# American Institute of Mathematical Sciences

ISSN:
1547-5816

eISSN:
1553-166X

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## Journal of Industrial & Management Optimization

July 2021 , Volume 17 , Issue 4

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2021, 17(4): 1531-1556 doi: 10.3934/jimo.2020033 +[Abstract](1256) +[HTML](572) +[PDF](869.51KB)
Abstract:

Markowitz proposes portfolio selection as a 2-objective model and emphasizes computing (whole) efficient sets and nondominated sets. Computing the sets has long been a topic in multiple-objective optimization. Researchers have gradually recognized other criteria in addition to variance and expected return. To formulate the additional criteria, researchers propose multiple-objective portfolio selection. However, computing the corresponding efficient set and nondominated set is not fully achieved. Moreover, discovering the sets' properties and utilizing the properties remain typically unanswered.

In this paper, we extend Sharpe's and Merton's model by adding a general linear objective and imposing equality constraints. To optimize the model, we analytically derive the minimum-variance surface (defined later), prove it as a nondegenerate paraboloid, and prove the nondominated set as a paraboloidal segment. We also analytically derive the efficient set and prove it as a 2-dimensional translated cone. We then prove that the set subsumes the efficient set of the corresponding traditional model, so the efficient set expands as the general linear objective is added. Furthermore, constraints can be changed or added. We utilize the translated-cone properties and readily compute the changing constraints' effect on the efficient sets by formulae or linear-equation systems.

2021, 17(4): 1557-1575 doi: 10.3934/jimo.2020034 +[Abstract](1480) +[HTML](648) +[PDF](3143.2KB)
Abstract:

In this paper, a new variant of the Solid Transportation Problem (STP) that incorporates both facility location and Fixed Charge Solid Transportation Problem (FCSTP) is presented with significant applications in logistics. It integrates decisions of diverse planning horizons: operational, tactical and strategic. The problem is termed Fixed Charge Solid Location and Transportation Problem (FCSLTP). Benchmark data obtained from the literature was extended for experimentation purposes. Solution to the FCSLTP was obtained using CPLEX commercial optimization solver. A Lagrange Relaxation Heuristic (LRH) was developed as an alternative solution for users not possibly having access to CPLEX. We further defined an equivalent FCSLTP in the main paper and termed this as FCSTP-EQ. The FCSTP-EQ was compared to our FCSLTP to investigate possible cost savings with both formulations. Results obtained showed CPLEX outperforming the Lagrange relaxation heuristic developed both in the upper bound and lower bound generation for the problem sizes considered. Additionally, the cost savings obtained using the FCSLTP was consistently better than the FCSTP-EQ. The upper bound generation capability of Lagrange relaxation could possibly be improved by using better search methods such as metaheuristics. Under certain conditions, the FCSTP could feasibly be used as a starting solution to solve the FCSLTP.

2021, 17(4): 1577-1591 doi: 10.3934/jimo.2020035 +[Abstract](1076) +[HTML](559) +[PDF](362.52KB)
Abstract:

Warranty service providers usually provide homogeneous warranty service to improve consumer satisfaction and market share. Considering the difference of consumers, some scholars have carried out studies on maintenance strategies, service pricing, payment method, claim behaviour and warranty cost analysis in recent years. However, few scholars have focused on the differentiated coverage of warranty service that considers usage rate and service option of consumers. On the basis of previous classification criteria on usage rate, this paper divides consumers into heavy, medium and light usage rate groups with clear boundaries. To avoid discrimination in warranty service, this study divides 2D warranty coverage into disjoint sub-regions and adopts different maintenance modes in each sub-region. By formulating and calculating warranty cost model under warranty cost constraints, we can obtain the maximum warranty coverage under usage rate \begin{document}$r$\end{document}. Therefore, differentiated warranty scope for consumers in the three groups can be proposed, whilst consumers can choose the most suitable warranty service according to their usage rate. Evidently, the proposed warranty strategy can provide flexible warranty service for consumers, meet the requirements of the warranty cost constraints of warranty service providers and enable enterprises to occupy a favourable position in the market competition.

