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A stochastic flow network composed of multistate arcs can be utilized to describe several practical systems such as computer networks, where transmission time taken for sending data to a sink is an important index. Determining a path with minimum transmission time is known as the quickest path problem (QPP). All algorithms addressing the QPP assume that the determined minimal paths (MPs) are disjoint. Further, for the general case of intersectional MPs, if a congestion phenomenon occurs during the transmission process, these algorithms will lead to an incorrect outcome. Moreover, in practical scenarios, as a budget limit is considered, spare routing is applied to consolidate the system. The objective is to develop an algorithm based on Monte Carlo simulations (MCSs) for evaluating the system reliability while considering the congestion phenomenon. The system reliability is the probability that a specific amount of data can be transmitted successfully through multiple MPs under both time and budget constraints. Furthermore, spare routing to increase the system reliability is established in advance to specify the main and spare MPs. Experiments validate the evaluation of system reliability based on MCSs. The credibility and efficiency of the proposed algorithm are also discussed.

We consider the problem of real-life evacuation of people at sea. The primary disaster response goal is to minimize the time to save all the people during the evacuation operation, taking into account different groups at risk (children, women, seniors etc.) and the evacuation processing time (including the routing time), subject to a budget constraint. There are different evacuation tools (e.g., lifeboats, salvage ships, sea robots, helicopters etc.) for rescuing groups at risk to some safe points (e.g., hospitals, other ships, police offices etc.). The evacuation processing time of a group at risk depends on the group and the evacuation tool used. The secondary goal is to minimize the cost among all the alternative optimal solutions for the primary goal. We present a new mathematical rescue-evacuation model and design a fast solution method for real-time emergency response for different population groups and different evacuation tools, based on iterative utilization of a modification of the scheduling algorithm introduced by Leung and Ng (Eur J Oper Res 260:507–513, 2017).

Information networks are critical media for various applications of cloud computing and the Internet of Things. How to maintain the system reliability of an information network is an important issue for system supervisors. The system reliability is influenced by the capacities of the lines assigned to the edges that connect the vertices of the information network, where each line should operate to provide a certain capacity or fail. This study aims to maximize the system reliability of an information network by determining the optimal binary-state line assignment under an assignment budget constraint. In contrast to previous studies related to system reliability maximization, this study considers a time threshold in system reliability evaluation. In addition, the information network is regarded as a stochastic information network, as each edge may include several binary-state lines and provide multiple states. A method based on minimal paths and genetic algorithm (GA) with elite replacement is developed to evaluate the system reliability and search for the optimal solution. Several numerical experiments are conducted to show that the GA not only determines the exact solution but also exhibits higher computational efficiency than several well-known meta-heuristic algorithms.

Vector control, particularly distribution of insecticide-treated bed nets (ITNs), constitutes one of the major pathways to prevent and reduce malaria transmission. ITN distribution campaigns face several challenges, such as inadequate funding, budgetary constraints, hard-to-reach areas, limited transportation, and market and price volatility. While long-term agreements and proper planning can effectively overcome such challenges, those options may not be available for all humanitarian organizations and governments. To gain a better understanding of such tradeoffs we develop a robust optimization model that minimize ITN distribution costs while taking into consideration protection against financial, market and logistical uncertainties. The simultaneous account of such uncertainties is rarely seen on the humanitarian supply chain design literature. The proposed robust model explores data-driven adaptive uncertainty sets that capture the dependence structure among procurement and distribution costs, leading to plausible uncertain scenarios. In addition, we develop a hierarchical optimization approach to ease the burden of setting a specific robustness level for each constraint, when uncertainties are related to the independent terms. We study a United Nations Children's Fund ITN distribution campaign in Ivory Coast, observing that (1) total costs increase with campaign robustness, as expected, and (2) campaign robustness comprises of improved supply chain flexibility, which might minimize efforts if it becomes necessary to adjust procurement and transportation plans when uncertainty arises. In addition, assessing robust solutions through Monte Carlo simulations against several realizations of uncertain parameter values indicates that, as desired, robust plan feasibility increases with the specified level of conservatism.

