer's profit,   , under the jth permutation,  = 1, 2, . . ., !.Subsequently, solving   /  = 0, we can get the optimal selling duration of product ,  *  ,  = 1, 2, . . ., , as well as the optimal selling duration and replenishment quantity of product ,  *  and  *  , according to the corresponding relationship between product  and product  under the jth permutation.Proposition 5.The optimal replenishment quantity of special component ,  *  ,  = 1, 2, . . ., , increases with the extending of the optimal selling duration of product ,  *  , while it is unrelated to the optimal selling duration of product , From ( 14), we can find that there are the demand of special component in the th duration,    ,  ≤ , or the demand of common component in the th duration    ,  ≤ , in every formula of the function of , as well as the first order derivative of  with respect to   , /  .Furthermore, from (1) and ( 2), we know that both    and    are equal to the potential total demand per unit of whole product family, , multiplied by a certain constant.Therefore, we can eliminate  from equation /  = 0, and get the optimal selling duration,  *  ,which is unrelated to  but related to the products' selling price,   , the purchase price and holding cost per unit of components,   ,   and   ,   .
Step 3. Determine the function of selling quantity of product  and replenishment quantity of special component  from Lemmas 3 and 4, and the ATO manufacturer's profit   under the th permutation,  = 1, 2, . . ., !.
Step 4. Solving   /  = 0 to get the optimal selling duration of product ,  *  ,  = 1, 2, . . ., , as well as the optimal selling duration and replenishment quantity of products ,  *  , and  *  , as well as the optimal profit,  *  , according to the corresponding relationship between product  and product  under the th permutation.
Step 6. Export  * and corresponding optimal selling duration of product ,  *  , the optimal replenishment quantity of special component  and common component ,  *  , and  *  ,  = 1, 2, . . ., .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 Table 1, we can find that  1 and  * 3 decrease while  * 1 ,  3 , and  * increase with the increase of  1 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  * 3 while extending  * 1 to earn more profit,  * .With the further increase of  1 , 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  3 gets a further expanding.

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,  = 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,  = 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 component .
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:  = 3000 units per day,  1 = 4050 yuan per unit,  2 = 4250 yuan per unit,  1 = 1275 yuan per unit,  2 = 1450 yuan per unit,   = 1800 yuan per unit,  1 = 7 yuan per unit per day,  2 = 7.5 yuan per unit per day, and   = 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 individually  * 1 = 57.5 days and  * 1 = 10566 units, and those of LingYa are individually  * 2 = 50 days and  * 2 = 3659 units.Therefore, JianShe should replenish special component  * 1 = 10566 units for LingYing,  * 2 = 3659 units for LingYa, as well as component  *  = 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,  * 1 /2 = 5283 units and  * 2 /2 = 1830 units, is close to the reality.

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.

Table 1 :
The optimal solutions with respect to  1 .