Stochastic Demand Research Articles

Joint optimization of overbooking and seat allocation for high-speed railways considering stochastic demand

To mitigate empty seat loss caused by random passenger no-show behavior, this study extends seat allocation to joint optimization of overbooking and seat allocation for high-speed railways (HSR). Assuming that stochastic passenger demand follows a specific distribution and considering various constraints, including train capacity, demand, and denied boarding rate constraints, a nonlinear stochastic programming model for joint optimization of overbooking and seat allocation for HSR is constructed with the aim of maximizing railway expected revenue. To solve this optimization model, a multi-level optimization algorithm is designed. Based on the sampling averaging approximation method, demand scenarios and passenger no-show scenarios are generated and the optimization problem is decomposed, including the joint optimization of overbooking and seat allocation under a single demand scenario, and the ticket adjustment under other demand scenarios. For the former, it is further divided into two sub-problems according to the stochastic nature of passenger no-show behavior, which is optimized iteratively. Finally, the effectiveness of the proposed model and algorithm is evaluated through numerical studies. The results demonstrate that the proposed joint optimization method effectively addresses the randomness of passenger demand and no-show behavior, thereby improving HSR expected revenue and making up for the empty seat loss resulting from passenger no-show behavior.

Dynamic pricing and inventory model for perishable products with reference price and reference quality

Reference price (quality) is a benchmark point used by consumers to make price (quality) judgements. Combining reference price and reference quality with dynamic pricing and inventory management, this paper applies differential equations theory to construct a optimal control model for perishable products when the quality of products declines exponentially. The demand of products depends on price, quality, reference price and reference quality, among which reference price and reference quality are influenced by consumers’ past memory. The aim is to maximize the retailer’s profit during the period. The conclusions are as follows. Firstly, a optimal dynamic pricing and inventory model for perishable products with reference price and reference quality is constructed, and the model is extended to dual decision variables, stochastic demand, time-dependent effects, competitive and infinite planing horizon setups. Secondly, through the Pontryagin maximum principle, the analytical expression of the optimal dynamic price is derived. Thirdly, a linear search method to solve the optimal static price is proposed. Finally, the sensitivity of the main parameters is analyzed and the corresponding management enlightenments are given. By comparison of dynamic and static pricing, we find that dynamic pricing can achieve more profits and take a shorter selling period. In addition, for the sales problem of perishable products affected by both reference price and reference quality, retailers should adopt skimming pricing strategy (the optimal price decreases with time). Furthermore, to obtain more profit, retails should strive to increase the sensitivity coefficients of two types of reference, and the reference quality memory coefficient, while to decrease the reference price memory coefficient.

Approximation Algorithms for Dynamic Inventory Management on Networks

We provide the first approximation algorithm for dynamic inventory management on a network with stochastic demand and backlogging. Specifically, under a mild cost condition, we prove the cost of a specially designed base-stock policy is less than 1.618 times the cost of an optimal policy. We develop a novel stochastic programming analysis to prove this result: We carefully calibrate two stochastic programs (providing upper and lower bounds on the optimal policy), and compare their objectives. The upper bound arises from a new class of base-stock policies we define to address the currently unresolved issue of how to assign and fulfill backlogs in a system with fulfillment flexibility. We show the optimal policy in this class takes a simple and intuitive form: backlogs are assigned to their lowest cost activities for replenishment, and the ordered resources for those activities are committed to the backlogs for fulfillment. Next, the lower bound stochastic program is derived through a novel cost accounting scheme that captures the tradeoff between current inventory decisions and future backlog decisions. We then exploit this tradeoff to bound the ratio of the two stochastic program objectives and prove our main result. We also demonstrate our policy’s practicality with numerical simulations that show it performs within 1% of optimal on average across a wide range of problem instances. Finally, we show that our techniques and results extend to more general settings, demonstrating their potential for broader applicability. Importantly, our approach extends to problems with lead times, where we prove an approximation guarantee for base-stock policies that is independent of both the lead time and network structure. Thus, our work provides the new managerial insight that properly designed base-stock policies can be effective in network settings. This paper was accepted by Victor Martínez-de-Albéniz, operations management. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2022.02965 .

