Stochastic Programming Model Research Articles

Green economic analysis for multi‐energy system with linearized CO2 electrolysis model

AbstractCO2 electrolysis system (CES), as a type of carbon capture, utilization, and storage, has significant potential for achieving carbon neutrality. By utilizing surplus renewable energy, CES can convert CO2 into valuable electrochemical products. This paper studies the advantages brought by the application of CES to the green economic operation of multi‐energy system. Firstly, a linear model based on the CES's operation data is developed to characterize its dynamic behaviour. In addition, a two‐stage stochastic programming model for the operation of CES within multi‐energy system is studied. Finally, a creative optimal‐operation‐based end‐of‐life net present value (OENPV) is proposed to assess the long‐term investment of CES. The results of the cases in different seasons and electrochemical products show that the application of CES not only brings significant carbon reduction to the multi‐energy system, but also alleviates the curtailment of renewable energy. Additionally, according to the OENPV analysis outcome, the production of formic acid gives the system the shortest return on investment, which is the fourth year of the 20‐year assessment cycle.

Market structure, ESG performance, and corporate efficiency: Insights from Brazilian publicly traded companies

AbstractUsing a sample of Brazilian listed companies, our study investigates the directional cause–effect relationship between market structure, ESG performance, and firm efficiency under a Stochastic Structural Relationship Programming (SSRP) model. Our empirical evidence is threefold. First, our findings indicate that firms with better environmental performance are more efficient, whereas lower ESG performance and poorer corporate governance practices are associated with a higher level of efficiency. Second, our study suggests that market structure measures (i.e., competition, concentration, and market power) have heterogeneous impacts on various ESG indexes. Specifically, higher market competition is associated with a lower concentration, better ESG performance and environmental performance, but worse corporate governance performance, although market power can only enhance the environmental and governance performance of firms. Third, the market structure proxies employed in this study are significantly attributed to firm efficiency. Our findings provide practical implications for various stakeholders and suggest avenues for future studies that can build on our evidence.

Multistage stochastic programming for the closed‐loop supply chain planning with mobile modules under uncertainty

AbstractTraditional supply chains usually follow fixed facility designs which coincide with the strategic nature of supply chain management (SCM). However, as the global market turns more volatile, the concept of mobile modularization has been adopted by increasingly more industrial practitioners. In mobile modular networks, modular units can be installed or removed at a particular site to expand or reduce the capacity of a facility, or relocated to other sites to tackle market volatility. In this work, we formulate a mixed‐integer linear programming (MILP) model for the closed‐loop supply chain network planning with modular distribution and collection facilities. To further deal with uncertain customer demands and recovery rates, we extend our model to a multistage stochastic programming model and efficiently solve it using a tailored stochastic dynamic dual integer programming (SDDiP) with Magnanti‐Wong enhanced cuts. Computational experiments show that the added Magnanti‐Wong cuts in the proposed algorithm can effectively close the gap between upper and lower bounds, and the benefit of mobile modules is evident when the temporal and spatial variability of customer demand is high.

An optimization model for express delivery with high-speed railway

With the expansion of the high-speed railway (HSR) network in China, high-speed rail express delivery (HSReD) is being used to satisfy the increasing demand for express cargo. The decisions on transportation resources arrangement and freight flow allocation are two of the key issues for practical implementation of HSReD. In this study, we examine the above key issues by developing a two-stage stochastic integer linear programming model to maximize the expected net operation profit of HSReD. A meta-heuristic solution approach introduced some tailored tactics is proposed to speed up the process of solving the above model in the large-scale instances. Numerical experiments based on different sizes and practical investigation on China Railway Nanchang Group are conducted to validate the effectiveness of the proposed model and solution approach. Some managerial implications are also obtained based on the sensitivity analysis, which may be potentially useful for optimizing the daily operation management of HSReD.

