Partial Distribution Information Research Articles

Designing a resilient supply chain network: A multi-objective data-driven distributionally robust optimization method

Uncertainty of supply chain disruption poses a significant challenge to decision-making in many real-life problems. In this paper, a new multi-objective data-driven distributionally robust optimization (MDDRO) model for the resilient supply chain network (RSCN) design problem under disruption scenario is developed, which includes minimizing the total cost, carbon emissions, and delivery time. In the context of supply chain disruption, it is important self-evidently that products can be supplied normally and the practical demand can be met. In handling the partial probability distribution information of the considered uncertain demand, this paper uses a data-driven Wasserstein-Moment ambiguity set (WMAS), which incorporate the Wasserstein metric and Moment information, and a robust counterpart to transform the developed model into a tractable approximation form. The finite-sample performance guarantee of the optimal solution of MDDRO model with Wasserstein-Moment ambiguous chance constraint is given. An accelerated Branch and cut algorithm (BCA) is constructed to solve the MDDRO model. Finally, through the investigation of a real-life case and sensitivity analysis of key parameters, some managerial insights are put forward.

Direct Mail or Bonded Warehouse? Logistics Mode Selection in Cross-Border E-Commerce under Exchange Rate Risk

The rapid development of cross-border e-commerce (CBEC) has enabled more suppliers to expand into overseas markets, meeting increasingly diverse consumer demands. Selecting an effective logistics mode is a crucial issue for suppliers, yet uncertainty in demand and exchange rate fluctuations make this problem challenging. This paper considers a supply chain consisting of a supplier and a CBEC platform. Using a distributionally robust optimization approach, we provide optimal overseas warehousing strategies and logistics mode selection for suppliers, given partial distribution information of market demand and exchange rate, such as means, variances, and covariances. It is found that when demand and exchange rate fluctuations are small, the supplier chooses the bonded warehouse logistics mode to reduce costs. Conversely, when demand and exchange rate fluctuations increase, the supplier opts for the direct mail mode to respond flexibly to market risk. The correlation between exchange rate and demand also affects the choice of logistics mode. Specifically, with low correlation, the preference is for the bonded warehouse mode, whereas high correlation leads suppliers to choose the direct mail mode. In addition, the impact of demand and exchange rate fluctuations on suppliers’ overseas warehousing volumes depends on the product’s profit margin. These findings provide guidance for the selection of CBEC logistics under exchange rate risk.

Consignment contract in a supply chain with price, demand and yield randomness: A distributionally robust approach

We investigate a supply chain consisting of a supplier and an e-commence platform who sign a consignment contract with uncertainty in price, demand, and yield. Due to the challenges of actually determining the joint distribution of random variables, we assume that only the partial distributional information is available for these three uncertainties. To address the issue of distributional ambiguity and to more accurately account for arbitrary correlations among these uncertainties, we introduce a moment-based distributionally robust optimization (DRO) approach that assumes only the first and second moments of these three uncertainties are known. Given an exogenous consignment rate, we derive the closed-form expression for the supplier’s optimal production decision and her worst-case expected profit. We study the impact of random price and random yield within the ambiguity set on the supplier’s production decision and the worst-case expected profit. From the e-commerce platform’s perspective, we further examine the contract design problem when the consignment rate is endogenous and analyze equilibrium solutions for both contracting parties. Our results show that the additional consideration of price or yield uncertainty within the ambiguity set does not always mitigate or enhance the degree of conservatism of the contracting parties, which is closely related to the correlation between any two types of uncertainty, as well as the degree of uncertainty itself.

Optimising hierarchical emergency logistics network under ambiguous demand and transportation cost

This article studies the response problem of an emergency logistics network with a decision hierarchy relationship under uncertainty. To account for the partial distribution information about uncertain demand and transportation costs, we construct a moment-based ambiguity set based on limited historical data, where the pivot variable method is employed to determine the confidence interval of the mean value. Based on the constructed ambiguity set, we develop a novel distributionally robust bi-level post-disaster emergency logistics location-routeing model. By exploiting the structural characteristic, chance-constrained models under box-ellipsoid and budget perturbation sets are reformulated as bi-level mixed-integer conic programming models. To accelerate the solution procedure, the bi-level models are further converted into single-level ones via Karush-Kuhn-Tucker condition, which can be directly solved to optimality using CPLEX software. Supply risk value for each supplier is obtained by applying analytic hierarchy process. We conduct extensive experiments using the Iranian flood as the case study to address the computational performance of our proposed optimisation method.

