# American Institute of Mathematical Sciences

ISSN:
1547-5816

eISSN:
1553-166X

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## Journal of Industrial & Management Optimization

October 2019 , Volume 15 , Issue 4

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2019, 15(4): 1517-1534 doi: 10.3934/jimo.2018107 +[Abstract](2844) +[HTML](1126) +[PDF](449.32KB)
Abstract:

In this paper, an interior point continuous path-following trajectory is proposed for linear programming. The descent direction in our continuous trajectory can be viewed as some combination of the affine scaling direction and the centering direction for linear programming. A key component in our interior point continuous path-following trajectory is an ordinary differential equation (ODE) system. Various properties including the convergence in the limit for the solution of this ODE system are analyzed and discussed in detail. Several illustrative examples are also provided to demonstrate the numerical behavior of this continuous trajectory.

2019, 15(4): 1535-1564 doi: 10.3934/jimo.2018109 +[Abstract](3121) +[HTML](1327) +[PDF](818.54KB)
Abstract:

Soccer is one of the most popular sports in the world with millions of fans that usually raise interesting questions when the competition is partially completed. One interesting question relates to the elimination problem which consists in checking at some stage of the competition if a team i still has a theoretical chance to finish first in a league or be within the first k teams in a tournament to qualify to the playoffs (e.g., become the champion if k = 1). Some other interesting problems from the literature are the guaranteed qualification problem, the possible qualification problem, the score vector problem, promotion and relegation problem. These problems are NP-complete for the actual FIFA pointing rule system (0 points-loss, 1 point-tie, 3 points-win). SABIO is an online platform that helps users discover information related to soccer by letting them formulate questions in form of constraints and go beyond the classical soccer computational problems. In this paper we considerably improve the performance of an existing constraint programming (CP) model and combine the use of mixed integer programming (MIP) and CP to answer general soccer queries in a real-time application.

2019, 15(4): 1565-1577 doi: 10.3934/jimo.2018110 +[Abstract](2857) +[HTML](1566) +[PDF](312.07KB)
Abstract:

We study a sports scheduling problem with the objective of minimizing carry-over effects in round robin tournaments.In the first part, focusing on tournaments that allow minimum number of breaks (at most one) for each team, we formulate an integer programming model and provide an efficient heuristic algorithm to solve this computationally expensive problem. We apply the algorithm to the current Turkish Professional Football League and present an alternative scheduling template. In the second part, we discuss how the carry-over effects can be further decreased if the number of breaks is allowed to be of slightly larger value and numerically represent this trade-off.

2019, 15(4): 1579-1597 doi: 10.3934/jimo.2018112 +[Abstract](2725) +[HTML](1229) +[PDF](465.83KB)
Abstract:

Recognizing that strategic consumers have become increasingly common in the perishable products market, we develop a two-stage pricing model for a monopolistic firm with two classes of inventory strategies: non-replenishment and replenishment. First, the retailer mapping out his pricing policy, and then consumers determining their buying behavior given the retailers policy. Our results indicate that the game equilibrium exists between retailers and consumers in both cases. For a given realized price and inventory policy, the consumer's space is split into several areas by the optimal threshold functions. Inventory replenishment decisions are affected by market demand and a decline factor of consumers reservation value. The retailers profit loss is not necessarily related to the consumers waiting behavior but results from the ignorance of this behavior when pricing.

2019, 15(4): 1599-1615 doi: 10.3934/jimo.2018113 +[Abstract](2346) +[HTML](1164) +[PDF](609.99KB)
Abstract:

This paper investigates the low-priority customers' strategic behavior in the single-server queueing system with general service time and two customer types. The priority system is preemptive resume, which means that if a high-priority customer enters the system that are serving a low-priority customer, the arriving customer preempts the service facility and the preempted customer returns to the head of the queue for his own class. The customer who is preempted resumes service at the point of interruption upon reentering the system. The low-priority customer's dilemma is whether to join or balk based on a linear reward-cost structure. Two cases are distinguished based on the different levels of information that the low-priority customers acquire before joining the system. The equilibrium threshold strategy in the observable case and the equilibrium balking strategy as well as the socially optimal balking strategy in the unobservable case for the low-priority customers are derived finally.