Quantitative Assessment of Green Inventory Management in Supply Chains: Simulation-Based Study of Economic and Environmental Outcomes Aligned with ISO 14083 Standard

This research employs numerical simulations and scenario analysis to assess a supply chain model’s economic and environmental performance operating under stochastic market demand, with inventory levels managed by a periodic review (R, s, S) inventory system. The inventory model in this research is designed to determine the minimal inventory levels required to achieve predefined fill rates across various operational constraints. The supply chain’s inventory model simulates optimal responses to normally distributed market demand within 365-day periods characterized by mean and two levels of demand variability through two fill rate levels, two workweek schedules, 15 review periods, and 16 lead times. By conducting an extensive analysis of the 192000 simulation experiments of the supply chain under periodic review (R, s, S) inventory system, complex influences between system variables and economic outcomes of supply chain operation measured by ordering, transportation, holding, penalty, and total costs along with greenhouse gas emissions arising from inventory-related transportation according to the ISO 14083 standard are analyzed. The insights from this research have significant practical implications, providing valuable guidance for supply chain managers, researchers, and freight companies offering guidance for improving economic and environmental performance.

A two-stage approach for modeling, performance evaluation, and decision support of disassembly processes

ABSTRACT This paper tackles an issue of disassembly lot-sizing within the framework of stochastic batch demand for multi-products featuring a two-level structure. The suggested methodology relies on a batch deterministic and stochastic Petri net as a modeling tool to address such problems in the context of stochastic processes. The primary objective is to determine the optimal sequencing of disassembly for each product type to meet stochastic batch demands of root items across multiple criteria. A novel mathematical model is introduced to minimize the cumulative costs associated with disassembly operations and extra costs, as well as holding costs for diverse products. In a subsequent stage, a decision support tool, namely the preference ranking organization method for enrichment evaluation method, is put forward for prioritizing different scenarios. For assessing the efficiency of the suggested approach, a real case study from the manufacturing industry is simulated. The simulation results prove the suggested approach’s capability to adapt to dynamic demands for leaf items throughout the planning horizon.

Stochastic measurement-based multi-cloud consolidation for efficient resource distribution

In distributed cloud systems, due to high computational complexity, existing methods cannot infinitely reduce the measurement interval to obtain the mean demand. However, considering both the unpredictability of demand and the diversity of pricing adds significant complexity to the problem. In response, we introduce an efficient algorithm called Stochastic Demand-oriented Resource consolidation based on Meta-heuristics (SDRM). The algorithm introduces the stochastic demand model and formulates a cross-cloud demand consolidation and resource scheduling problem, and tries to solve it efficiently by combining differential evolution algorithm with dynamic programming optimization. The results are provided for experiments utilizing both simulated and real-world data. It demonstrates that compared with traditional algorithms, increasing gain by up to 53%, SDRM just incurs an affordable time expense (i.e., 1.10-2.51 times that of existing algorithms). Therefore, SDRM stands as a compelling solution for resource distribution in heterogeneous cloud environments.

Supply chain network equilibrium with outsourcing for fresh agricultural products under stochastic demands

Abstract Accepted by: M. Zied Babai This paper examines a supply chain network focusing on the distribution of fresh agricultural products, incorporating outsourcing and stochastic demands considerations. Initially, we establish a generalized Nash equilibrium model with stochastic demands among fresh produce firms. Subsequently, we transform this model into a mixed complementarity system using the Karush–Kuhn–Tucker conditions. Utilizing the Fischer–Burmeister function, we further convert the mixed complementarity system into a set of nonlinear equations, amenable to solution via GAMS software. We conduct numerical experiments and perform sensitivity analysis on key parameters. Our findings suggest that higher subsidy rates incentivize firms towards production outsourcing, particularly benefiting low-income farmers, thereby potentially increasing profits. Moreover, market fluctuations play a pivotal role in ensuring the stability and profitability of firms, with moderate fluctuations presenting opportunities for fresh enterprises to adapt, innovate and capitalize on evolving market conditions. These results offer valuable insights for effective management of fresh produce supply chains.

Fast heuristics for the time‐constrained immobile server problem

AbstractWe propose easy‐to‐implement heuristics for time‐constrained applications of a problem referred to in the literature as the facility location problem with immobile servers, stochastic demand, and congestion, the service system design problem, or the immobile server problem (ISP). The problem is typically posed as one of allocating capacity to a set of M/M/1 queues to which customers with stochastic demand are assigned with the objective of minimizing a cost function composed of a fixed capacity‐acquisition cost, a variable customer‐assignment cost, and an expected‐waiting‐time cost. The expected‐waiting‐time cost results in a nonlinear term in the objective function of the standard binary programming formulation of the problem. Thus, the solution approaches proposed in the literature are either sophisticated linearization or relaxation schemes, or metaheuristics. In this study, we demonstrate that an ensemble of straightforward, greedy heuristics can rapidly find high‐quality solutions. In addition to filling a gap in the literature on ISP heuristics, new stopping criteria for an existing cutting plane algorithm are proposed and tested, and a new mixed‐integer linear model requiring no iterating algorithm is developed. In many cases, our heuristic approach finds solutions of the same or better quality than those found by exact methods implemented with expensive, state‐of‐the‐art mathematical programming software, in particular a commercial nonlinear mixed‐integer linear programming solver, given a five‐minute time limit.