Towards a responsive-sustainable-resilient tea supply chain network design under uncertainty using big data

Nowadays, issues such as limited natural resources, environmental problems, social matters, and significance of resilience in agricultural supply chain (ASC) have dragged considerable attention worldwide. In this research, a five-level multi-objective stochastic mixed-integer linear programming model is designed for tea supply chain (TSC) in Iran. The objective functions of the suggested network are minimizing total costs of the supply chain (SC), the total water consumption, and non-resilience measures, and maximizing job opportunities of facilities. Literally, considering uncertainty for SC networks is extremely beneficial due to the existence of some variations in different parameters like demand. As a consequence, a robust possibilistic optimization (RPO) is implemented to manage the uncertainty. Due to the nature of the multi-objective optimization problem, the weighted-normalized-extended goal programming (WNEGP) approach is employed to solve the model. In order to credit the model, real data is collected from the tea organization of Iran. It is worth mentioning that parameters are gathered according to three aspects of big data: volume, velocity, and variety. The results validated the functionality of the model regarding planning strategy. In addition, it showed applying more costs on SC triggers an effective sustainable-resilient-responsive network. In terms of managerial insights, this study offers a far-reaching perspective to managers especially in ASC to develop their industries. Finally, some sensitivity analyses are discussed on key parameters such as demand, robustness coefficients, and also the value of the objective functions in various states. It is worth mentioning that sensitivity analyses on different states of the problem show how sustainability and resiliency affect the supply chain efficiency.

Medium-term multi-stage distributionally robust scheduling of hydro–wind–solar complementary systems in electricity markets considering multiple time-scale uncertainties

Joint trading of hydro–wind–solar complementary systems (HWSCSs) in the electricity market (EM) helps to reduce the imbalance cost and increase profits. However, multiple energy resources and market price uncertainties affect the trading strategies. Existing medium-term (MT) scheduling approaches assume that the probability distribution of the random variable is perfectly known. Short-term variations were also ignored, which led to revenue loss and trading risk. To address the above issues, this paper proposes an MT multi-stage distributionally robust optimization (MDRO) scheduling approach for a price-taking HWSCS in the EM. Firstly, hourly unit commitment (HUC) constraints are incorporated into the MT scheduling model to accurately capture short-term variations. A novel ambiguity set is designed based on the modified chi-square distance to address probability distribution uncertainties at two different time scales. Subsequently, an MDRO scheduling model is proposed to optimize the trading strategy. Finally, the proposed MDRO model is converted to a large-scale multi-stage integer programming problem based on linearization and reformation. The stochastic dual dynamic integer programming algorithm is modified to ensure computational tractability. Xiluodu-Xiangjiaba HWSCS, located in the Jinsha River in China, was selected as a case study. The results show that: 1) the MDRO model is more robust to distributional uncertainties than the multi-stage stochastic programming (MSSP) model. When the probability distribution of the random variable changes, the MDRO model yields a higher expected revenue (+2.43%) and a lower standard deviation (-60.8%) of revenue, which illustrates lower trading risk. 2) Compared with MSSP, deterministic, two-stage stochastic programming, and distributionally robust optimization models, the MDRO model exhibits the best out-of-sample performance in terms of the highest expected revenue and lowest trading risk. 3) Incorporating HUC constraints into the MDRO model helps to increase the total revenue (+3.53%) and energy generation (+3.31%) at the expense of increasing the computational burden.

Review And Synthesis of Sustainable Land Management on Small Family Farms in Developing Countries

Despite efforts to restore land and fight food insecurity, sustainable land management (SLM) technologies are not yet widely adopted in developing countries. This article reviews and synthesizes studies that have been conducted in developing countries over the last few decades. A total of 145 documents were used in this synthesis. A key finding of this paper is the identification of a theoretical gap that provides information for future studies on SLM practices.Another one is that, studies focusing on adoption in this synthesis have not considered awareness of technology as a key part of the adoption process. Most econometric studies focus on econometric regression and consider SLM adoption as a dichotomous choice. The use of mathematical models such as existing dynamic Discrete Stochastic Programming (DSP) models could provide more robust results to overcome this limitation observed in statistical regression results. Additionally, it would be interesting to analyze smallholder farmers’ preferences for SLM practices because very little is currently known about their preferences.