2019, 15(4): 1617-1630 doi: 10.3934/jimo.2018114 +[Abstract](2118) +[HTML](1110) +[PDF](363.72KB)
Abstract:

Suppliers always provide free-shipping for retailers whose total order value exceeds or equals an explicit promotion threshold. This paper incorporates a shipping fee in the discrete multi-period newsvendor problem and applies Weak Aggregating Algorithm (WAA) to offer explicit online ordering strategy. It further considers an extended case with salvage value and shortage cost. In particular, online ordering strategies are derived based on return loss function. Numerical examples serve to illustrate the competitive performance of the proposed ordering strategies. Results show that newsvendors' cumulative return losses are affected by the threshold of the order value-based free-shipping. Moreover, the introduction of salvage value and shortage cost greatly improves the competitive performance of online ordering strategies.

2019, 15(4): 1631-1651 doi: 10.3934/jimo.2018115 +[Abstract](2812) +[HTML](1430) +[PDF](773.46KB)
Abstract:

This paper considers a three-echelon supply chain system with one raw-material supplier, one manufacturer and one retailer in which both the manufacturer and the raw-material supplier are exposed to the risk of production disruptions. The market demand is assumed to be uncertain but sensitive to the retail price. The objective is to determine the optimal lot sizes of the supplier and the manufacturer, and the selling price of the retailer when the wholesale prices of the upstream entities are prescribed and the retailer's order quantity is chosen before the actual demand is realized. As the benchmark case, the expected total profit of the centralized channel is maximized. The decentralized supply chain is coordinated under pairwise and spanning revenue sharing mechanisms. Numerical study shows that disruptions have remarkable impact on supply chain decisions.

2019, 15(4): 1653-1675 doi: 10.3934/jimo.2018116 +[Abstract](2308) +[HTML](1226) +[PDF](583.6KB)
Abstract:

We consider the sample average approximation method for a stochastic multiobjective programming problem constrained by parametric variational inequalities. The first order necessary conditions for the original problem and its sample average approximation (SAA) problem are established under constraint qualifications. By graphical convergence of set-valued mappings, the stationary points of the SAA problem converge to the stationary points of the true problem when the sample size tends to infinity. Under proper assumptions, the convergence of optimal solutions of SAA problems is also obtained. The numerical experiments on stochastic multiobjective optimization problems with variational inequalities are given to illustrate the efficiency of SAA estimators.

2019, 15(4): 1677-1699 doi: 10.3934/jimo.2018117 +[Abstract](2153) +[HTML](972) +[PDF](491.84KB)
Abstract:

In this paper, we study an extended trust region subproblem (eTRS) in which the unit ball intersects with \begin{document}$m$\end{document} linear inequality constraints. In the literature, Burer et al. proved that an SOC-SDP relaxation (SOCSDPr) of eTRS is exact, under the condition that the nonredundant constraints do not intersect each other in the unit ball. Furthermore, Yuan et al. gave a necessary and sufficient condition for the corresponding SOCSDPr to be a tight relaxation when \begin{document}$m = 2$\end{document}. However, there lacks a recovering algorithm to generate an optimal solution of eTRS from an optimal solution \begin{document}$X^*$\end{document} of SOCSDPr when rank \begin{document}$(X^*)≥ 2$\end{document} and \begin{document}$m≥ 3$\end{document}. This paper provides such a recovering algorithm to complement those known works.