A new traffic network equilibrium problem with continuously distributed bounds on path weights is introduced in this paper, as an emerging modeling tool for evaluating traffic networks in which the route choice behavior of individual motorists is subject to some physical or psychological upper limit of a travel weight. Such a problem may arise from at least two traffic network instances. First, in a traffic network serving electric vehicles, the driving range of these vehicles is subject to a distance constraint formed by onboard battery capacities and electricity consumption rates as well as network-wide battery-recharging opportunities, which cause the range anxiety issue in the driving population. Second, in a tolled traffic network, while drivers take into account both travel time and road toll in their route choice decisions, many of them implicitly or explicitly set a budget constraint in their mental account for toll expense, subject to their own income levels and other personal and household socio-economic factors. In both cases, we model the upper limit of the path travel weight (i.e., distance or toll) as a continuously distributed stochastic parameter across the driving population, to reflect the diverse heterogeneity of vehicle- and/or motorist-related travel characteristics. For characterizing this weight-constrained network equilibrium problem, we proposed a convex programming model with a finite number of constraints, on the basis of a newly introduced path flow variable named interval path flow rate. We also analyzed the problem’s optimality conditions for the case of path distance limits, and studied the existence of optimal tolls for the case of path toll limits. A linear approximation algorithm was further developed for this complex network equilibrium problem, which encapsulates an efficient weight-constrained k-minimum time path search procedure to perform the network loading. Numerical results obtained from conducting quantitative analyses on example networks clearly illustrate the applicability of the modeling and solution methods for the proposed problem and reveal the mechanism of stochastic weight limits reshaping the network equilibrium.

In this paper, we consider capacity expansion for network models subject to uncertainty and budget constraints. We use a scenario tree approach to handle the uncertainty and construct a multi-stage stochastic mixed-integer programming model for the network capacity expansion problem. We assume that permanent capacity and spot market capacity are available, which can be used permanently or temporarily by the decision maker respectively. By relaxing the budget constraints, we propose a heuristic Lagrangian relaxation method to solve the problem. Two algorithms are developed to find tight upper bounds for the Lagrangian relaxation procedure. The experimental results show superior performance of the proposed Lagrangian relaxation method compared with CPLEX.

This paper develops a multiproduct economic production quantity inventory model for a vendor–buyer system in which several products are manufactured on a single machine. The vendor delivers the products to customer in small batches. The number of orders must be a discrete value. Moreover, benefitting from a just-in-time policy, the buyer decides the size of the delivered batches. Due to the fact that several products are manufactured on one machine, this makes that the production capacity be considered as a constraint. The aim of this study is to determine the optimal cycle length and the number of delivered batches for each product so that the total inventory cost is minimized. The problem under study is modeled as a mixed integer nonlinear programing problem considering maximum number of orders, capacity and budget constraints. Three different methods are developed and employed to solve this problem: an exact method, a heuristic algorithm and a hybrid genetic algorithm. Based on the results, the three algorithms have near efficiency with different running times. The results shows that the heuristic algorithm obtains a good solution in a short time and the hybrid genetic algorithm finds solutions with higher quality in an acceptable time. Finally, a sensitivity analysis is done to evaluate the effect of changes in the parameters of problem.

The primary concern of managers in logistics management is how market demand can be satisfied. The inventory in the transfer center of a distribution network plays a key role in meeting customer demands. This study focuses on evaluating the system reliability of a stochastic delivery-flow distribution network (SDDN) with inventory. An SDDN is composed of nodes and routes, where each node denotes a vendor, transfer center, or market and each route connects a pair of nodes. Along each route, there is a carrier whose available capacity is stochastic. Different from previous issues regarding system reliability, this study does not consider the vendor (source) and market (sink) but also the amount of stocks in transfer center is included. If the market demand cannot be met by inventory in transfer center, then vendors must meet the remaining demand to satisfy the market. Therefore an algorithm is developed in terms of minimal paths to evaluate system reliability, which is defined as the probability that the SDDN with stocks can satisfy the market demand from multiple vendors and transfer centers to the market under budget constraints. A numerical example is given to illustrate this proposed algorithm.