A multi-stage stochastic programming approach for pre-positioning of relief supplies considering returns

A dynamic pre-positioning problem is proposed to efficiently respond to victims’ need for relief supplies under uncertain and dynamic demand in humanitarian relief. The problem is formulated as a multi-stage stochastic programming model that considers pre-positioning with the dynamic procurement and return decisions about relief supplies over a time horizon. To validate the advantages of dynamic pre-positioning, three additional pre-positioning strategies are presented: pre-positioning with one-time procurement and without returns, pre-positioning with one-time procurement and returns, and pre-positioning with dynamic procurement and without returns. Using data from real-world disasters in the United States in the Emergency Events Database, we present a numerical analysis to study the applicability of the proposed models. We develop a sample average approximation approach to solving the proposed model in large-scale cases. Our main contribution is that we integrate dynamic procurement and return strategies into pre-positioning to decrease both costs and shortage risks in uncertain and dynamic contexts. The results illustrate that dynamic pre-positioning outperforms the other three strategies in cost savings. It also indicates that a higher return price is particularly helpful for decreasing unmet demand. The proposed models can help relief agencies evaluate and choose the solutions that will have the greatest overall effectiveness in the context of different relief practices.

Geographical relevance-based multi-period optimization for e-commerce supply chain resilience strategies under disruption risks

With the fierce competition in e-commerce, e-tailers are required to rapid responses to a variety of customised orders with multiple frequencies and strict delivery times. The delay or insufficient supply caused by disruptions might result in lost sales during long-term processes. To address this problem, a two-stage stochastic programming model considering profits, consumer service level (CSL) as well as market priorities is developed to manage long-term disruptions. We analyse multi-period consumer transaction data and formulate geographical relevance (GR) to link each marketplace with historical data in related regions and then prioritise market segments. A GR-based two-stage stochastic programming with multi-period is proposed, which (1) considers both proactive mitigation decisions before disruption and reactive recovery plans after disruption; (2) collaborates three resilience strategies; (3) optimises the e-tailer's profits considering market priorities during long-term disruptions. Using a real case of Chinese e-commerce under the COVID-19 pandemic, it is illustrated (1) the applicability and performance of the proposed GR-based model for multi-period resilience optimisation improving both the CSL and the total profit; (2) the efficiency and robustness of the developed sequential particle swarm optimisation with social structures algorithm. The proposed method could optimise e-tailers' response strategies for managing long-term disruptions in practice.

Modeling Multi-Objective Optimization with Updating Information on Humanitarian Response to Flood Disasters

Unpredictable natural disasters brought by extreme climate change compound difficulties and cause a variety of systemic risks. It is thus critical to provide possibilistic scheduling schemes that simultaneously involve emergency evacuation and relief allocation. But the existing literature seldom takes emergency evacuation and relief supplies as a joint consideration, nor do they explore the impact of an unpredictable flood disaster on the scheduling scheme. A multi-stage stochastic programming model with updating information is constructed in this study, which considers the uncertainty of supply and demand, road network, and multiple types of emergency reliefs and vehicles. In addition, a fuzzy algorithm based on the objective weighting of two-dimensional Euclidean distance is introduced, through moderating an effect analysis of the fuzzy number. Computational results show that humanitarian equity for allocating medical supplies in the fourth period under the medium and heavy flood is about 100%, which has the same as the value of daily and medical supplies within the first and third period in the heavy scenarios. Based on verifying the applicability and rationality of the model and method, the result also presents that the severity of the flood and the fairness of resources is not a simple cause-and-effect relationship, and the consideration of survivor is not the only factor for humanitarian rescue with multi-period. Specifically, paying more attention to a trade-off analysis between the survival probability, the timeliness, and the fairness of humanitarian service is essential. The work provides a reasonable scheme for updating information and responding to sudden natural disasters flexibly and efficiently.