2021, 17(4): 1593-1612 doi: 10.3934/jimo.2020036 +[Abstract](1119) +[HTML](564) +[PDF](400.17KB)
Abstract:

Enterprises are aware that bundling strategies can improve profitability in the highly competitive marketplace. This study evaluates an online to offline (O2O) supply chain system made up of a supplier and an e-retailer who can sell two products independently or bundled through online and offline channels, and discuss the influence of pricing strategy and channel choice on profit under different market-dominant powers. Based on a game theory model, we derive an optimal wholesale price for the supplier, an optimal sale price for the e-retailer, and their respective profit. We demonstrate that a Stackelberg leader is more profitable, irrespective of whether independent sales or bundling are chosen. Regardless of who the leader is, the whole supply chain receive equal profit. For a market leader, independent sales or bundling decisions should be made according to market size. Sensitivity analysis show that as the self-price sensitivity coefficient increases, the profit monotonically decreases for both independent sales and bundling; this occur for both the market dominated by the supplier and that dominated by the e-retailer. For independent sales, as the cross-price sensitivity coefficient increases, the profit monotonically increases; for bundled sales, the profit of the game players is not affected.

2021, 17(4): 1613-1637 doi: 10.3934/jimo.2020037 +[Abstract](1071) +[HTML](541) +[PDF](772.0KB)
Abstract:

The paper introduces the worst-case portfolio optimization models within the robust optimization framework for maximizing return through either the mean or median metrics. The risk in the portfolio is quantified by Gini mean difference. We put forward the worst-case models under the mixed and interval+polyhedral uncertainty sets. The proposed models turn out to be linear and mixed integer linear programs under the mixed uncertainty set, and semidefinite program under interval+polyhedral uncertainty set. The performance comparison of the proposed models on the listed stocks of Euro Stoxx 50, Dow Jones Global Titans 50, S & P Asia 50, consistently exhibit advantage over their conventional non-robust counterpart models on various risk parameters including the standard deviation, worst return, value at risk, conditional value at risk and maximum drawdown of the portfolio.

2021, 17(4): 1639-1661 doi: 10.3934/jimo.2020038 +[Abstract](1086) +[HTML](578) +[PDF](19487.79KB)
Abstract:

At present, electric vehicles (EVs), small-scale wind power, and solar power have been increasingly integrated into modern power system via the combined cooling heating and power based microgrid (CCHP-MG). However, inside the microgrid the uncertainties of EVs charging, wind power, and solar power significantly impact the economy of CCHP-MG operation. Therefore to improve the economy deteriorated by the uncertainties, this paper presents a two-stage adjustable robust optimization to achieve the minimal operational cost for CCHP-MG. Before the realizations of the uncertainties, the day-ahead stage as the first stage decides an operational strategy that can withstand the worst-case uncertainties. As long as the uncertainties are observed, the real-time stage as the second stage adjusts the operational units to compensate the errors caused by the day-ahead operational strategy. Due to the difficulties of the model solution, this paper further adopts the duality theory, Big-M method, and column-and-constraint generation (C & CG) decomposition to convert the model into two tractable mixed integer linear programming (MILP) problems. Further, C & CG iteration algorithm is also employed to solve the MILPs, which can ultimately provide an optimal economic day-ahead dispatch strategy capable of handling uncertainties. The experimental results demonstrate the effectiveness of the presented approach.

2021, 17(4): 1663-1680 doi: 10.3934/jimo.2020039 +[Abstract](1086) +[HTML](480) +[PDF](557.89KB)
Abstract:

In this paper, a new multiperiod mean semivariance portfolio selection with the transaction costs, borrowing constraints, threshold constraints and cardinality constraints is proposed. In the model, the return and risk of assets are characterized by mean value and semivariance, respectively. Because the semivariance operator is not separable, the optimal solution of the model is not time-consistent. The time-consistent strategy for this model can be obtained by using game approach. The time-consistent strategy, which is a mix integer dynamic optimization problem with path dependence, is approximately turned into a dynamic programming problem by approximate dynamic programming method. A novel discrete approximate iteration method is designed to obtain the optimal time-consistent strategy, and is proved linearly convergent. Finally, the comparison analysis of trade-off parameters is given to illustrate the idea of our model and the effectiveness of the designed algorithm.