Pricing and bundling decisions for complementary products in an agency selling platform with risk‐averse suppliers and stochastic demand

AbstractThe exponential rise of e‐commerce has prompted many companies to utilize online platforms (e‐tailers) to sell their products. Agency selling is a prevalent policy among e‐tailers where suppliers determine retail prices of their products, and the e‐tailer charges an agency rate. This paper investigates the selling format and pricing problem of two risk‐averse suppliers who sell complementary products on the e‐tailer's platform. The e‐tailer can offer products separately and as a bundle. A game‐theoretic analysis reveals that the equilibrium prices of individual products and the bundle uniquely exist. The study explores that the e‐tailer does not always benefit from a higher agency rate. Furthermore, the agency rate can influence the superiority of mixed bundling over separate selling. Moreover, a mixed‐bundling strategy may hurt the supplier's profitability which is sensitive to marginal costs and the degree of risk aversion. Finally, when demand correlation increases, the supplier sets a lower price to mitigate the risk cost.

A deterministic fluid model for production and energy mode control of a single machine

Improving machines’ energy efficiency through dynamic energy mode control to meet demand requirements with minimal energy consumption is a promising approach. This study considers a machine operating in working, idle, off, and warmup energy modes with different energy consumption in each mode. A deterministic fluid model is developed to analyze an energy mode control policy that determines when to keep the machine working, off or idle, and switch to other modes based on the inventory/backlog level to minimize the total energy, inventory, and backlog costs. This approach facilitates the derivation of closed-form expressions for the optimal thresholds and the associated costs. This modeling approach allows us to prove that a policy that operates the machine between the working and off modes or the working and idle modes is always better than a hybrid policy that operates the machine in working, off, and idle modes simultaneously. We use the solution of the deterministic fluid model to propose an approximate policy for machines with stochastic production, warmup, and demand processes. We compare the results of the proposed approximation method with the optimal solution of a stochastic system where the production and warmup times are exponential and the demand inter-arrival times have Erlang distribution. The optimal solution for the stochastic system is determined by solving a Markovian Decision Process (MDP). Our numerical experiments show that the proposed approximation method predicts the optimal policy type for the stochastic case with a 89.3% accuracy, and the average error between the optimal cost and the cost obtained with the approximation method is 1.37% for 729 different cases tested. Furthermore, the computational efficiency of the proposed approximation is around 250 times better than the effort to determine the optimal policy using an MDP approach. We propose this approximation method where the control parameters are given in closed form as an easy-to-implement and effective policy to control energy modes to minimize the total energy, inventory, and backlog costs. Furthermore, we present the deterministic fluid modeling approach as a versatile approach to analyze energy mode control problems.

Fast and efficient heuristics for the (s, S, T) inventory policy

We consider a single-echelon inventory system facing stochastic demand controlled by the standard ( s , S , T ) periodic review policy. While an exact algorithm for optimising all three policy variables exists, high computational requirements prohibit real industrial applications. To deal with this problem, here we propose three search heuristic algorithms that can approximate the optimal policy solution very fast and accurately. All heuristics make use of the empirical observation that the optimisers of the ( s , S , T ) policy are closely related to those of the classical ( r , nQ , T ) policy. For the latter, however, a fast near-optimal heuristic exists. Therefore, starting with the ( r , nQ , T ) optimal solution, all algorithms employ local neighbourhood search to determine the required ( s , S , T ) policy solution approximation. Experiments with well-known meta-heuristics such as Simulated Annealing, Tabu Search, and others show that the heuristics compare favourably in both solution time and quality. Moreover, compared with the existing exact algorithm heuristic solutions are very satisfactory in quality and are obtained in a fraction of CPU-time.

Last mile delivery with drones: A carbon emissions comparison

The development and potential adoption of drones or unmanned aerial vehicles as delivery vehicles creates incredible opportunities and unique challenges for last mile delivery. This research first presents a last mile delivery fleet model with drones that can be further modified and expanded over time. The model shows the optimal number of drones needed based on deterministic or stochastic demand using both traditional charging and battery swapping. The research then compares the carbon emissions of four delivery modes: traditional internal combustion delivery vehicles, all-electric vehicles, plug-in hybrids, and drones within the context of last mile delivery. Findings reveal that the breakdown of carbon emissions by delivery modality depends on parameter assumptions, ambient temperature, delivery radius, electric grid pollution rate, and number of customers.

