View source for Assemble-to-order system

In [[Applied probability|applied probability]], an '''assemble-to-order system''' is a model of a warehouse operating a [[Finance:Build to order|build to order]] policy where products are assembled from components only once an order has been made.
The time to assemble a product from components is negligible, but the time to create components is significant (for example, they must be ordered from a supplier).<ref name="song">{{Cite book | last1 = Song | first1 = J. S. | last2 = Zipkin | first2 = P. | chapter = Supply Chain Operations: Assemble-to-Order Systems | doi = 10.1016/S0927-0507(03)11011-0 | title = Supply Chain Management: Design, Coordination and Operation | series = Handbooks in Operations Research and Management Science | volume = 11 | pages = 561–596| year = 2003 | isbn = 9780444513281 | url = https://faculty.fuqua.duke.edu/~jssong/bio/Publications/AtoctoSurvey28Jan03.pdf| pmid =  | pmc = }}</ref>

Research typically focuses on finding good policies for inventory levels and on the impact of different configurations (such as having more shared parts). The special case of only one product is an assembly system, the case of just once component is a distribution system.<ref name="song" />

==Model definition==

===Single period model===

This case is a generalisation of the [[Finance:Newsvendor model|newsvendor model]] (which has only one component and one product). The problem involves three stages and we give one formation of the problem below<ref>{{Cite journal | last1 = Gerchak | first1 = Y. | last2 = Henig | first2 = M. | doi = 10.1016/0167-6377(86)90089-1 | title = An inventory model with component commonality | journal = Operations Research Letters | volume = 5 | issue = 3 | pages = 157 | year = 1986 | pmid =  | pmc = }}</ref>

# components acquired
# demand realized
# components allocated, products produced

We use the following notation<ref name="song" />
{| class="wikitable"
|-
! Symbol
! Meaning
|-
| ''m''
| total number of components
|-
| ''n''
| total number of products
|-
| ''a''<sub>''i''</sub><sup>''j''</sup> 
| units of component ''i'' required to make one unit of product ''j''
|-
| ''d''<sub>''j''</sub>
| demand for product ''j''
|-
| ''y''<sub>''i''</sub>
| supply for component ''i''
|-
| ''p''<sub>''j''</sub>
| penalty cost for unit shortage of product ''j''
|-
| ''h''<sub>''i''</sub>
| cost for unit excess of component ''i''
|-
| ''z''<sub>''j''</sub>
| production level of product ''j''
|-
| ''w''<sub>''j''</sub>
| shortage of product ''j''
|-
| ''x''<sub>''i''</sub>
| excess of component ''i''
|}

In the final stage when demands are known the optimization problem faced is to 

:<math>\begin{align}
\text{minimize } G(\mathbf y, \mathbf d) &= \mathbf h' \mathbf x+ \mathbf p' \mathbf w\\
\text{subject to } 
A \mathbf z + \mathbf x &= \mathbf y\\
\mathbf z + \mathbf w &= \mathbf d\\
\mathbf w, \mathbf x, \mathbf z &\geq 0,\end{align}</math>

and we can therefore write the optimization problem at the first stage as

:<math>\begin{align}
\text{minimize } & c(\mathbf y - \mathbf x_0) + \mathbb E_{\mathbf d} [ G(\mathbf y, \mathbf d) ] \\
\text{subject to } & \mathbf y \geq \mathbf x_0,\end{align}</math>

with '''x'''<sub>0</sub> representing the starting inventory vector and ''c'' the cost function for acquiring the components.

===Continuous time===

In continuous time orders for products arrive according to a Poisson process and the time required to produce components are independent and identically distributed for each component. Two problems typically studied in this system are to minimize the expected backlog of orders subject to a constraint on the component inventory, and to minimize the expected component inventory subject to constraints on the rate at which orders must be completed.<ref>{{Cite journal | last1 = Song | first1 = J. S. | last2 = Yao | first2 = D. D. | doi = 10.1287/opre.50.5.889.372 | title = Performance Analysis and Optimization of Assemble-to-Order Systems with Random Lead Times | journal = Operations Research| volume = 50 | issue = 5 | pages = 889 | year = 2002 | jstor = 3088488| pmid =  | url = https://faculty.fuqua.duke.edu/~jssong/bio/Publications/SongYao_OR.pdf| pmc = }}</ref>

==References==

{{Reflist}}

[[Category:Inventory optimization]]

{{Sourceattribution|Assemble-to-order system|1}}