2021, 17(4): 1681-1711 doi: 10.3934/jimo.2020040 +[Abstract](1316) +[HTML](522) +[PDF](896.82KB)
Abstract:

In this paper, we focus on a splitting method called the \begin{document}$\theta$\end{document}-scheme proposed by Glowinski and Le Tallec in [17,20,27]. First, we present an elaborative convergence analysis in a Hilbert space and propose a general convergent inexact \begin{document}$\theta$\end{document}-scheme. Second, for unconstrained problems, we prove the convergence of the \begin{document}$\theta$\end{document}-scheme and show a sublinear convergence rate in terms of objective value. Furthermore, a practical inexact \begin{document}$\theta$\end{document}-scheme is derived to solve \begin{document}$l_2$\end{document}-loss based problems and its convergence is proved. Third, for constrained problems, even though the convergence of the \begin{document}$\theta$\end{document}-scheme is available in the literature, yet its sublinear convergence rate is unknown until we provide one via a variational reformulation of the solution set. Besides, in order to relax the condition imposed on the \begin{document}$\theta$\end{document}-scheme, we propose a new variant and show its convergence. Finally, some preliminary numerical experiments demonstrate the efficiency of the \begin{document}$\theta$\end{document}-scheme and our proposed methods.

2021, 17(4): 1713-1727 doi: 10.3934/jimo.2020041 +[Abstract](1109) +[HTML](496) +[PDF](445.91KB)
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In this paper we present a mathematical model describing the physical dynamics of a building maintenance unit (BMU) equipped with reaction jets. The momentum provided by reaction jets is considered as the control variable. We introduce an objective functional based on the deviation of the BMU from its equilibrium state due to external high-wind forces. Pontryagin minimum principle is then used to determine the optimal control policy so as to minimize possible deviation from the rest state thereby increasing the stability of the BMU and reducing the risk to the workers as well as the public. We present a series of numerical results corresponding to three different scenarios for the formulated optimal control problem. These results show that, under high-wind conditions the BMU can be stabilized and brought to its equilibrium state with appropriate controls in a short period of time. Therefore, it is believed that the dynamic model presented here would be potentially useful for stabilizing building maintenance units thereby reducing the risk to the workers and the general public.

2021, 17(4): 1729-1752 doi: 10.3934/jimo.2020042 +[Abstract](897) +[HTML](473) +[PDF](540.4KB)
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This paper develops a lattice method for option evaluation in the presence of regime shifts in the correlation structure of assets, aiming at investigating whether the option prices reflect such shifts. We try to investigate whether option prices reflect switches in the correlation between the underlying asset of an option and risk-free rates.We develop and test two models.In the first model we allow all the parameters to follow a regime-switching process while in the second model, in order to isolate the regime-switching correlation effect on the option prices, we allow only the correlation to follow a regime-switching process. We use pentanomial lattices to represent the evolution of the regime-switching underlying assets. This is then applied in our empirical analysis, which focuses on crude oil. We use grid- and patternsearch based techniques to fit our models. Our findings suggest that prices of market traded options reflect the regime-switches and that a model which considers these switches produces significantly more accurate results than a single-regime model. We demonstrate that there is an asymmetry between parameter values obtained from historical data (backward looking) and those that are implied by traded options (for- ward looking) by employing the Kim filter to estimate our model.

2021, 17(4): 1753-1769 doi: 10.3934/jimo.2020044 +[Abstract](1160) +[HTML](639) +[PDF](802.0KB)
Abstract:

In this paper we propose a metaheuristic approach to solve a customer priority based location-allocation problem in presence of obstacles and location-dependent supplier capacities. In many network optimization problems presence of obstacles prohibits feasibility of a regular network design. This includes a wide range of applications including disaster relief and pandemic disease containment problems in healthcare management. We focus on this application since fast and efficient allocation of suppliers to demand nodes is a critical process that impacts the results of the containment strategy. In this study, we propose an integrated mixed-integer program with location-based capacity decisions that considers customer priorities in the network design. We propose an efficient multi-stage genetic algorithm that solves the problem in continuous space. The computational findings show the best allocation strategies derived from proposed algorithms.

2021, 17(4): 1771-1793 doi: 10.3934/jimo.2020045 +[Abstract](1230) +[HTML](669) +[PDF](1056.29KB)
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Freight bus is a new public transportation means for city logistics, and each freight bus can deliver and pick up goods at each customer/supplier location it passes. In this paper, we study the route planning problem of freight buses in an urban distribution system. Since each freight bus makes a tour visiting a set of pickup/delivery locations once at every given time interval in each day following a fixed route, the route planning problem can be considered a new variant of periodic vehicle routing problem with pickup and delivery. In order to solve the problem, a Mixed-Integer Linear Programming (MILP) model is formulated and an Adaptive Large Neighborhood Search (ALNS) algorithm is developed. The development of our algorithm takes into consideration specific characteristics of this problem, such as fixed route for each freight bus, possibly serving a demand in a later period but with a late service penalty, etc. The relevance of the mathematical model and the effectiveness of the proposed ALNS algorithm are proved by numerical experiments.