In this paper, we develop a supply chain network game theory model consisting of retailers and demand markets with retailers competing noncooperatively in order to maximize their expected profits by determining their optimal product transactions as well as cybersecurity investments subject to nonlinear budget constraints that include the cybersecurity investment cost functions. The consumers at the demand markets reflect their preferences through the demand price functions, which depend on the product demands and on the average level of cybersecurity in the supply chain network. We identify the supply chain network vulnerability to cyberattacks as well as that of the individual retailers. We demonstrate that the governing Nash equilibrium conditions can be formulated as a variational inequality problem and we provide a novel alternative formulation, along with the accompanying theory. We also propose an algorithm for the alternative formulation, which yields, at each iteration, closed form expressions in product transactions, security levels, and Lagrange multipliers associated with the budget constraints. We then apply the algorithm to compute solutions to a spectrum of numerical supply chain network cybersecurity investment examples. The examples broaden our understanding of the impacts of the addition of retailers, changes in budgets, demand price functions, and financial damages, on equilibrium product transactions and cybersecurity investments, as well as on the supply chain network vulnerability and retailer vulnerability under budget constraints.

This paper deals with the supply chain network design and planning for a multi-commodity and multi-layer network over a planning horizon with multiple periods in which demands of customer zones are considered to be price dependent. These prices determine the demands using plausible price–demand relationships of customer zones. The net income of the supply chain is maximized, while satisfying budget constraints for investment in network design. In addition, a new approach is considered for capacity planning to make the problem more realistic. In this regard, when production plants are opened and expanded, capacity options are taken into account for manufacturing operations. Furthermore, several aspects relevant to real world applications are captured in the problem. Different interconnected time periods in the planning horizon are considered for strategic and tactical decisions in the problem and then, a mixed-integer linear programming (MILP) model is developed. The performance and applications of the model are investigated by several test problems with reasonable sizes. The numerical results illustrate that obtained solutions after solving the MILP model by using CPLEX solver are acceptable. Moreover, using an alternative pricing approach, a tight upper bound is provided for the problem. In addition, a deep sensitivity analysis is conducted to show the validity and performance of the proposed model.

This paper studies pricing strategies of a seller with budget constraints facing two types of strategic consumers with different search costs, and proposes three pricing strategies to motivate all consumers to visit his shop. These are the basic price strategy, differentiated compensation strategy and an improved differentiated compensation strategy. Based on the rational expectations paradigm, we characterize the rational expectations equilibrium in the game and propose a basic pricing strategy. In order to address the interplay between price and demand, we further propose a differentiated compensation strategy to improve the basic model. We then compare the differentiated compensation strategy to the basic pricing strategy when both are feasible. We find that selection of the optimal strategy is independent of composition of consumers but is dependent on the seller’s budget level and the difference between the two search costs. If the budget is large enough and the difference between the search costs is small enough, a differentiated compensation strategy can further improve the seller’s profitability. In addition to these findings, we first propose an improved differentiated compensation strategy to further enhance the firm’s profit. We find that the optimal strategy is to implement the improved differentiated compensation strategy when all three strategies are feasible. Interestingly, the firm may benefit from paying a high compensation to the consumers.