2019, 15(4): 1701-1727 doi: 10.3934/jimo.2018118 +[Abstract](2740) +[HTML](1446) +[PDF](536.82KB)
Abstract:

In current competitive market, the products and their demand's uncertainty are high. In order to reduce these uncertainties the coordination of supply chain is necessary. Supply chain can be managed under two viewpoints typically: 1) centralized supply chain and 2) decentralized supply chain, and the coordination can be done in both types of chains. In the centralized supply chain there exists a global decision maker who takes all the best decisions in order to maximize the profit of the whole supply chain. Here, the useful information required to make the best decisions is open to all members of the chain. On the other hand, in the decentralized supply chain all members decide in a separate and sequential way, how to maximize their profits. In order to coordinate efficiently the supply chain, both supplier and retailer are involved in a coordination contract that makes it possible for the decentralized decisions to maximize the profit of the entire supply chain. In this context, the situation that the supplier-retailer chain faces is a two-stage decision model. In the first stage the supplier, based on former knowledge about the market, decides the production capacity to reserve for the retailer. In the second stage, after that demand information is updated, the retailer determines the bundle price and the quantity of bundles to order. This paper considers a supply chain comprised of one supplier and one retailer in which two complementary fashion products are manufactured and sold as a bundle. The bundle has a short selling season and a stochastic price dependent on demand with a high level of uncertainty. Therefore, this research considers that the demand rates are uncertain and are dependent on selling prices and on a random noise effect on the market. Profit maximization models are developed for centralized and decentralized supply chains to determine decisions on production capacity reservation, order quantity of bundled products and the bundle-selling price. The applicability of the developed models and solution method are illustrated with a numerical example.

2019, 15(4): 1729-1731 doi: 10.3934/jimo.2018119 +[Abstract](1813) +[HTML](913) +[PDF](286.85KB)
Abstract:

This paper proposes a dynamic programming algorithm for the NRCSRP with multiple crews. This algorithm also improves the existing algorithm for the problem with a single crew.

2019, 15(4): 1733-1751 doi: 10.3934/jimo.2018120 +[Abstract](2748) +[HTML](899) +[PDF](538.62KB)
Abstract:

This paper considers a logistics distribution network with multiple suppliers that each replenish a set of retailers having constant demand rates. The underlying optimization problem is the Cyclic Inventory Routing Problem (CIRP), for which a heuristic solution method is developed. Further, horizontal collaboration through a third party Logistics Service Provider (LSP) is considered and the collaborative savings potential is analyzed. A design of experiments is performed to evaluate the impact of some relevant cost and network structure factors on the collaborative savings potential. The results from the design of experiments show that for some factor combinations there is in fact no significant savings potential.

2019, 15(4): 1753-1772 doi: 10.3934/jimo.2018121 +[Abstract](2534) +[HTML](1097) +[PDF](411.97KB)
Abstract:

This paper addresses a vendor-managed inventory (VMI) supply chain with a loss-averse manufacturer and a risk-neutral retailer. Market demand faced by the retailer is stochastic and dependent on product quality level and marketing effort level. We propose a combined contract composed of option and cost-sharing to investigate coordination and profit allocation issues of the supply chain. To model loss aversion of the manufacturer, we employ multiple mental accounts and apply the utility function to upside and downside potentials of manufacturer's production decision separately. We derive the optimal strategy for each member with a Stackelberg game in which the retailer acts as the leader. It is proved that both coordination of the supply chain and Pareto-improvement can be achieved synchronously by the combined contract. In the premise of coordination, the system-wide profit can be allocated arbitrarily only by option price. Through negotiation, the retailer and the manufacturer just need to confirm an appropriate option price to obtain that neither of them becomes worse off. We also find that the manufacturer's loss aversion is a significant element for contract design and profit allocation, and the manufacturer could benefit from its own loss aversion behavior under certain condition.