2021, 17(4): 1795-1807 doi: 10.3934/jimo.2020046 +[Abstract](1145) +[HTML](472) +[PDF](368.07KB)
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Network data envelopment analysis (DEA) concerns using the DEA technique to measure the relative efficiency of a system, taking into account its internal structure. The results are more meaningful and informative than those obtained from the conventional DEA models. This work proposed a new network DEA model based on the fuzzy concept even though the inputs and outputs data are crisp numbers. The model is then extended to investigate the network DEA with fuzzy non-discretionary variables. An illustrative application assessing the impact of information technology (IT) on firm performance is included. The results reveal that modeling the IT budget as a fuzzy non-discretionary factor improves the system performance of firms in a banking industry.

2021, 17(4): 1809-1823 doi: 10.3934/jimo.2020047 +[Abstract](874) +[HTML](468) +[PDF](322.55KB)
Abstract:

We present a mathematical model in an integer programming (I.P.) framework for non-linear delay costing in the airline industry. We prove the correctness of the model mathematically. Time is discretized into intervals of, for example, 15 minutes. We assume that the cost increases with increase in the number of intervals of delay in a piecewise linear manner. Computational results with data obtained from Sydney airport (Australia) show that the integer programming non-linear cost model runs much slower than the linear cost model; hence fast heuristics need to be developed to implement non-linear costing, which is more accurate than linear costing. We present a greedy heuristic that produces a solution only slightly worse than the ones produced by the I.P. models, but in much shorter time.

2021, 17(4): 1825-1843 doi: 10.3934/jimo.2020048 +[Abstract](838) +[HTML](494) +[PDF](522.68KB)
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In this paper, we focus on the viability and attraction for switched nonlinear systems with nonsmooth Lyapunov functions. We determine the viable set and region of attraction for switched systems in which Lyapunov functions are piecewise smooth. The switching law is constructed by using the directional derivatives of a piecewise smooth Lyapunov function along the trajectories of the subsystems. Sufficient conditions are derived to guarantee the viability and attraction of switched nonlinear systems on the level set of a piecewise smooth Lyapunov function. We further extend the method to switched systems involving possible sliding motions. The approach in the paper provides a unified framework for studying viability and attraction with a systematic consideration of sliding motions. Finally, considering two certain classes of piecewise smooth functions, the related conditions of the viability and attraction for the level set are developed.

2021, 17(4): 1845-1861 doi: 10.3934/jimo.2020049 +[Abstract](1044) +[HTML](468) +[PDF](602.67KB)
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Blockchain is well known as a database technology supporting digital currencies such as Bitcoin, Ether and Ripple. For the purpose of maximizing the overall revenue of the blockchain system, we propose a pricing policy to impose on transactions. Regarding the mining process as a vacation, and the block-verification process as a service, we establish a type of non-exhaustive queueing model with a limited batch service and a possible zero-transaction service. By selecting the beginning instant of a block-verification process as a Markov point and using the method of a generating function, we obtain the stationary probability distribution for the number of transactions in the system at the Markov points and analyze the elapsed time for the mining cycle. Based on the model analysis results, we derive the average latency of transactions and demonstrate how the average latency of transactions changes in relation to the arrival rate of transactions. With a reward-cost structure, we construct an individual benefit function and a social benefit function. By improving the Grasshopper Optimization Algorithm (GOA), we search for the Nash equilibrium and the socially optimal arrival rates of transactions. Numerical results show that the Nash equilibrium arrival rate of transactions is always higher than the socially optimal arrival rate of transactions for a given mining parameter and a specific block capacity. For this, we propose a pricing policy that forces the transactions to accept the socially optimal arrival rate and maximize the overall revenue of the blockchain system, including all transactions and miners.