Nonlinear robust distribution planning model for perishable products based on sustainable development

With the development of social economy and the improvement of people living standards, the distribution of cold chain products has become increasingly prominent. In the process of distribution of cold chain products, due to its perishable nature, it is not only necessary to consider the efficiency and economy of transportation, but also to pay attention to risk aversion factors. Therefore, the stochastic programming of cold chain logistics based on risk aversion has become an urgent problem to be solved. Aiming to minimize the economic cost and transform the carbon emission level of cold chain transportation into carbon emission cost, a distributionally robust mean-conditional value-at-risk optimization model for the perishable goods distribution management problem is established, in which the partial distribution information of uncertain demand and transportation environment temperature was known. Then a computationally tractable equivalence model is obtained under the box ambiguity set. For the proposed nonlinear model, a trust region sequential quadratic programming method with filter is proposed to solve it. Finally, through a case study, the relationship between the stochastic programming model and the distributionally robust model is analysed.

Distributionally Robust Optimization Under Distorted Expectations

Optimal Decision Making Under Distorted Expectation with Partial Distribution Information Decision makers who are not risk neutral may evaluate expected values by distorting objective probabilities to reflect their risk attitudes, a phenomenon known as distorted expectations. This concept is widely applied in behavioral economics, insurance, finance, and other business domains. In “Distributionally Robust Optimization Under Distorted Expectations,” Cai, Li, and Mao study how decision makers using distorted expectations can optimize their decisions when only partial information about objective probabilities is available. They show that decision makers who are ambiguity averse can optimize their decisions as if they are risk averse with their risk attitudes characterized by a convex distortion function. This finding demonstrates why even non–risk-averse decision makers, such as those studied in the celebrated cumulative prospect theory, may consider it optimal to take risk-averse decisions when facing uncertainty about objective probabilities. Leveraging this finding, the authors show that a large class of distributionally robust optimization problems involving the use of distorted expectations can be tractably solved as convex programs.

Distributionally robust facility location with uncertain facility capacity and customer demand

This study explores a capacitated facility location problem where facility capacity and customer demand are subject to uncertainties simultaneously. This problem decides the subset of facilities to open at the system design phase to serve customers during the operational phase. The objective is to minimize the total cost, including the first-stage location cost and the second-stage recourse cost, and guarantee the system’s reliability, i.e., meeting demand as much as possible when uncertainties arise. A distributionally robust optimization (DRO) framework is utilized to model the problem. A scenario-wise ambiguity set with partial distributional information of random variables is constructed, which can capture uncertainties caused by different random events or different magnitudes of the same event type and explicitly represent the correlation between facilities’ uncertain capacity and customers’ uncertain demand. We apply an adaptation policy to the DRO model and reformulate it to a mixed-integer linear programming model, which is solvable by off-the-shelf solvers. Numerical results show that the scenario-wise DRO framework can provide a better trade-off between cost and service level than the stochastic programming model and the DRO model with a marginal moment-based ambiguity set, demonstrating that the proposed scenario-wise DRO model offers a practical decision-making tool for enhancing supply chain robustness via facility location.

Joint pricing and inventory management under minimax regret

We study the problem of jointly optimizing the price and order quantity for a perishable product in a single selling period, also known as the pricing newsvendor problem, under demand ambiguity. Specifically, the demand is a function of the selling price and a random factor of which the distribution is unknown. We employ the minimax regret decision criterion to minimize the worst‐case regret, where the regret is defined as the difference between the optimal profit that could be obtained with perfect/complete information and the realized profit using the decision made with ambiguous demand information. First, given the interval in which the random factor lies with high probability, we characterize the optimal pricing and ordering decisions under the minimax regret criterion and compare their properties with those in the classical models that seek to maximize the expected profit. Specifically, we explore the impact of inventory risk by comparing the optimal price and the risk‐free price and study comparative statics with respect to the degree of demand ambiguity and the unit ordering cost. We further show that the minimax regret approach avoids the high degree of conservativeness that is often incurred in the application of the commonly used max–min robust optimization approach. Second, when partial distributional information of the random factor is available, we adopt the Wasserstein distance to depict the distributional ambiguity and characterize the set of worst‐case distributions and the maximum regret given the selling price and order quantity. Third, we compare the minimax regret approaches with the traditional profit‐maximization approach in a data‐driven setting. We show via a numerical study that the minimax regret approaches outperform the traditional profit‐maximization approach, especially when the data are scarce, the demand has high volatility, and the number of exercised prices is small. Furthermore, leveraging the partial distributional information of the random factor can further improve the performance of the minimax regret approach.

