This paper presents an optimization decision model for a production system that comprises the hybrid maketostock/assembletoorder (MTS/ATO) organization mode with demand uncertainty, which can be described as a twostage decision model. In the first decision stage (i.e., before acquiring the actual demand information of the customer), we have studied the optimal quantities of the finished products and components, while in the second stage (i.e., after acquiring the actual demand information of the customer), we have made the optimal decision on the assignment of components to satisfy the remaining demand. The optimal conditions on production and inventory decision are deduced, as well as the bounds of the total procurement quantity of the components in the ATO phase and final products generated in the MTS phase. Finally, an example is given to illustrate the above optimal model. The findings are shown as follows: the hybrid MTS and ATO production system reduces uncertain demand risk by arranging MTS phase and ATO phase reasonably and improves the expected profit of manufacturer; applying the strategy of component commonality can reduce the total inventory level, as well as the risk induced by the lower accurate demand forecasting.
Over recent years, assembly manufacturing enterprises have faced fierce competition in the market as a result of individual and diverse needs of customers as well as delivery uncertainty among suppliers. To address these challenges, these enterprises have to apply the hybrid operator mode that comprises the maketostock/assembletoorder (MTS/ATO) production organization mode.
This hybrid mode has two features. First, before the actual demand is observed, the assembly manufacturer procures a certain quantity of each component required for assembling the final product. This quantity is determined based on the forecasted demand and assembly capacity. A certain quantity of the final product may have to be assembled in advance. Second, after confirming the actual customer demand, the manufacturer may need to assemble more final products to fulfil the needs of customers as much as possible [
The following managerial questions often arise in MTS and ATO production systems. How can a manufacturer make a reasonable production decision to evade uncertainty demand as much as possible? How can a manufacturer rationally allocate the limited components in the inventory for assembling additional final products?
Several studies have investigated the optimal decision model of inventoryproduction for assembly systems. These studies can be classified into the following aspects.
Eynan and Rosenblatt [
Hillier [
The recent work of Xiao et al. [
The preceding analysis shows us that the inventoryproduction optimal decision issue on the MTS/ATO hybrid production mode has been rarely investigated in the existing literature.
Our model differs from that of Xiao et al. [
The rest of this paper is organised as follows. Section
This paper considers an optimal production and inventory decision method for a hybrid MTS/ATO production system with uncertain demand. As shown in Figure
BOM of products 1 and 2.
As shown in Figure
Production procedure under the MTS/ATO hybrid production mode.
At time
Phase
At time
Phase
At time
The definitions and notations are denoted as follows:
To facilitate the presentation of the essential ideas without loss of generality, this study makes the following basic assumptions:
the actual demands for the two types of final products are irrelevant and determined at the same time.
given that the purchase lead time is too long for the assembly of the corresponding final products after confirming the actual demand information of the customer, the manufacturer makes the procurement decision at time
without loss of generality, the unit profit of final product 1 is greater than that of final product 2, which satisfies the following expressions:
the cost of assembling units in advance in the MTS phase is lower than that for the normal assembly of products in the ATO phrase, which satisfies the following expressions:
compared with product demand, production capacity is infinite and product storage is allowed.
the salvage value of the unused components is smaller than the original procurement cost. Similarly, the salvage value of the redundant final product is smaller than the revenue of such products.
We formulate our optimisation problem in a backward order. At time
Therefore, the decision optimisation problem (
The objective function (
We always use the onhand final products that are generated in the MTS phase to satisfy the demands of customers. In other words, we assemble additional products only when the amount of preassembled products in phase 1 does not satisfy the actual demand of customers. The addedassembly quantities of the two products in the ATO phrase are constrained by the limited inventory of components and the demand fulfilment shortage.
With regard to the product structure, the quantities of exclusive components do not need to exceed those of common components. Meanwhile, the sum of exclusive components must be larger than the volume of common components or the common components will be left over, which does not comply with our objective (see [
Therefore, the addedassembly quantities of products 1 and 2 satisfy (
We also investigate the optimal extra production decisions in the ATO phase. After confirming the demand information of the customers, the manufacturer must determine whether or not additional assembly is necessary and how to allocate the inventory of common component (
After observing the actual demand for products 1 and 2, we have 10 possible cases that are denoted as
Demand space in the MTS/ATO production system.
We now discuss the additionalassembly decisions in each area by allocating common components according to product priority. The optimal combinations of the assembly quantities and actual sales quantities of products 1 and 2 are analysed as follows:
In domain
In domain
In domain
In domain
In domain
In domain
In domain
In domain
In domain
In domain
From the preceding analysis, under the strategy of commonality with product priority, the additional quantity of assembled products and the actual selling amount of both products 1 and 2 are shown in Table
Additional assembly arrangements and actual sales of commonality strategy with product priority in separate demand areas.
Demand area  Product 1 additional assembly volume  Product 2 additional assembly volume  Product 1 actual sales  Product 2 actual sales 


