Case Study
Schumann, a modeling framework for supply chain management under uncertainty

https://doi.org/10.1016/S0377-2217(98)00366-XGet rights and content

Abstract

We present a modeling framework for the optimization of a manufacturing, assembly and distribution (MAD) supply chain planning problem under uncertainty in product demand and component supplying cost and delivery time, mainly. The automotive sector has been chosen as the pilot area for this type of multiperiod multiproduct multilevel problem, but the approach has a far more reaching application. A deterministic treatment of the problem provides unsatisfactory results. We use a 2-stage scenario analysis based on a partial recourse approach, where MAD supply chain policy can be implemented for a given set of initial time periods, such that the solution for the other periods does not need to be anticipated and, then, it depends on the scenario to occur. In any case, it takes into consideration all the given scenarios. Very useful schemes are used for modeling balance equations and multiperiod linking constraints. A dual approach splitting variable scheme is been used for dealing with the implementable time periods related variables, via a redundant circular linking representation.

Introduction

Decision making is inherent to all aspects of industrial, business and social activities. In all of them, difficult tasks must be accomplished. One of the most reliable decision support tools available today is Optimization, a field at the confluence of Mathematics and Computer Science. The purpose of the field is to build and solve effectively realistic mathematical models of the situation under study, allowing the decision makers to explore a huge variety of possible alternatives. As reality is complex, many of these models are large (in terms of the number of decision variables), and stochastic (there are parameters whose value cannot be controlled by the decision maker and are uncertain). The last fact makes the problems difficult to tackle, yet its solution is critical for many leading organizations in fields such as supply chain planning among many other areas.

Manufacturing, Assembly and Distribution (MAD) Supply Chain Management is concerned with determining supply, production and stock levels in raw materials, subassemblies at different levels of the given Bills of Material (BoM), end products and information exchange through (possibly) a set of factories, depots and dealer centres of a given production and service network to meet fluctuating demand requirements. If resources can be acquired as needed and plant capacity is infinitely expandable and contractible at no cost, then the optimal production schedule consists of producing end products according to the demand schedule, and producing and transporting subassemblies exactly when needed as input to the next assembly process. However, in many supply chain systems, the supply of some raw materials is tightly constrained, with long production and/or procurement lead times. The demand for products fluctuates, both in total volume and in product mix. As a result, just-in-time production is not usually feasible, and when feasible, may result in poor utilization of the supply chain. Four key aspects of this problem are identified as time, uncertainty, cost and customer service level. In these circumstances, the supply chain management optimisation consists of deciding on the best utilization of the available resources in suppliers, factories, depots and dealerships given the different scenarios for the stochastic parameters along the planning horizon.

Problems with the characteristics given above are transformed into mathematical optimization models. Often there are tens of thousands of constraints and variables for a deterministic situation. The problems can be modeled as large-scale linear programs. Given today's Operations Research state-of-the-art tools, deterministic logistics scheduling optimization problems should not present major difficulties for not very large-scale problem solving, at least. However, it has long been recognized (Beale, 1955; Dantzig, 1955) that traditional deterministic optimization is not suitable for capturing the truly dynamic behavior of most real-world applications. The main reason is that such applications involve data uncertainties which arise because information that will be needed in subsequent decision stages is not available to the decision maker when the decision must be made. MAD supply chain planning applications, such as those that this work deals with, exhibit uncertain product demand as well as uncertain procurement and production availability, supply costing and lag time and others. Additionally, the problem has a large-scale nature that makes it difficult, even in its deterministic version.

The aim of this work is to present a novel modeling approach for the MAD supply chain planning optimization problem under uncertainty for very large-scale instances. Although the scheme has been primarily designed for tackling MAD supply chain planning problems in the automotive sector, the approach has a far more reaching application to the very broad supply chain area that deals with multiperiod, multiproduct and multilevel types of problems in manufacturing, assembly and distribution.