Designing a centralized distribution system for omni‐channel retailing

We study the optimal supply chain design for a dual‐channel retailer that operates physical and web‐based stores and has traditionally managed each one separately. With increased pressures from customers and decreased profits due to marked‐down leftover inventories in each channel, the retailer considers integrating the supply chain operations of both channels and using in‐store inventory more effectively across different channels. In contrast to the existing literature, we focus on the firm's choice of the best omni‐channel strategies considering demand segmentation, cost structure, and, more importantly, the inventory rationing ability of the firm. We formulate our problem as a two‐stage stochastic programming model and use first‐order optimality conditions to study the optimal inventory ordering decisions. Based on different omni‐channel strategy decisions, we explore four supply chain design options. We identify the optimal inventory ordering policy under each option and explicitly describe the optimality conditions on cost and demand under perfect and imperfect demand information. When the demand information is imperfect, the demand fulfillment decisions can be characterized as nested protection‐level policies. First, we show that omni‐channel strategies are not necessarily profitable under all settings as dictated by market conditions. Second, we demonstrate that when the company can allocate its inventory perfectly, running omni‐channel strategies may help better serve web‐based customers and in‐store customers. Finally, we show that, in the worst case, the imperfect demand information may result in losing all the claimed inventory integration benefits when running omni‐channel strategies.

A two-stage stochastic location–routing problem for electric vehicles fast charging

Electric cars are projected to become the vehicles of the future. A major barrier for their expansion is range anxiety stemming from the limited range a typical electric vehicle can travel. Electric vehicle batteries’ performance and capacity are affected by many factors. In particular, the decrease in ambient temperature below a certain threshold will adversely affect the battery’s efficiency. This paper develops a two-stage stochastic program model for charging stations’ optimal location to facilitate the routing decisions of delivery services that use electric vehicles while considering the variability inherent in climate and customer demand. A novel solution approach based on the progressive hedging algorithm is presented to solve the resulting mathematical model and to provide high-quality solutions within reasonable running times for problems with many scenarios. To evaluate the proposed formulation and solution approach’s performance, Fargo city in North Dakota is selected as a testbed. We observe that the location–routing decisions are susceptible to the electric vehicle logistics underlying climate, signifying that decision-makers of the direct current fast charging electric vehicle logistic network for cities that suffer from high-temperature fluctuations would not overlook the effect of climate to design and manage the respective logistic network efficiently.

A two-stage stochastic optimization framework to allocate operating room capacity in publicly-funded hospitals under uncertainty.

Surgery demand is an uncertain parameter in addressing the problem of surgery block allocations, and its typical variability should be considered to ensure the feasibility of surgical planning. We develop two models, a stochastic recourse programming model and a two-stage stochastic optimization (SO) model with incorporated risk measure terms in the objective functions to determine a planning decision that is made to allocate surgical specialties to operating rooms (ORs). Our aim is to minimize the costs associated with postponements and unscheduled demands as well as the inefficient use of OR capacity. The results of these models are compared using a case of a real-life hospital to determine which model better copes with uncertainty. We propose a novel framework to transform the SO model based on its deterministic counterpart. Three SO models are proposed with respect to the variability and infeasibility of the measures of the objective function to encode the construction of the SO framework. The analysis of the experimental results demonstrates that the SO model offers better performance under a highly volatile demand environment than the recourse model. The originality of this work lies in its use of SO transformation framework and its development of stochastic models to address the problem of surgery capacity allocation based on a real case.

