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Publication# Modeling and mitigating supply chain disruptions as a bilevel network flow problem

Abstract

Years of globalization, outsourcing and cost cutting have increased supply chain vulnerability calling for more effective risk mitigation strategies. In our research, we analyze supply chain disruptions in a production setting. Using a bilevel optimization framework, we minimize the total production cost for a manufacturer interested in finding optimal disruption mitigation strategies. The problem constitutes a convex network flow program under a chance constraint bounding the manufacturer's regrets in disrupted scenarios. Thus, in contrast to standard bilevel optimization schemes with two decision-makers, a leader and a follower, our model searches for the optimal production plan of a manufacturer in view of a reduction in the sequence of his own scenario-specific regrets. Defined as the difference in costs of a reactive plan, which considers the disruption as unknown until it occurs, and a benchmark anticipative plan, which predicts the disruption in the beginning of the planning horizon, the regrets allow measurement of the impact of scenario-specific production strategies on the manufacturer's total cost. For an efficient solution of the problem, we employ generalized Benders decomposition and develop customized feasibility cuts. In the managerial section, we discuss the implications for the risk-adjusted production and observe that the regrets of long disruptions are reduced in our mitigation strategy at the cost of shorter disruptions, whose regrets typically stay far below the risk threshold. This allows a decrease of the production cost under rare but high-impact disruption scenarios.

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Globalization, outsourcing and cost optimization have all contributed to increased supply chain vulnerability, yet our understanding of effective mitigation strategies remains limited. In our research, we study the effects of disruptions on supply chain networks. To do so, we develop in the first research project a bilevel optimization model to analyze supply chain disruptions in a production setting. This results in a convex network flow problem in which total production cost is minimized under a chance constraint. This chance constraint imposes a bound on the regret of disrupted scenarios with high pre-determined probability, where this regret is defined as a cost surplus which results from a comparison between a reactive setting, where we consider the disruption unknown until it occurs, and an anticipative setting, which assumes the disruption scenario to be known at the beginning of the planning horizon. A generalized Benders decomposition approach which makes use of the problem structure is developed to solve the problem efficiently.
In the second research project we study a similar model in which an additional chance constraint on service level is introduced to account for demand uncertainty. We derive an approximation of this model and derive a bound on the approximation error. This approximation model is then solved with the same Benders decomposition procedure as the first model discussed. We obtain managerial insights from both models by means of numerical experimentation. We demonstrate a relationship between the stochastic demand and service level requirements. Moreover, we observe that unused production capacity is a key driver for mitigation inventory.
In the last research project we shift our focus towards gaining a more holistic understanding of the supply chain network disruption literature. The number of articles written in the area has increased significantly in the last few years, and with the advent of the Covid-19 pandemic the interest in the area has expanded even further. We perform a literature review with a particular focus on recognizing research gaps. We observe a surprising lack of articles studying assembly supply chains, despite their ubiquity in real world applications. A similar lack of articles is observed in the area of multi-product supply chains as well. Finally, in light of the ongoing Covid-19 pandemic we shift our attention towards the disruptive effect of pandemics on supply chains. We observe that most of the mathematical models of supply chain networks under disruptions discussed in the literature are incapable of accounting for the fact that pandemics disrupt several aspects of supply chain networks simultaneously. Moreover, we observe that a large number of articles studies problems stemming directly from real world applications. The resulting models are often challenging to solve mathematically, so we perform a comprehensive study of solution methods used in the supply chain network literature and highlight multi-objective optimization as an area of utmost importance for current and future research.

Daniel Kuhn, Wolfram Wiesemann

The purpose of software partitioning is to assign code segments of a given computer program to a range of execution locations such as general-purpose processors or specialist hardware components. These execution locations differ in speed, communication characteristics, and size. In particular, hardware components offering high speed tend to accommodate only few code segments. The goal of software partitioning is to find an assignment of code segments to execution locations that minimizes the overall program run time and respects the size constraints. In this paper we demonstrate that an additional speedup is obtained if we allow code segments to be instantiated on more than one location. We further show that the program run time not only depends on the execution frequency of the code segments but also on their execution order if there are multiply instantiated code segments. Unlike frequency information, however, sequence information is not available at the time when the software partition is selected. This motivates us to formulate the software-partitioning problem as a robust optimization problem with decision-dependent uncertainty. We transform this problem to a mixed-integer linear program of moderate size and report on promising numerical results.

2012Robust dynamic optimization problems involving adaptive decisions are computationally intractable in general. Tractable upper bounding approximations can be obtained by requiring the adaptive decisions to be representable as linear decision rules (LDRs). In this paper we investigate families of tractable lower bounding approximations, which serve to estimate the degree of suboptimality of the best LDR. These approximations are obtained either by solving a dual version of the robust optimization problem in LDRs or by utilizing an inclusion-wise discrete approximation of the problem's uncertainty set. The quality of the resulting lower bounds depends on the distribution assigned to the uncertain parameters or the choice of the discretization points within the uncertainty set, respectively. We prove that identifying the best possible lower bounds is generally intractable in both cases and propose an efficient procedure to construct suboptimal lower bounds. The resulting instance-wise bounds outperform known worst-case bounds in the majority of our test cases.