The paper is organized as follows. Section 2presents the MAD supply chain planning problem to solve. Section 3gives the notation and the meaning of the main parameters and variables. Section 4presents a concept-oriented mathematical representation of the model. Section 5introduces our modeling framework to treat the uncertainty via scenario analysis. 6 Parameters and variables for the stochastic approach, 7 An implementable-oriented 2-stage stochastic splitting variable modelgive the parameters and variables as well as the implementation-oriented mathematical representation of the deterministic equivalent model for the stochastic version of the problem.

Section snippets

Current state-of-the-art

A global multinational player (e.g., in the automotive sector) would ideally like to take business decisions which span sourcing, manufacturing, assembly and distribution. Thus, a company with multiple suppliers at different levels of the BoM production plants and multiple markets may seek to allocate demand quantities to different plants over the next month, next quarter or next year time horizon. Its objective is to minimize the sum of manufacturing, assembly and distribution supplying costs

Sets

Tset of time periods in the planning horizon
Jset of products
JE⊆Jset of end products
JS⊆Jset of subassemblies (JEJS=∅ and J=JEJS)
JSDJSset of subassemblies with external demand
PGset of product groups
Jh⊆Jset of products that belong to group h, for h∈PG
DSjset of (external) demand sources for product j, for j∈JEJSD, such that DSjDSj=∅ for j, jJEJSD|j≠j
DSDSjfor all j∈JEJSD
Iset of components
IR⊆Iset of raw components, such that IRJS=∅ and I=IRJS
ITIRset of transferable raw components (Note:

Objective functions

Option 1: Optimizing the system resources usage by minimizing the total production costs (by using prime and alternate components in the products' BoM), and the procurement standard and expediting costs as well as lost demand penalization.z1=minj∈Jτ∈TPCj,τZj,τ+j∈Jτ∈Ti∈Ijf∈Ii,jγ3PAi,jf,τZAi,jf,τ+i∈IRτ=1,TsSCsi,τXi,τi∈I+τ=1,TeSCei,τEi,τ+d∈DSτ∈TρdLd,τ,where γ3 is given in Section 3.5, andTs≡|T|−lsi,Te≡|T|−leisubject to , , , , , , , , , , , , , , , , , , , .

Option 2: Optimizing the

General approach

The model described in the previous section can be compacted in the following model structuring:mincTzs.t.Az=p,z⩾0,where c is the vector of the objective function coefficients, A the m × n constraint matrix, p the right-hand side (r.h.s.) m-vector and z the n-vector of the decision variables to optimise. It must be extended in order to deal properly with uncertainty on the values of some parameters. We may employ a technique so-called scenario analysis, where the uncertainty is modelled via a set

Constraint and objective functions related parameters

G is set of scenarios, T1 set of implementable time periods, and T2=TT1 set of non-implementable time periods.

Remark. It is assumed that all parameters are deterministic (i.e., known values) for time period set T1, but the assumption can be very easily removed.

The uncertain parameters to be considered in the model below are related to the production/procurement costs and availability, demand volume and lost fraction, prime and alternate components' effective periods segment, product and

Objective functions

Option 1: Optimizing the system resource usage by minimizing the expected total production cost (by using prime and alternate components in the products' BoM) and the procurement standard and expediting cost as well as the lost demand penalization.z1=ming∈Gwgj∈Jτ∈TPCgj,τZgj,τ+j∈Jτ∈Ti∈Ijf∈Ii,jγg3PAi,jgf,τZAi,jgf,τ+i∈IRτ=1,TsSCsgi,τXgi,τ+i∈Iτ=1,TeSCegi,τEgi,τ+d∈DSτ=TρdδgdBgd,τd∈DSτ=1,TdρdδgdYgd,τ,where γg3 is referred to in Section 6.2, andTs≡|T|−ls,gi,Te≡|T|−le,gi,Td≡|T|−ld

Conclusions

A modeling framework for MAD supply chain management optimization under uncertainty has been presented. The mathematical expressions of certain types of variables have been used to reduce the problem's dimensions. In spite of the constraint matrix density increase, it seems to be a very good implementation-oriented model. However, the DEM for the 2-stage stochastic problem has still such big dimensions that it is impractical to solve it without using some type of decomposition approach. An

Acknowledgements

This work has been partially supported by the Europe Commission within the ESPRIT program HPCN domaine, project ES26267.

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