Application of a Distributionally Robust Optimization Approach for Single-Period Stochastic Inventory Problems with Different Carbon Policies

While single-period stochastic inventory problems have been studied in academia and industry, the impacts of both carbon emissions reduction and partial distribution information have not been investigated. In this paper, we consider carbon emission reduction in a single-period inventory problem where both product price and demand are stochastic. When only partial information on the first and second moments of the two stochastic elements is specified, we use the max–min criterion to construct two optimization models under cap-and-trade and carbon tax policies. For each model, we prove the existence of the optimal robust decisions on production quantity and cleaning equipment investment. By comparing the profit and carbon emission between these two models, we identify certain conditions that improve the economic and environmental performance of the manufacturer. Finally, we conduct a numerical study to test the robustness of the optimal operational decisions under each carbon policy.

Distributionally robust equilibrious hybrid vehicle routing problem under twofold uncertainty

A novel nonlinear hybrid vehicle routing problem is examined, and its nonlinear component is linearised considering the transportation costs associated with electricity and traditional fuel-based driving. Due to the fact that the transportation cost and fuel consumption involve the twofold uncertainty of randomness and fuzziness, and only partial probability distribution information may be available. Therefore, these two parameters are considered as random fuzzy variables with ambiguous probability distributions. An ambiguous equilibrium risk value objective function and an ambiguous equilibrium chance constraint are formulated. Accordingly, a distributionally robust equilibrium approach is proposed, in which the ambiguity sets are used to characterize the ambiguous probability distributions of the random fuzzy variables. Specifically, this paper first applies the central limit theorem to construct the ambiguity sets. Subsequently, the inner ambiguous probability constraint and the outer credibility constraint are derived into their equivalent counterparts, respectively. In this manner, the proposed model is successfully converted into a mixed integer second-order cone programming model, where the conventional branch-and-cut algorithm is adopted to obtain the optimal routing. Finally, the performance of the proposed model and its price of distributional robustness are verified in the numerical experiments. Overall, the main achievements of this paper are summarized as (1) the proposal of a distributionally robust equilibrium optimization model for a nonlinear hybrid vehicle routing problem, (2) the definitions of an ambiguous equilibrium risk value objective function and an ambiguous equilibrium chance constraint under twofold uncertainty, and (3) the derivation of an equivalent second-order cone programming model for computationally solvable.

An integrated approach for a new flexible multi-product disassembly line balancing problem

Flexible disassembly line design for end-of-life (EOL) products is a key issue in the remanufacturing industry. However, existing studies for disassembly line balancing have not simultaneously considered multiple EOL products, the identical parts of these products and uncertainty during disassembly, which are important characteristics of flexible disassembly lines. The present study addresses a new flexible multi-product disassembly line balancing problem in which (1) disassembly schemes need to be selected, (2) a workstation can disassemble multiple EOL products, (3) identical parts of multiple products can be treated as identical tasks, and (4) only partial probability distribution information of processing times is known. For the problem, an integrated approach is developed, which is composed of a chance-constrained program, a distribution-free model, efficient valid inequalities and an exact lifted cut-and-solve method. Numerical experiments are conducted on an illustrative example, 10 instances based on realistic products and 480 randomly generated instances with up to 20 products, 400 tasks and 86 workstations. Computational results show that the proposed valid inequalities can reduce about 75% computational time of the original model, and the lifted cut-and-solve method needs only 17.53% and 40.65% of the computational times required by the CPLEX and the classic cut-and-solve method, respectively.

MD-GAN-Based UAV Trajectory and Power Optimization for Cognitive Covert Communications

This article investigates the covert performance of an unmanned aerial vehicle (UAV) jammer-assisted cognitive radio (CR) network. In particular, the covert transmission of secondary users can be effectively protected by UAV jamming against the eavesdropping. For practical consideration, the UAV is assumed to only know certain partial channel distribution information (CDI), whereas not to know the detection threshold of an eavesdropper. For this sake, we propose a model-driven generative adversarial network (MD-GAN)-assisted optimization framework, consisting of a generator and a discriminator, where the unknown channel information and the detection threshold are learned weights. Then, a GAN-based joint trajectory and power optimization (GAN-JTP) algorithm is developed to train the MD-GAN optimization framework for covert communication, which results in the joint solution of the UAV’s trajectory and transmits power to maximize the covert rate and the probability of detection errors. Our simulation results show that the proposed GAN-JTP with a rapid convergence speed can attain near-optimal solutions of the UAV’s trajectory and transmit power for the covert communication.