2021, 17(4): 1863-1886 doi: 10.3934/jimo.2020050 +[Abstract](1029) +[HTML](484) +[PDF](393.06KB)
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A stochastic mathematical program model with second-order cone complementarity constraints (SSOCMPCC) is introduced in this paper. It can be considered as a non-trivial extension of stochastic mathematical program with complementarity constraints, and could arise from a hard-to-handle class of bilivel second-order cone programming and inverse stochastic second-order cone programming. By introducing the Chen-Harker-Kanzow-Smale (CHKS) type function to replace the projection operator onto the second-order cone, a smoothing sample average approximation (SAA) method is proposed for solving the SSOCMPCC problem. It can be shown that with proper assumptions, as the sample size goes to infinity, any cluster point of global solutions of the smoothing SAA problem is a global solution of SSOCMPCC almost surely, and any cluster point of stationary points of the former problem is a C-stationary point of the latter problem almost surely. C-stationarity can be strengthened to M-stationarity with additional assumptions. Finally, we report a simple illustrative numerical test to demonstrate our theoretical results.

2021, 17(4): 1887-1912 doi: 10.3934/jimo.2020051 +[Abstract](1030) +[HTML](546) +[PDF](491.08KB)
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In this paper, we investigate the optimal investment and reinsurance strategies for a mean-variance insurer when the surplus process is represented by a Cramér-Lundberg model. It is assumed that the instantaneous rate of investment return is stochastic and follows an Ornstein-Uhlenbeck (OU) process, which could describe the features of bull and bear markets. To solve the mean-variance optimization problem, we adopt a backward stochastic differential equation (BSDE) approach and derive explicit expressions for both the efficient strategy and efficient frontier. Finally, numerical examples are presented to illustrate our results.

2021, 17(4): 1913-1941 doi: 10.3934/jimo.2020052 +[Abstract](1187) +[HTML](639) +[PDF](562.06KB)
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In this paper, an Economic Production Quantity model for deteriorating items with time-dependent demand and shortages including partially back-ordered is developed under a cloudy-fuzzy environment. At first, we develop a crisp model by considering linearly time-dependent demand with constant deterioration rate, constant inflation rate and shortages under partially back-ordered, then we fuzzify the model to archive a decision under the cloudy-fuzzy (extension of fuzziness) demand rate, inflation rate, deterioration rate and the partially back-ordered rate which are followed by their practical applications. In this model, we assume ambiances where cloudy normalized triangular fuzzy number is used to handle the uncertainty in information which is coming from the data. The main purpose of our study is to defuzzify the total inventory cost by applying Ranking Index method of fuzzy numbers as well as cloudy-fuzzy numbers and minimize the total inventory cost of crisp, fuzzy, and cloudy-fuzzy model. Finally, a comparative analysis among crisp, fuzzy and cloudy-fuzzy total cost is carried out in this paper. Numerical example, sensitivity analysis, and managerial insights are elaborated to justify the usefulness of the new approach. A comparative inquiry of the numerical result with a new existing paper is also carried out. This paper ends with a conclusion along with advantages and limitations of our solution approach, and an outlook towards possible future studies.

2021, 17(4): 1943-1971 doi: 10.3934/jimo.2020053 +[Abstract](1065) +[HTML](481) +[PDF](557.6KB)
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The symmetric alternating direction method of multipliers is an efficient algorithm, which updates the Lagrange multiplier twice at each iteration and the variables are treated in a symmetric manner. Considering that the convergence range of the parameters plays an important role in the implementation of the algorithm. In this paper, we analyze the convergence rate of the symmetric ADMM with a more relaxed parameter range for solving the two block nonconvex separable optimization problem under the assumption that the generated sequence is bounded. Two cases are considered. In the first case, both components of the objective function are nonconvex, we prove the convergence of the augmented Lagrangian function sequence, and establish the \begin{document}$O(1/\sqrt{k})$\end{document} worst-case complexity measured by the difference of two consecutive iterations. In the second case, one component of the objective function is convex and the error bound condition is assumed, then we can prove that the iterative sequence converges locally to a KKT point in a R-linear rate; and an auxiliary sequence converges in a Q-linear rate. Furthermore, a practical inexact symmetric ADMM with relative error criteria is proposed, and the associated convergence analysis is established under the same conditions.