Cataloged from PDF version of article. Search-based advertising allows the advertisers to run special campaigns targeted to different groups of potential consumers at low costs. Google, Yahoo and Microsoft advertising programs allow the advertisers to bid for an ad position on the result page of a user's query when the user searches for a keyword that the advertiser relates to its products or services. The expected revenue generated by the ad depends on the ad position, and the ad positions of the advertisers are concurrently determined after an instantaneous auction based on the bids of the advertisers. The advertisers are charged only when their ads are clicked by the users. To avoid excessive ad expenditures due to sudden surges in the keyword-search activities, each advertiser reserves a fixed finite daily budget, and the ads are not shown in the remainder of the day when the budget is depleted. Arrival times of keyword-search instances, ad positions, ad selections, and sales generated by the ads are random. Therefore, an advertiser faces a dynamic stochastic total net revenue optimization problem subject to a strict budget constraint. Here we formulate and solve this problem using dynamic programming. We show that there is always an optimal dynamic bidding policy. We describe an iterative numerical approximation algorithm that uniformly converges to the optimal solution at an exponential rate of the number of iterations. We illustrate the algorithm on numerical examples. Because dynamic programing calculations of the optimal bidding policies are computationally demanding, we also propose both static and dynamic alternative bidding policies. We numerically compare the performances of optimal and alternative bidding policies by systematically changing each input parameter. The relative percentage total net revenue losses of the alternative bidding policies increases with the budget loading, but were never more than 3.5 % of maximum expected total net revenue. The best alternative to the optimal bidding policy turned out to be a static greedy bidding policy. Finally, statistical estimation of the model parameters is visited.

The paper investigates the impacts of trade credit (i.e., open account payment) and limited liability on the performance of a two-level supply chain with budget constraints. It shows that limited liability accounts for the reason why the retailer with a lower initial budget initiates a higher ordering level under trade credit contract. Furthermore, this paper finds that trade credit contract could create value in a supply chain with budget constraints, and partly coordinate the supply chain.

In this paper we propose a range of dynamic data envelopment analysis (DEA) models which allow information on costs of adjustment to be incorporated into the DEA framework. We first specify a basic dynamic DEA model predicated on a number of simplifying assumptions. We then outline a number of extensions to this model to accommodate asymmetric adjustment costs, non-static output quantities, non-static input prices, and non-static costs of adjustment, technological change, quasi-fixed inputs and investment budget constraints. The new dynamic DEA models provide valuable extra information relative to the standard static DEA models—they identify an optimal path of adjustment for the input quantities, and provide a measure of the potential cost savings that result from recognising the costs of adjusting input quantities towards the optimal point. The new models are illustrated using data relating to a chain of 35 retail department stores in Chile. The empirical results illustrate the wealth of information that can be derived from these models, and clearly show that static models overstate potential cost savings when adjustment costs are non-zero.

Several stochastic optimization models for planning capacity expansion for convenience store chains (or other similar businesses) are developed that incorporate uncertainty in future demand. All of these models generate schedules for capacity expansion, specifying the size, location, and timing of these expansions in order to maximize the expected profit to the company and to remain within a budget constraint on available resources. The models differ in how uncertainty is incorporated, specifically they differ in the point in the decision-making process that the uncertainty in the demand is resolved. Several measures of the value of information are defined by comparing the results from the different models. Two sample problems are given and their solutions for the various approaches compared.

This paper presents a two-stage multi-period decision model for allocation of the individual's savings into several investment plans. Although the U.S. economy is used as the background, the modelling methods are general enough to accommodate any tax law. The first stage of the model uses an asset-allocation method based on the single-index model. Because this method is static and does not provide for tax considerations and other constraints, it alone is not enough. The output of this optimal selection is used as exogenous parameters and controls for the second stage of the model which is an integer program. The IP includes fixed charges, statutory and budgetary constraints, a discount rate, and the risk level. We provide an example of this approach to illustrate how an individual can achieve his goals of terminal accumulations while maintaining the risk level, measured by the aggregate beta, he prefers. A linear programming relaxation of the IP model is utilized for sensitivity analysis to examine whether future adjustments in investment strategies are required. The model remains tractable enough for implementation by individuals who may not be experts in mathematical programming and financial planning.