Optimizing a robust capital-constrained dual-channel supply chain under demand distribution uncertainty

Dual-channel supply chain (DCSC) is one type of patterns that combines offline channel and online channel in a common market. It is known that demand plays an important role in DCSC. In the literature, however, the demand is assumed to be deterministic or stochastic with completely known probability distribution. In contrast, this paper addresses the uncertain demand from a new perspective by considering the distribution uncertainty, and the uncertainty source comes from subjectivity. When only partial demand distribution information is available, this paper introduces a new uncertainty distribution set to characterize the ambiguous demand distribution. Based on the proposed uncertainty distribution set, a novel distributionally robust bilevel optimization modeling framework is developed for our capital-constrained DCSC. In order to address the upstream manufacturer’s capital constraint, three financing strategies, i.e., bank financing, trade credit and hybrid financing (a combination of bank and equity financing) are considered. An analytically tractable method is developed to obtain the corresponding robust equilibrium solutions under the three financing strategies. Numerical analysis are conducted to demonstrate how the demand ambiguity and the equity ratio affect the manufacturer’s equilibrium financing strategy. The numerical results show that the change of the uncertainty perturbation parameters can change the manufacturer’s financing strategy. When the values of uncertainty perturbation parameters are relatively small, the equilibrium financing strategy is either trade credit or hybrid financing. When the values of uncertainty perturbation parameters are medium or large, bank financing is the equilibrium financing strategy. Also, the manufacturer’s equilibrium financing strategy is affected by the equity financing ratio. When the equity financing ratio is small, the equilibrium financing strategy is either trade credit or hybrid financing. When the equity financing ratio is medium or large, the equilibrium financing strategy is always hybrid financing. The demand uncertainty can affect the manufacturer’s financing strategy. As a result, the capital-constrained manufacturer should take into account the demand uncertainty to make her informed financing decisions.

Distributionally robust ramp metering under traffic demand uncertainty

This study presents a distributionally robust optimization model to address the ramp metering problem with uncertain traffic demand flows. The aim of this model is to minimize the total travel delay of the system based on the macroscopic cell transmission model (CTM) of traffic flow. In our model, the only required data is the partial distributional information of stochastic demand flows. Using the Worst-Case Conditional Value-at-Risk (WCVaR) constraints to approximate the distributionally robust chance constraints, the proposed problem can be conservatively approximated as a semidefinite programming (SDP), which is computationally efficient. The performances of our proposed model are illustrated by practical applications. Experimental results show that the distributionally robust control strategy can achieve reliable performances over a range of uncertain scenarios.

Distributionally Robust MPC for Nonlinear Systems

Classical stochastic model predictive control (SMPC) methods assume that the true probability distribution of uncertainties in controlled systems is provided in advance. However, in real-world systems, only partial distribution information can be acquired for SMPC. The discrepancy between the true distribution and the distribution assumed can result in sub-optimality or even infeasibility of the system. To address this, we present a novel distributionally robust data-driven MPC scheme to control stochastic nonlinear systems. We use distributionally robust constraints to bound the violation of the expected state-constraints under process disturbance. Sequential linearization is performed at each sampling time to guarantee that the system's states comply with constraints with respect to the worst-case distribution within the Wasserstein ball centered at the discrete empirical probability distribution. Under this distributionally robust MPC scheme, control laws can be efficiently derived by solving a conic program. The competence of this scheme for disturbed nonlinear systems is demonstrated through two case studies.