2021, 17(4): 1973-1991 doi: 10.3934/jimo.2020054 +[Abstract](885) +[HTML](512) +[PDF](951.42KB)
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In this paper, we consider a manufacturing system with two-machine no-wait flowshop scheduling problem where setup times are uncertain. The problem with the performance measure of maximum lateness was addressed in the literature (Computational and Applied Mathematics 37, 6774-6794) where dominance relations were proposed. We establish a new dominance relation and show that the new dominance relation is, on average, about 90\begin{document}$\%$\end{document} more efficient than the existing ones. Moreover, since it is highly unlikely to find optimal solutions for problems of reasonable size by utilizing dominance relations and since there exist no heuristic in the literature for the problem, we propose constructive heuristics to solve real life problems. We conduct extensive computational experiments to evaluate the proposed heuristics. Computational experiments indicate that the performance of the worst proposed heuristic is at least 20\begin{document}$\%$\end{document} better than a benchmark solution. Furthermore, they also indicate that the best proposed heuristic is about 130\begin{document}$\%$\end{document} better than the worst one. The average CPU time of the heuristics is significantly less than a second.

2021, 17(4): 1993-2011 doi: 10.3934/jimo.2020055 +[Abstract](968) +[HTML](529) +[PDF](569.06KB)
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It is worth noting that the conventional maximally decimated M-channel mirrored paraunitary linear phase finite impulse response condition is defined in the frequency domain. As the frequency domain is a continuous set, it is expressed as a matrix functional (a continuous function of the frequency) equation. On the other hand, this paper expresses the condition as a finite number of discrete (a set of functions of the sampled frequencies) equations. Besides, this paper proposes to sample the magnitude responses of the filters with the total number of the sampled frequencies being more than the filter lengths. Hence, the frequency selectivities of the filters can be controlled more effectively. This filter bank design problem is formulated as an optimization problem in such a way that the total mirrored paraunitary linear phase error is minimized subject to the specifications on the magnitude responses of the filters at these sampling frequencies. However, this optimization problem is highly nonconvex. To address this difficulty, a norm relaxed sequential quadratic programming approach is applied for finding its local optimal solution. By iterating the above procedures using different initial conditions, a near global optimal solution is obtained. Computer numerical simulation results show that our proposed design outperforms the existing designs.

2021, 17(4): 2013-2030 doi: 10.3934/jimo.2020056 +[Abstract](978) +[HTML](510) +[PDF](1362.75KB)
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In the \begin{document}$k$\end{document}-facility location problem, an important combinatorial optimization problem combining the classical facility location and \begin{document}$k$\end{document}-median problems, we are given the locations of some facilities and clients, and need to open at most \begin{document}$k$\end{document} facilities and connect all clients to opened facilities, minimizing the facility opening and connection cost. This paper considers the squared metric \begin{document}$k$\end{document}-facility location problem with linear penalties, a robust version of the \begin{document}$k$\end{document}-facility location problem. In this problem, we do not have to connect all clients to facilities, but each client that is not served by any facility must pay a penalty cost. The connection costs of facilities and clients are supposed to be squared metric, which is more general than the metric case. We provide a constant approximation algorithm based on the local search scheme with add, drop, and swap operations for this problem. Furthermore, we improve the approximation ratio by using the scaling technique.

2021, 17(4): 2031-2049 doi: 10.3934/jimo.2020057 +[Abstract](900) +[HTML](512) +[PDF](726.14KB)
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In a product market with price-sensitive demand, we examine a supply chain consisting of one manufacturer and one capital-constrained retailer. The retailer may purchase by borrowing by securing confirmed warehouse financing (CWF) from a competitive bank or the manufacturer's trade credit financing (TCF), provided that it is also to the latter's benefit to extend TCF. We obtain the manufacturer's optimal pricing decision in the two financing modes under the wholesale price contract that coordinates the supply chain given a wider range of wholesale prices in CWF. We find that CWF is more suitable for the customer segment for which the unit retail price is low and demand is stable, so the product can be monetized quickly; TCF is suitable for the customer segment for which the unit retail price is high and has a high price elasticity of demand and hence for long-term investment. The repurchase price is an important factor affecting the participants' selection of CWF.

2021, 17(4): 2051-2072 doi: 10.3934/jimo.2020058 +[Abstract](1926) +[HTML](543) +[PDF](1227.89KB)
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Nurse Rostering is the activity of assigning nurses to daily shifts in order to satisfy the cover requirements, taking into account the operational requirements and nurses' preferences. The problem is usually modeled as sets of hard and soft constraints with an objective function to minimize violations of soft constraints. The nurse rostering problem is known to be NP-hard. Many metaheuristics were used to tackle this problem. One of the frequently used heuristics is the Variable Neighbourhood Search (VNS). The VNS is usually used as a standalone method or in integration with another exact or heuristic method. In this paper, a new hybrid VNS and Dynamic Programming based heuristic approach is proposed to handle the nurse rostering problem. In the proposed approach, two perturbation mechanisms are adopted simultaneously. The proposed approach is tested on two different benchmark data sets. A comparison with state-of-the-art methods from literature revealed the competitive performance of the proposed approach.