Mitigating Consumer Returns through Omnichannel Retail Operations: An EOQ Model under Stochastic Demand

Mitigating Consumer Returns through Omnichannel Retail Operations: An EOQ Model under Stochastic Demand

Dynamic resource allocation and scheduling for appointment-based systems with walk-ins

Effectively managing constrained and perishable capacity amid fluctuating demands is a common challenge in service industries, often addressed through appointment-based systems that incorporate walk-ins. Balancing appointments and walk-ins is crucial for enhancing system value and maintaining stability. In this article, we study an integrated decision problem for appointment scheduling and resource allocation across multiple facilities, considering stochastic demands from three segments: priority customers, walk-in customers, and appointment customers. We formulate the problem as a sequential scheduling-allocation challenge using stochastic dynamic programming. Leveraging the anti-multimodularity of the value function, we fully characterize the optimal dynamic resource allocation policy and its monotone structural properties under any appointment schedule. Given the complexity and intractability of optimal scheduling with dynamic resource allocation, we introduce two suboptimal scheduling policies—vertical scheduling and horizontal scheduling—and a heuristic allocation policy. Additionally, we create an Upper Bound (UB) for the original problem as a surrogate, establishing an asymptotically optimal UB for the scaled problem through its objective value and a corresponding lower bound through its solution. This UB solution serves as an efficient and effective heuristic, demonstrating superior performance compared with the combined performance of suboptimal scheduling policies with optimal dynamic resource allocation, especially as problem size increases. It also holds significant practical implications for integrated pooling systems.

Advance selling policies with stochastic consumer valuations and advertising effects: dynamic pricing and inventory decisions

Advance selling enables consumers to order new products before their release, ensuring prompt deliveries upon release. Retailers benefit from advance selling by boosting total sales and mitigating excess or insufficient inventory costs. In practice, various advance selling policies are designed and applied. Standard Advance Selling, Advance Selling with Price Guarantee, and Advance Selling with a Deposit are often seen and successfully implemented policies by different retailers. However, given the stochastic demands and consumer valuations, whether advance selling is always desirable for retailers is unclear. If advance selling is beneficial, what are the optimal prices and ordering quantities and which policy is the best remain interesting questions. This paper aims to provide analytical solutions for optimal pricing and inventory decisions under the three advance selling policies mentioned above. The results show that the performance of the advance selling policies relies on the proportion of high-valuation consumers. The critical values of the proportion of high-valuation consumers are identified. Further, we extend the problem to consider the effects of advertising. The advertising affects the information diffusion and indirectly affects the demands of the product. Moreover, the analytical and numerical study compares the performances of different advance selling policies. Finally, multiple managerial insights are summarized.

Modelling and optimization of ATM cash-loading under stochastic demand

A set of ATMs in commercial areas were considered for business transactions. Customer demand for money requires prompt cash supply; to harmonize optimality options for ATM cash loading to sustain customers. We formulated this tis decision as a multi-stage stochastic process with Markovian demand of ATM cash among customers. The loading and operations of the ATMs were governed by cash demand and supply; with unit (loading, operational, and shortage) costs. The study considered the ATM cash demand of ten ATMs in regions within commercial centres of Kampala city. The relevant data was collected every hour throughout four months. The task focused on optimizing cash-loading decisions that minimize the operational costs of managing ATMs. The optimality of cash loading decisions were determined using the Markov decision processes. The tested model revealed optimal state-dependent options and operational costs of managing ATMs cost-effectively. As a cost optimization strategy for managers in the banking industry, the productivity of ATMs towards improved efficiency and utilization of resources for cash provision to customers emerged; supporting a larger customer base for business transactions.