0  0 




0 



0 









0 

























0 


For the nonidentical profits of the two types of final products, the optimal solution for the ATO phrase can be easily obtained in the form of a greedy algorithm. Obviously, we are interested in making optimal decisions for AIA and components inventory. By considering all possible combinations of future demand and component inventory, the manufacturer determines how many final products must be assembled in advance and how many extra inventories of each component must be prepared at time
The expected revenue function
Given that the objective function equation (
Beginning with the quantity of final product 1 that is produced in the MTS phase,
Likewise, the expression for the derivative with respect to the quantity of final product 2 that is produced in the MTS phase (
The expression for the derivative with respect to the quantity of component 3 that is acquired in the MTS phase (
The expression for the derivative with respect to the quantity of component 5 that is acquired in the MTS phase (
By equating and then rearranging (
Relations (
We determine the bounds for the quantity of the two kinds of products that are produced before confirming the demands of customers and the total amount of each component that is acquired at the beginning of the period.
We can rewrite (
Equation (
Given that
The salvage value of a final product must not exceed the sum of the salvage value that is required for assembling the ordered product because the manufacturer prefers to reserve components than final products with high demand uncertainty. Therefore, the quantity of final products that are produced in the MTS phase is maintained at a low level.
Equation (
The second, third, and fourth parts on the lefthand side of (
When the manufacturer can produce goods after receiving the demands of customers, lower bounds are given for the quantity of final products that are generated in the MTS phase.
According to (
We replace the second and third probabilities on the righthand side of (
Thereafter, we obtain the following:
We now determine the bounds for the total procurement quantity of each component. The optimality condition (
By disregarding the nonnegative second term on the righthand side, we obtain the upper bound for the optimal value of
Similarly, (
Therefore, we can obtain the lower bound
The gap of fractals between the upper and lower bounds can be computed as follows:
In many practical situations, the gap is closed in cases where the difference between the unit cost of the common component and the salvaged unit of the common component is small. Thus, tight bounds can be obtained.
Equation (
Apparently, the term on the lefthand side of (
Given that
Intuitively, the lefthand side of (
This bound becomes tighter for a highly negatively correlated demand.
By using the lower bounds of the specific components and relation
We set about bounding the total procurement quantity of component 5. By using (
By excluding the second probability on the righthand side of (
Equation (
The structure of (
By excluding the nonnegative second term on the righthand side of (
In a hybrid MTS/ATO production system, the components acquired at the beginning of the period are divided into two parts. Some of these components are used for production and the others are saved for use in later production. We have set bounds for the total procurement quantity of these components and the amount of final products that are generated in the MTS phase. Therefore, the bounds for the components that are saved for ATO can be obtained easily.
In this section, we report the results of a numerical experiment that is designed to demonstrate the advantage of the hybrid MTS and ATO production system. The base parameters used in our numerical experiments are the following.
Let
As is shown in Table
Optimal inventoryproduction decisions in different production system.
Production system 






MTS and ATO (with common components)  250  167  503  749  560 


MTS and ATO (without common components)  250  167  407  840  433 


ATO and MTS  581  516  657  1257  600 
As is shown in Table
Expected profit of the manufacturer in different production system.
Production system  Expected profit  Profit improvement (compared to MTS) 


MTS and ATO (with common components)  10927  16.3% 


MTS and ATO (without common components)  10490  11.6% 


ATO  10057  7.03% 


MTS  9396 
According to the statistics above, we can conclude the following.
The hybrid MTS and ATO production system reduces uncertain demand risk by the method combined assembleinadvance in the MTS phase with assembleinadvance and ATO phase; meanwhile, the expected profit of manufacturer is improved in return. This implies the assembleinadvance strategy is an efficient way to cater for the risk from the uncertainties of both the demand.
Our example shows that commonality can bring about a higher profit than would be attainable without commonality and contribute to the reduction of forecast error. In the context of this paper, after applying commonality strategy, the inventory level of common component will decrease while those exclusive components will increase; owing to the riskpooling effect of commonality, total inventories
We have addressed in this paper the optimal product and inventory decision problem that arises from a hybrid MTS/ATO production system with commonality strategy under demand uncertainty. By analysing the balance between MTO at a lower cost and ATO at a higher cost operation, an optimal decision model is presented based on the twostage production and inventory decision. Consequently, a balanced tradeoff between the low unit production cost of MTS and the flexibility of ATO arises. According to the KarushKuhnTucker condition, the optimality conditions are found in a set of adjusted newsvendorlike solutions in which two products share one component. We have also studied the bounds and properties for the total procurement quantity of the components that are acquired at the beginning of the period and the amount of final products that are manufactured in the MTS phase.
Finally, we conduct numerical experiments to validate the model and the advantages of the hybrid MTS and ATO production system with common components. Research results show that a hybrid MTS and ATO production system can not only effectively respond to emergent orders and market demands using product inventory in MTS phase, but also reduce demand uncertainty risk by smoothing customers’ demand using components in the ATO phase. And the riskpooling effect of commonality strategy effectively reduces inventory cost.
Although our paper recommends some management techniques to assembly manufacturing enterprises, our findings can be extended in the following ways:
finding optimal productioninventory decisions in a multiperiod condition with an uncertain context;
exploring from a whole supply chain perspective the collaboration among material suppliers, manufacturers, wholesalers, and retailers as well as the related inventory control problems;
considering the randomness of supplier output and supply uncertainty; inventory control and optimisation of multisupplier and multiproduct collaborative delivery that is also a promising research direction.
For any given
Note that,
For any
Then
Hence
The authors declare that there is no conflict of interests regarding the publication of this paper.
The work that is described in this paper was supported by the Talented Person Foundation of Central South Forestry University of Science and Technology, Forestry Engineering Postdoctoral Research Center, and Graduate Education and Degree Innovation Foundation of Central South University (2014). The authors would like to thank the guest editors and the anonymous referees for their helpful comments and constructive suggestions on an earlier version of the paper.