2021, 17(4): 2073-2095 doi: 10.3934/jimo.2020059 +[Abstract](1068) +[HTML](519) +[PDF](341.81KB)
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In this paper, we study three types of heterogeneous fixed fleet vehicle routing problems, which are capacitated vehicle routing problem, open vehicle routing problem and split delivery vehicle routing problem. We propose new multiobjective linear binary and mixed integer programming models for these problems, where the first objective is the minimization of a total routing and usage costs for vehicles, and the second one is the vehicle type minimization, respectively. The proposed mathematical models are all illustrated on test problems, which are investigated in two groups: small-sized problems and the large-sized ones. The small-sized test problems are first scalarized by using the weighted sum scalarization method, and then GAMS software is used to compute efficient solutions. The large-sized test problems are solved by utilizing the tabu search algorithm.

2021, 17(4): 2097-2118 doi: 10.3934/jimo.2020060 +[Abstract](846) +[HTML](471) +[PDF](523.62KB)
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This paper studies the valuation problem of European call options when the presence of distressed selling may lead to further endogenous volatility and correlation between the stock issuer's asset value and the price of the stock underlying the option, and hence influence the option price. A change of numéraire technique, based on Girsanov Theorem, is applied to derive the analytical pricing formula for the European call option when the price of underlying stock is subject to price pressure triggered by the stock issuer's own distressed selling. Numerical experiments are also provided to study the impacts of distressed selling on the European call option prices.

2021, 17(4): 2119-2138 doi: 10.3934/jimo.2020061 +[Abstract](931) +[HTML](496) +[PDF](475.04KB)
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This paper considers a distributed convex optimization problem over a time-varying multi-agent network, where each agent has its own decision variables that should be set so as to minimize its individual objective subject to local constraints and global coupling constraints. Over directed graphs, we propose a distributed algorithm that incorporates the push-sum protocol into dual sub-gradient methods. Under the convexity assumption, the optimality of primal and dual variables, and the constraint violation are first established. Then the explicit convergence rates of the proposed algorithm are obtained. Finally, numerical experiments on the economic dispatch problem are provided to demonstrate the efficacy of the proposed algorithm.

2021, 17(4): 2139-2159 doi: 10.3934/jimo.2020062 +[Abstract](1022) +[HTML](483) +[PDF](413.55KB)
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The present paper investigates an optimal reinsurance-investment problem with Hyperbolic Absolute Risk Aversion (HARA) utility. The paper is distinguished from other literature by taking into account the interests of both an insurer and a reinsurer. The insurer is allowed to purchase reinsurance from the reinsurer. Both the insurer and the reinsurer are assumed to invest in one risk-free asset and one risky asset whose price follows Heston's SV model. Our aim is to seek optimal investment-reinsurance strategies to maximize the expected HARA utility of the insurer's and the reinsurer's terminal wealth. In the utility theory, HARA utility consists of power utility, exponential utility and logarithmic utility as special cases. In addition, HARA utility is seldom studied in the optimal investment and reinsurance problem due to its sophisticated expression. In this paper, we choose HARA utility as the risky preference of the insurer. Due to the complexity of the structure of the solution to the original Hamilton-Jacobi-Bellman (HJB) equation, we use Legendre transform to change the original non-linear HJB equation into its linear dual one, whose solution is easy to conjecture in the case of HARA utility. By calculations and deductions, we obtain the closed-form solutions of optimal investment-reinsurance strategies. Moreover, some special cases are also discussed in detail. Finally, some numerical examples are presented to illustrate the impacts of our model parameters (e.g., interest and volatility) on the optimal reinsurance-investment strategies.

2021, 17(4): 2161-2180 doi: 10.3934/jimo.2020063 +[Abstract](1215) +[HTML](623) +[PDF](204.22KB)
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In this paper, we propose and study a multi-step iterative algorithm that comprises of a finite family of asymptotically \begin{document}$k_i$\end{document}-strictly pseudocontractive mappings with respect to \begin{document}$p,$\end{document} and a \begin{document}$p$\end{document}-resolvent operator associated with a proper convex and lower semicontinuous function in a \begin{document}$p$\end{document}-uniformly convex metric space. Also, we establish the \begin{document}$\Delta$\end{document}-convergence of the proposed algorithm to a common fixed point of finite family of asymptotically \begin{document}$k_i$\end{document}-strictly pseudocontractive mappings which is also a minimizer of a proper convex and lower semicontinuous function. Furthermore, nontrivial numerical examples of our algorithm are given to show its applicability. Our results complement a host of recent results in literature.