2019, 15(4): 1773-1793 doi: 10.3934/jimo.2018122 +[Abstract](3004) +[HTML](1469) +[PDF](603.69KB)
Abstract:

Memoryless quasi-Newton methods are studied for solving large-scale unconstrained optimization problems. Recently, memoryless quasi-Newton methods based on several kinds of updating formulas were proposed. Since the methods closely related to the conjugate gradient method, the methods are promising. In this paper, we propose a memoryless quasi-Newton method based on the Broyden family with the spectral-scaling secant condition. We focus on the convex and preconvex classes of the Broyden family, and we show that the proposed method satisfies the sufficient descent condition and converges globally. Finally, some numerical experiments are given.

2019, 15(4): 1795-1807 doi: 10.3934/jimo.2018123 +[Abstract](2268) +[HTML](1018) +[PDF](420.54KB)
Abstract:

In this paper, we utilize a new homotopy method to solve generalized Nash equilibrium problem with equality and inequality constraints on unbounded sets. Based on the existing homotopy method, we establish a new homotopy equation by introducing a suitable perturbation on the equality constraint, the existence and the global convergence of homotopy path under certain assumptions have also been proved. In the proposed method, the initial point only needs to satisfy the inequality constraints. Compared with the existing homotopy method, this method expands the scope of the initial points and provides the convenience for solving generalized Nash equilibrium problem. The numerical results illustrate the effectiveness of this method.

Li Xue and
2019, 15(4): 1809-1830 doi: 10.3934/jimo.2018124 +[Abstract](2733) +[HTML](1115) +[PDF](501.04KB)
Abstract:

In real life, investors face background risk which may affect their portfolio selection decision. In addition, since the security market is too complex, there are situations where the future security returns cannot be reflected by historical data and have to be given by experts' estimations according to their knowledge and judgement. This paper discusses a portfolio selection problem with background risk in such an uncertain environment. In the paper, in order to reflect different attitudes towards risk that vary by goal in one portfolio investment, we apply mental accounts to the investment. Using uncertainty theory, we propose an uncertain portfolio selection model with mental accounts and background risk and provide the determinate form of the model. Moreover, we discuss the shape and location of efficient frontier of the subportfolios with background risk and without background risk. Further, we present the conditions under which the optimal aggregate portfolio is on the efficient frontier when return rates of security and background asset are all normal uncertain variables. Finally, a real portfolio selection example is given as an illustration.

2019, 15(4): 1831-1856 doi: 10.3934/jimo.2018125 +[Abstract](2875) +[HTML](1014) +[PDF](1035.03KB)
Abstract:

A Supply Chain Finance (SCF) system involving and a commercial bank and a capital-constrained retailer is designed in the imperfect capital market with non-zero bankruptcy costs. A decentralized borrower-lender game is analyzed, and the optimal centralized strategy is developed for SCF from the perspective of multi-attribute utility (MAU) maximization, including maximizing the expected profit and the service level, as well as minimizing the bankruptcy cost. Furthermore, we analytically and numerically explore the coordination condition for SCF and conclude that the bank financing scheme with a suitable combination of decision preferences can realize coordination, even super coordination. Through sensitivity analyses and numerical experiments, we discuss the impacts of the borrower's initial capitals on the upstream firm's pricing decision and dig out why he has incentives to support the retailer's choice of adopting SCF. The findings of this study reveal that the capital-constrained retailer would require more initial capital when maximizing MAU than maximizing the expected profit, and thus the equilibrium order quantity and the bankruptcy risk would also be higher. Moreover, based on a suitable combination of decision preferences, our proposed bank financing scheme can realize coordination, even super coordination.

2019, 15(4): 1857-1879 doi: 10.3934/jimo.2018126 +[Abstract](1331) +[HTML](465) +[PDF](802.94KB)
Abstract:

This paper presents an inventory model for deteriorating items with variable demand when shortage is permitted and quantity discount in purchase cost, and rework on deteriorating products are also allowed. The main idea of this research is to study the effects of the discount and the rework on the inventory costs. In this paper, it is assumed that for a certain quantity of purchased items, the seller would offer a discount and the manager would have the choice to either accept the discount or dismiss. On the other hand, there is also a similar decision-making scenario, where the manager makes a decision to reduce the total costs by using the rework and reducing the shortage periods or reducing the total costs by ignoring the rework cost and increasing the shortage periods. The implementation of the mathematical model is illustrated with a numerical example and sensitivity analysis describes the effects of the parameters on the total costs. The results show that the rework will decrease the total costs of the inventory system, significantly.

