Yehua Wei

Understanding the Performance of the Long Chain and Sparse Designs in Process Flexibility

The long chain has been an important concept in the design of flexible processes. This design concept, as well as other sparse designs, have been applied by the automotive and other industries as a way to increase flexibility in order to better match available capacities with variable demands. Numerous empirical studies have validated the effectiveness of these designs. However, there is little theory that explains the effectiveness of the long chain, except when the system size is large, i.e., by applying an asymptotic analysis. Our attempt in this paper is to develop a theory that explains the effectiveness of long chain designs for finite size systems. First, we uncover a fundamental property of long chains, supermodularity, that serves as an important building block in our analysis. This property is used to show that the marginal benefit, i.e., the increase in expected sales, increases as the long chain is constructed, and the largest benefit is always achieved when the chain is closed by adding the last arc to the system. Then, supermodularity is used to show that the performance of the long chain is characterized by the difference between the performances of two open chains. This characterization immediately leads to the optimality of the long chain among 2-flexibility designs. Finally, under independent and identically distributed i.i.d. demand, this characterization gives rise to three developments: i an effective algorithm to compute the performances of long chains using only matrix multiplications; ii a result that the gap between the fill rate of full flexibility and that of the long chain increases with system size, thus implying that the effectiveness of the long chain relative to full flexibility increases as the number of products decreases; iii a risk-pooling result implying that the fill rate of a long chain increases with the number of products, but this increase converges to zero exponentially fast.

Identifying Risks and Mitigating Disruptions in the Automotive Supply Chain

Firms are exposed to a variety of low-probability, high-impact risks that can disrupt their operations and supply chains. These risks are difficult to predict and quantify; therefore, they are difficult to manage. As a result, managers may suboptimally deploy countermeasures, leaving their firms exposed to some risks, while wasting resources to mitigate other risks that would not cause significant damage. In a three-year research engagement with Ford Motor Company, we addressed this practical need by developing a novel risk-exposure model that assesses the impact of a disruption originating anywhere in a firms supply chain. Our approach defers the need for a company to estimate the probability associated with any specific disruption risk until after it has learned the effect such a disruption will have on its operations. As a result, the company can make more informed decisions about where to focus its limited risk-management resources. We demonstrate how Ford applied this model to identify previously unrecognized risk exposures, evaluate predisruption risk-mitigation actions, and develop optimal postdisruption contingency plans, including circumstances in which the duration of the disruption is unknown.

Belief Propagation for Min-Cost Network Flow: Convergence and Correctness

Distributed, iterative algorithms operating with minimal data structure while performing little computation per iteration are popularly known as message passing in the recent literature. Belief propagation (BP), a prototypical message-passing algorithm, has gained a lot of attention across disciplines, including communications, statistics, signal processing, and machine learning as an attractive, scalable, general-purpose heuristic for a wide class of optimization and statistical inference problems. Despite its empirical success, the theoretical understanding of BP is far from complete. With the goal of advancing the state of art of our understanding of BP, we study the performance of BP in the context of the capacitated minimum-cost network flow problem---a cornerstone in the development of the theory of polynomial-time algorithms for optimization problems and widely used in the practice of operations research. As the main result of this paper, we prove that BP converges to the optimal solution in pseudopolynomial time, provided that the optimal solution of the underlying network flow problem instance is unique and the problem parameters are integral. We further provide a simple modification of the BP to obtain a fully polynomial-time randomized approximation scheme (FPRAS) without requiring uniqueness of the optimal solution. This is the first instance where BP is proved to have fully polynomial running time. Our results thus provide a theoretical justification for the viability of BP as an attractive method to solve an important class of optimization problems.

Worst-Case Analysis of Process Flexibility Designs

Theoretical studies of process flexibility designs have mostly focused on expected sales. In this paper, we take a different approach by studying process flexibility designs from the worst-case point of view. To study the worst-case performances, we introduce the plant cover indices PCIs, defined by bottlenecks in flexibility designs containing a fixed number of products. We prove that given a flexibility design, a general class of worst-case performance measures can be expressed as functions of the designs PCIs and the given uncertainty set. This result has several major implications. First, it suggests a method to compare the worst-case performances of different flexibility designs without the need to know the specifics of the uncertainty sets. Second, we prove that under symmetric uncertainty sets and a large class of worst-case performance measures, the long chain, a celebrated sparse design, is superior to a large class of sparse flexibility designs, including any design that has a degree of two on each of its product nodes. Third, we show that under stochastic demand, the classical Jordan and Graves JG index can be expressed as a function of the PCIs. Furthermore, the PCIs motivate a modified JG index that is shown to be more effective in our numerical study. Finally, the PCIs lead to a heuristic for finding sparse flexibility designs that perform well under expected sales and have lower risk measures in our computational study.

A criterion for the half-plane property

We establish a convenient necessary and sufficient condition for a multiaffine real polynomial to be stable, and use it to verify that the half-plane property holds for seven small matroids that resisted the efforts of Choe, Oxley, Sokal, and Wagner [Y.-B. Choe, J.G. Oxley, A.D. Sokal, D.G. Wagner, Homogeneous polynomials with the half-plane property, Adv. Appl. Math. 32 (2004) 88-187].

Increasing Supply Chain Robustness through Process Flexibility and Inventory

We study a hybrid strategy that uses both process flexibility and inventory for supply chain risk mitigation. The interplay between process flexibility and inventory is modeled as a two-stage robust optimization problem; in the first stage, the firm allocates inventory, and in the second stage, after uncertainties are realized, the firm schedules its flexible production to minimizes its cost. By taking advantage of the structure of the second stage recourse problem, we develop an effective constraint generation algorithm for solving the two-stage robust optimization problem. To further improve the algorithms performance, we develop a method for generating Pareto cuts. We apply our algorithm numerically to study risk mitigation problems in automotive and clothing industries. Our numerical analysis provides insights regarding the impact of process flexibility on total inventory level and inventory allocation pattern. Finally, we verify our insights for a general class of uncertainty sets under a canonical family of flexibility designs known as the K-chains.

Analyzing Process Flexibility: A Distribution-Free Approach with Partial Expectations

Abstract We develop a distribution-free model to evaluate the performance of process flexibility structures when only the mean and partial expectation of the demand are known. We characterize the worst-case demand distribution under general concave objective functions, and apply it to derive tight lower bounds for the performance of chaining structures under the balanced systems (systems with the same number of plants and products). We also derive a simple lower bound for chaining-like structures under unbalanced systems with different plant capacities.

Retailing with Opaque Products

An opaque product is a product where the retailer hides one or more attributes of the product until after it has been sold. Historically, opaque products have been used in the hotel and airline industries where customers can purchase a room or flight without knowing the brand name. In this work, we show that using opaque products in the retail industry, where there are nonperishable goods and supply chain costs, may also be beneficial. Although opaque products have traditionally been used as a vehicle for price discrimination, we show that opaque products can also be used as a vehicle for inventory management and therefore lower supply chain costs.In our model, customers arrive dynamically and choose between two substitutable products and an opaque product, which is sold at a discount. Customers that purchase the opaque product will be allocated one of the two substitutable products at the discretion of the retailer. For serving demand, joint replenishment and holding costs are incurred by the retailer. We prove that offering an opaque product can strictly improve the profit of the retailer, even without altering the prices of the original products. We also provide a simple method to estimate demand in an opaque selling environment based on the demand without an opaque product, which may be of independent interest. Finally, we analytically show that using opaque products to balance inventory has dramatic cost advantages, even when the amount of opaque products sold is relatively small.