2021, 17(4): 2181-2202 doi: 10.3934/jimo.2020064 +[Abstract](1084) +[HTML](515) +[PDF](208.27KB)
Abstract:

Classical Littlewood's rule (1972) for the two-period static revenue management of a single perishable resource is extended to a generic \begin{document}$T$\end{document}-period model with monotonically increasing fixed fares, ending with standby customers with a special fare. The expected revenue in the entire period is expressed explicitly in terms of multiple definite integrals involving the distribution function of the demand in each period. The exact optimal protection level in each period is calculated successively, resulting in the maximized total expected revenue. The Brumelle-McGill's theorem for the optimal booking limits in the \begin{document}$T$\end{document}-period model is also extended to a similar model with standby customers. We show some numerical examples with comments on the effects of accepting standby customers on the optimal booking limits and the increase in the expected revenue.

Ru Li and
2021, 17(4): 2203-2215 doi: 10.3934/jimo.2020065 +[Abstract](940) +[HTML](463) +[PDF](348.84KB)
Abstract:

In this paper, we consider a kind of proper efficiency, namely strict efficiency, of a multi-product supply-demand network equilibrium model. We prove that strict equilibrium pattern flows with both a single criterion and multiple criteria are equivalent to vector variational inequalities. In the case of multiple criteria, we provide necessary and sufficient conditions for strict efficiency in terms of vector variational inequalities by using Gerstewitz's function without any convexity assumptions.

2021, 17(4): 2217-2242 doi: 10.3934/jimo.2020066 +[Abstract](922) +[HTML](501) +[PDF](570.54KB)
Abstract:

In this paper, a three-level supply chain network equilibrium problem with direct selling and multi-commodity flow is considered. To this end, we first present equilibrium conditions which satisfy decision-making behaviors for manufacturers, retailers and consumer markets, respectively. Based on this, a nonlinear complementarity model of supply chain network equilibrium problem is established. In addition, we propose a new projection-type algorithm to solve this model without the backtracking line search, and global convergence result and \begin{document}$R-$\end{document}linearly convergence rate for the new algorithm are established under weaker conditions, respectively. We also illustrate the efficiency of given algorithm through some numerical examples.

2021, 17(4): 2243-2264 doi: 10.3934/jimo.2020067 +[Abstract](1001) +[HTML](509) +[PDF](394.32KB)
Abstract:

We consider an \begin{document}$M^X/G/1$\end{document} retrial queue with impatient customers subject to disastrous failures at which times all customers in the system are lost. When the server finishes serving a customer and finds the orbit empty, the server becomes dormant until \begin{document}$N$\end{document} or more customers accumulate. If a coming batch of customers finds the server idle, one of the arriving customers begins his service immediately and the rest join the orbit to repeat their request later. Otherwise, if the server is dormant or busy or down, all customers of the coming batch enter the orbit. When the server is under repair, customers in the orbit can become impatient after waiting a random amount of time and leave the system. By using the characteristic method for partial differential equations, the steady-state distributions of the server state and the number of customers in the orbit are obtained along with various performance measures. In addition, the reliability of the system is analyzed detailed. Finally, an application to telecommunication networks is provided and the effects of various parameters on the system performance are demonstrated numerically.

2021, 17(4): 2265-2275 doi: 10.3934/jimo.2020068 +[Abstract](903) +[HTML](519) +[PDF](292.61KB)
Abstract:

In this paper, we study convergence properties of the inexact Levenberg-Marquardt method under the Hölderian local error bound condition and the Hölderian continuity of the Jacobian. The formula of the convergence rates are given, which are functions with respect to the Levenberg-Marquardt parameter, the perturbation vector, as well as the orders of the Hölderian local error bound and Hölderian continuity of the Jacobian.

2021, 17(4): 2277-2277 doi: 10.3934/jimo.2021064 +[Abstract](211) +[HTML](107) +[PDF](114.37KB)
Abstract:

2019  Impact Factor: 1.366