Minimizing the Expected Renewable Resource Costs in a Project with Stochastic Resource Availability

In this paper, we study the well-known resource availability cost problem with stochastic resource availability. The objective is to determine the initial levels of all renewable resources and establish a schedule corresponding to each scenario such that the expected resource availability cost is minimized. We assume that resource shortfalls can be compensated externally but at a noticeable higher cost. We formulate the problem as a two-stage stochastic programming model (TSSPM). We also develop an exact decomposition-based algorithm (DBA) for the particular case of the problem with at most two resources, which also functions as a heuristic for the original problem. Since the number of scenarios influences the performance of the developed solution approaches, we utilize a fast scenario reduction method to reduce the number of scenarios. Computational results indicate that the DBA outperforms the TSSPM formulation in solution quality and CPU runtime.

Integrating pre and post-disaster activities for designing an equitable humanitarian relief supply chain

The design of an efficient humanitarian relief supply chain (HRSC) for disaster management depends on the integration of pre- and post-disaster decisions, which reduces the suffering of the victims and saves lives. The use of strategies such as quantity flexibility contracts in the pre-disaster phase and evacuation management in the post-disaster phase significantly increase the efficiency of HRSCs for disaster management. Therefore, in this paper, for the first time, a comprehensive two-stage stochastic mixed-integer linear programming model is presented for integrating the pre- and post-disaster activities for possible disasters such as floods and earthquakes by considering quantity flexibility contract and equitable relief goods distribution. In addition, the proposed model determines the location of warehouses and inventory levels in these warehouses in the pre-disaster phase. It also deals with public donations and budget planning, planning vehicles and helicopters, locating hospitals and shelters, and optimizing the inventory flow between echelons in the post-disaster phase. In the proposed model, the quantity flexibility contract is applied to decrease the inventory level in the pre-disaster phase, and reduce the supply risk in the post-disaster phase. The efficiency and performance of the proposed model are evaluated using the data related to a possible earthquake in Karaj (one of the metropolises of Iran).

Handling uncertainty in the quay crane scheduling problem: a unified distributionally robust decision model

AbstractQuay cranes are among the most important resources in port terminals, and their efficient use is crucial for port terminals to remain competitive in the market. The problem of determining the sequence of tasks performed by the quay cranes that minimizes the turnaround time of a vessel is an NP‐hard problem known as the quay crane scheduling problem (QCSP).In this paper, we consider the unidirectional QCSP under uncertain processing times, where the cranes are only allowed to move in a specific direction. We start by presenting the first distributionally robust optimization (DRO) model for this problem and an exact decomposition algorithm to solve it. The proposed DRO model makes it possible to derive a stochastic programming model and a robust optimization model for the unidirectional QCSP by an appropriate choice of the risk‐averse parameter of the model. Through extensive numerical results, we compare these three approaches—stochastic programming, robust optimization, and DRO—to investigate whether significant differences among the solutions obtained exist. Finally, we propose a new method for helping practitioners to determine a representative set of different DRO solutions.

A recipe for an omnichannel warehouse storage system: Improving the storage efficiency by integrating block stacking and racking

Warehouses in multi-channel retailing are expected to possess storage capacity based on the purpose of operations. However, the inefficient use of storage capacity due to omnichannel retail restructuring, wherein mixed-storage items feature both online and offline, has prompted us to revisit the nature of existing storage systems (block stacking and racking). Most warehouses continue to improvise and undertake critical decision-making with regard to profiling balanced storage systems between block stacking and racking, including the efficient slotting of unit-load items across storage systems. This study aimed to establish a framework to systemically improve storage efficiency by achieving an optimum balance between block stacking and racking under inventory uncertainty – lot size and item dimensions. This study ingeniously proposes a two-stage stochastic programming model for robust optimisation during long-run profiling (redesign) and the multi-dimensional knapsack problem for achieving minimum space wastage during slotting (reshuffle). The study adopted a two-fold approach. First, the hypothetical directions for a balanced system, combining increases in buildings’ ceiling heights with its tendency to store low-volume, high-mix items, were examined. Second, the applicability of the two methods – redesign and reshuffle – was examined by considering a real-life case study and the life cycle of warehouse storage design. It was found that the reshuffling approach can resourcefully serve as a short-term solution for warehouses with imminent space issues. Redesigning also demonstrated a long-term space gain in floor space, thus minimising space wastage.

