Assemble-to-order system

Last updated December 17, 2020

In applied probability, an assemble-to-order system is a model of a warehouse operating a 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).^[1]

Model definition

Single period model

This case is a generalisation of the newsvendor model (which has only one component and one product). The problem involves three stages and we give one formation of the problem below^[2]

components acquired
demand realized
components allocated, products produced

We use the following notation^[1]

Symbol	Meaning
m	total number of components
n	total number of products
a_i^j	units of component i required to make one unit of product j
d_j	demand for product j
y_i	supply for component i
p_j	penalty cost for unit shortage of product j
h_i	cost for unit excess of component i
z_j	production level of product j
w_j	shortage of product j
x_i	excess of component i

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

{\begin{aligned}{\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{aligned}}

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

{\begin{aligned}{\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{aligned}}

with x₀ 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.^[3]

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References

1 2 3 Song, J. S.; Zipkin, P. (2003). "Supply Chain Operations: Assemble-to-Order Systems". Supply Chain Management: Design, Coordination and Operation (PDF). Handbooks in Operations Research and Management Science. 11. pp. 561–596. doi:10.1016/S0927-0507(03)11011-0. ISBN 9780444513281.
↑ Gerchak, Y.; Henig, M. (1986). "An inventory model with component commonality". Operations Research Letters. 5 (3): 157. doi:10.1016/0167-6377(86)90089-1.
↑ Song, J. S.; Yao, D. D. (2002). "Performance Analysis and Optimization of Assemble-to-Order Systems with Random Lead Times" (PDF). Operations Research . 50 (5): 889. doi:10.1287/opre.50.5.889.372. JSTOR 3088488.

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[song-1] 1 2 3 Song, J. S.; Zipkin, P. (2003). "Supply Chain Operations: Assemble-to-Order Systems". Supply Chain Management: Design, Coordination and Operation (PDF). Handbooks in Operations Research and Management Science. 11. pp. 561–596. doi:10.1016/S0927-0507(03)11011-0. ISBN 9780444513281.

[2] Gerchak, Y.; Henig, M. (1986). "An inventory model with component commonality". Operations Research Letters. 5 (3): 157. doi:10.1016/0167-6377(86)90089-1.

[3] Song, J. S.; Yao, D. D. (2002). "Performance Analysis and Optimization of Assemble-to-Order Systems with Random Lead Times" (PDF). Operations Research . 50 (5): 889. doi:10.1287/opre.50.5.889.372. JSTOR 3088488.

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