Robust decisions for regulated sustainable manufacturing with partial demand information: Mandatory emission capacity versus emission tax

While emission tax and mandatory emission capacity regulations are widely implemented to control greenhouse gas emissions, it remains unclear which will lead to better performance in an uncertain environment. Furthermore, a regulated sustainable manufacturer with only partial demand information may be challenged by more uncertainties than a manufacturer would be in a free market without regulation. These factors motivate us to examine the effects of two types of emission reduction regulations on the optimal production decisions of a manufacturer facing stochastic demand with partial distribution information. Based on distributionally robust newsvendor models, we use the Hurwicz decision criterion to build robust optimization models for a regulated sustainable manufacturer. We first solve the closed-form expressions of the optimal production quantities for two optimization models; then, we compare their expected profits and carbon emissions. The comparisons show that an emission tax may be more favorable than mandatory emission capacity regulation in terms of the manufacturer’s environmental performance. A numerical study with a Taguchi experiment is performed to compare the two policies and to test the robustness. We find interesting insights: For manufacturers, a decision maker with an optimistic attitude to demand uncertainty, can gain higher profits and emit less carbon, which differs from the results of some previous research. For regulators, with a large carbon cap, mandatory emission capacity regulation can lead manufacturers to gain higher profits and emit more carbon than those achieved under emission tax regulation, which provides new insights for policy makers.

Two-Stage Distributionally Robust Optimization for a Two-Allocation p-Hub Median Problem

This paper presents a novel two-stage distributionally robust optimization model of the two-allocation p-hub median problem with different hub link scales. With the objective of minimizing overall costs of building and operating the hub network, the choices of hub locations and hub link scales are decided in the first stage, while the optimal flows are determined in the second stage once the uncertain demands have been realized. Before establishing the hub network, we just have partial distribution information about the uncertain flow demands, which can be described by a given perturbation set based on the historical information. Due to the ambiguous distributions leading to a computationally intractable model, we reformulate the proposed model into the tractable robust counterpart forms under two types of uncertainty sets (Box[Formula: see text]ellipsoidal perturbation set and Generalized ellipsoidal perturbation set). Finally, to demonstrate the effectiveness and applicability for our model, we conduct a case study for the express network system in the Beijing–Tianjin–Hebei region.

Distributionally robust and risk-averse optimisation for the stochastic multi-product disassembly line balancing problem with workforce assignment

Existing works usually focus on the single-product disassembly line balancing problem (DLBP). In practice, end-of-life (EOL) products to be disassembled may be heterogeneous, and the actual processing time of each task may vary with its assigned worker. This work studies a stochastic multi-product DLBP with workforce assignment, to minimise the system cost. Due to historical data scarcity, we assume that only partial distributional information of uncertain task processing times is known. Exceeding the preset cycle time may lead to a disassembly performance reduction, thus we control the cycle time violation via conditional Value-at-Risk (CVaR) constraints, i.e. in a risk-averse fashion. For the problem, we first propose a novel formulation with distributionally robust CVaR constraints. Then some valid inequalities are proposed, leading to an improved model. Two solution approaches, i.e. an exact cutting-plane method and an approximation method, are further proposed and compared, via numerical experiments. Some managerial insights are also drawn.

A distributionally robust optimisation model for last mile relief network under mixed transport

The last mile relief network is the final stage of the relief chain but the most critical stage for ensuring the timely delivery of relief supplies after a disaster. Due to the suddenness of the disaster, balancing the shortages of relief supplies and the high demands of victims is a serious problem. We introduce a mixed transport way of relief supply transportation between points of distributions and demand nodes in our problem to face the manpower and resource limitations. We establish a bi-objective distributionally robust optimisation model to balance transportation time and transportation safety, where the demand, transportation time, freight and safety coefficient are assumed to be uncertain variables with partial distribution information. We also deduce the refinement robust counterparts under the ambiguous sets to prove the safe tractable approximations of chance constraints. Finally, we conduct a case study of Tonghai county earthquake to illustrate the efficiency of our proposed distributionally robust model.

Multi-period dynamic distributionally robust pre-positioning of emergency supplies under demand uncertainty

The pre-positioning problem is an important part of emergency supply. Pre-positioning decisions must be made before disasters occur under high uncertainty and only limited distribution information. This study proposes a distributionally robust optimization model for the multi-period dynamic pre-positioning of emergency supplies with a static pre-disaster phase and a dynamic post-disaster phase. In the post-disaster phase, the uncertain demands are time varying and have partial distribution information that belongs to a given family of distributions. The family of distributions is described by a given perturbation set based on historical information. Therefore, the proposed model forms a semi-infinite programming problem with ambiguous chance constraints, which typically would be computationally intractable. We refine the bounded perturbation sets (box, box-ball and box-polyhedral) and develop computationally tractable safe approximations of the chance constraints. Finally, a realistic application to the Circum-Bohai Sea Region of China is presented to illustrate the effectiveness of the robust optimization model.