An evaluation of common modeling choices for the vehicle routing problem with stochastic demands

We investigate three common modeling choices for the vehicle routing problem with stochastic demands: (i) the total expected demand of customers on a route may not exceed the capacity of the vehicle, (ii) the number of routes is fixed, and (iii) demand is distributed with a support that contains negative-valued realizations. We prove that modeling choices (i) and (ii) result in an arbitrarily large increase of the optimal objective value in the worst case. Additionally, we provide lower and upper bounds on the change of the optimal objective value following from (iii) in case the actual distribution of demand is censored, truncated or folded. We also evaluate the consequences of these choices numerically, by employing a state-of-the-art integer L-shaped method to solve the vehicle routing problem with stochastic demands to optimality, which we modify to deal with the alternative choices. We find that restricting the expected demand of a route to the vehicle’s capacity has a limited effect on the optimal objective value for most, but not all, benchmark instances from the literature, while drastically reducing the computation times of the integer L-shaped method. When restricting the number of routes, a similar effect occurs when the total expected demand on a route is not restricted. Otherwise, the computation time decreases only slightly, and even increases for some benchmark instances. For instances from the literature, despite admitting negative realizations, the normal distributions used to model demand are an adequate approximation for censored, truncated and folded normal distributions that have nonnegative supports.

A closed-loop supply chain inventory model with stochastic demand, exchange rate, green investment, and carbon tax

In this paper, a mathematical inventory model for a closed-loop supply chain consisting of a single manufacturer and a single retailer is investigated under a stochastic environment and imperfect production. The manufacturer adopts a remanufacturing policy to recover returned products from the market. The model takes into account exchange rate disparities because the supply chain’s participants are in separate countries. The model also considers carbon emissions, which are expected to be produced from storage, transportation, and production. A carbon policy, namely carbon tax, is implemented to manage the overall emissions resulting from the supply chain. To cope with carbon restrictions, the manufacturer has an opportunity to invest in green technologies. The objective of the study is to find optimal shipment quantity, number of deliveries, safety factor, and green investment such that the joint total cost is minimized. An iterative procedure is suggested to solve the proposed problem, and a numerical example is presented for model validation. The findings imply that variations in the production rate and carbon tax have a significant influence on the model’s performance. Aside from that, the results reveal that uncertainty in the exchange rate and production flaws are important factors that must be considered by managers in making inventory decisions, especially those related to the number of deliveries, frequency of deliveries, and the amount of money invested in green technology. Finally, a sensitivity analysis is provided to explain the behavior of the model.

From operational to strategic modelling: A continuous multi-scale approach for last-mile analysis

Supply chain planning requires decision-making at all levels, especially in the new normal of intervened supply networks. The integration of strategic network design and operational routing decisions has been widely studied in the literature as the location-routing problem (LRP). However, the LRP does not consider the differences in the planning horizon of each sub-problem, nor have geospatial elements been included in the distribution of demand. This paper aims to redesign the conceptual modelling of the last mile to inform strategic network design problems. A continuous multi-scale approach (CMA) is proposed by taking elements from the districting problem (DP) and continuous approximation (CA). This approach includes stochastic demand in the analysis and the effects of time windows and failed deliveries. The validation of CMA in a case of rural parcel deliveries in Belgium shows an estimate of the distance travelled similar to traditional routing algorithms in scenarios with high demand density. Likewise, the effects of time windows on the spatial configuration of multi-scale districting are explored. This approach provides insights for decision-making in strategic and tactical planning, such as sonification, differentiated services to satisfy consumer preferences, and fleet management. The limitations of the CMA lie in its implementation in purely operational scenarios since it does not offer detailed routing information. Subsequent research aims to exploit the potential of strategic last-mile modelling with effective integration into network design problems.

Optimal Dynamic Production Planning for Supply Network with Random External and Internal Demands

This paper focuses on joint production/inventory optimization in single and multiple horizons, respectively, within a complicated supply network (CSN) consisting of firm nodes with coupled demands and firm nodes with coupled demands. We first formulate the single-epoch joint optimal output model by allowing the production of extra quantity for stock underage, considering the fixed costs incurred by having inventory over demand and shortfalls. Then, the multi-temporal dynamic joint production model is further investigated to deal with stochastic demand fluctuations among CSN nodes by constructing a dynamic input–output model. The K-convexity defined in Rn space is proved to obtain the optimal control strategy. According to physical flow links, all demands associated to the nodes of CSN are categorized into the inter-node demand inside CSN (intermediate demand) and external demand outside CSN (final demand). We exploit the meliorated input–output matrix to describe demand relations, building dynamic input–output models where demand fluctuates randomly in single-cycle CSN and finite multi-cycle CSN. The novel monocyclic and multicyclic dynamic models have been developed to minimize system-wide operational costs. Unlike existent literature, we consider fixed costs incurred by overdemand and underdemand inventory into system operational cost functions and then demonstrate the convexity of objective functions. The cost function with two fixed penalty costs due to excess and shortage of inventory is developed in a multicycle model, and the K-convexity defined in Rn is proved to find out the optimal strategy for joint dynamic production of CSNs in the case of multi-products and multicycles.