Optimal and heuristic policies for production and inventory controls in dual supply chains with fluctuating demands

AbstractIn the context of fast‐fashion business, we study the joint control of production, inventory, and unilateral transshipment in dual supply chains in which one supply chain is a responsive supply chain for fashion products and the other is an efficient supply chain for basic products in a fluctuating demand environment (FDE). Our research focuses on: (1) formulating an appropriate model for this complex problem; (2) characterizing the structured optimal policy, which can be used to develop effective heuristic policies; and (3) generalizing our studies for more applications. To address these issues, we formulate a discrete‐time finite‐horizon stochastic dynamic program model in which FDE is represented by a world state evolving in accordance with the discrete‐time Markov chain. The supply chain for the fashion product is modeled by a two‐stage tandem production system with a make‐to‐stock (MTS) stage for the fabric and a subsequent MTS stage for finished fashions. In parallel, the supply chain for the basic products is also modeled by a two‐stage tandem production system with a MTS stage for the fabric and a subsequent make‐to‐order (MTO) stage for finished basics. Further, the two supply chains are coupled via the unilateral transshipment of fabric with random transshipment time. By value iteration, we characterize the optimal policy for the single‐product problem with a set of monotone switching surfaces. The results are extended to the multi‐product multi‐supplier case. Further, based on the structure of the optimal policy, we develop a heuristic policy for the multi‐dimensional dynamic program model. From a practice perspective, we demonstrate the value of transshipment and controlled transshipment, analytically and numerically, and therefore justify the establishment of two separate local manufacturing bases in dual supply chains. Additionally, to achieve both quick response and efficiency, we show that the total cost can be reduced significantly if sufficient fabric is prepared just in time (JIT) for in‐season production at the beginning of a season.

Rice irrigation water efficiency improvement: An AquaCrop-based optimization modeling approach

Rice is the cereal crop that consumes the largest amount of water in the world. To improve the water use efficiency of rice farming and promote the sustainable development of rice production, this study proposes a novel AquaCrop-based optimization modeling approach. First, an AquaCrop-based multiobjective model is constructed to generate dynamic water production functions (WPFs). The NSGA-II algorithm is employed to solve the model, and quadratic dynamic WPFs for rice under different hydrological years (wet, normal and dry) are generated based on the obtained Pareto fronts, which can improve the modeling efficiency and help reflect the complicated reality of water flows. Second, a fuzzy credibility-constrained stochastic multiobjective programming (FCC-SMOP) model is developed for irrigation water allocation based on the acquired WPFs. The model addresses the randomness of precipitation and runoff, the fuzziness of decision-makers' preferences, and conflicting objectives of water savings, benefit increments, and water demand during the critical period of rice growth. The proposed simulation-optimization approach was applied to the Changgang irrigation district in northeast China. Optimal irrigation amounts under different decision-making preference scenarios (including conservative, moderate and positive) and different hydrological years were simultaneously generated. The following conclusions can be drawn: (1) The AquaCrop-based multiobjective method is capable of generating WPFs of different hydrological years with sparse discontinuous field experiments; (2) the decision-making preferences of decision-makers significantly impact the economic benefits, water use efficiency, and practicality of schemes; and (3) the FCC-SMOP model successfully makes tradeoffs among the three objectives under multiple uncertainties and can further improve the system efficiency and increase planting income in the Changgang irrigation district when its irrigation facilities become optimal.

A robust stochastic possibilistic programming model for dynamic supply chain network design with pricing and technology selection decisions

A robust stochastic possibilistic programming model for dynamic supply chain network design with pricing and technology selection decisions