2019, 15(4): 1881-1896 doi: 10.3934/jimo.2018127 +[Abstract](1991) +[HTML](1119) +[PDF](545.12KB)
Abstract:

The sparse probabilistic Boolean network (SPBN) model has been applied in various fields of industrial engineering and management. The goal of this model is to find a sparse probability distribution based on a given transition-probability matrix and a set of Boolean networks (BNs). In this paper, a partial proximal-type operator splitting method is proposed to solve a separable minimization problem arising from the study of the SPBN model. All the subproblem-solvers of the proposed method do not involve matrix multiplication, and consequently the proposed method can be used to deal with large-scale problems. The global convergence to a critical point of the proposed method is proved under some mild conditions. Numerical experiments on some real probabilistic Boolean network problems show that the proposed method is effective and efficient compared with some existing methods.

2019, 15(4): 1897-1920 doi: 10.3934/jimo.2018128 +[Abstract](2015) +[HTML](1002) +[PDF](469.47KB)
Abstract:

In this work, we propose an approximating scheme based on the proximal point algorithm, for solving generalized fractional programs (GFP) by their continuous reformulation, also known to as partial dual counterparts of GFP. Bundle dual algorithms are then derived from this scheme. We prove the convergence and the rate of convergence of these algorithms. As for dual algorithms, the proposed methods generate a sequence of values that converges from below to the minimal value of \begin{document}$(P)$\end{document}, and a sequence of approximate solutions that converges to a solution of the dual problem. For certain classes of problems, the convergence is at least linear.

2019, 15(4): 1921-1936 doi: 10.3934/jimo.2018129 +[Abstract](2330) +[HTML](942) +[PDF](359.87KB)
Abstract:

Currency option is an important risk management tool in the foreign exchange market, which has attracted the attention of many researchers. Unlike the classical stochastic theory, we investigate the valuation of currency option under the assumption that the risk factors are described by uncertain processes. Considering the long-term fluctuations of the exchange rate and the changing of the interest rates from time to time, we propose a mean-reverting uncertain currency model with floating interest rates to simulate the foreign exchange market. Subsequently, European and American currency option pricing formulas for the new currency model are derived and some mathematical properties of the formulas are studied. Finally, some numerical algorithms are designed to calculate the prices of these options.

2019, 15(4): 1937-1953 doi: 10.3934/jimo.2018130 +[Abstract](2468) +[HTML](1081) +[PDF](397.88KB)
Abstract:

This paper investigates the problem of dynamic investment and consumption in a market, where a risky asset evolves with jumps and capital gains are taxed. In addition, the investor's behavior of tax evasion is taken into account, and tax evasion is subject to penalty when it is uncovered by audits. Using dynamic programming approach, we derive an analytical solution for an investor with the CRRA utility. We find the following: (1) jumps in the risky asset do not affect the optimal tax evasion strategy; (2) jump risk lessens the optimal fraction of wealth in the risky asset; (3) tax evasion can be reduced by increasing the fine and/or the frequency of tax audits; (4) the effects of the jumps, audits and penalty on the optimal consumption are determined by the degree of risk aversion of the investor.

2019, 15(4): 1955-1964 doi: 10.3934/jimo.2018131 +[Abstract](2003) +[HTML](1059) +[PDF](327.43KB)
Abstract:

In this paper, we consider the scheduling with step-deteriorating jobs. The objective is to minimize the makespan. We first propose a property of any optimal schedule after analyzing the NP-hardness for the parallel-machine scheduling with a common deteriorating date, and then present a fully polynomial time approximation scheme for the case where the number of machines $m$ is a constant and jobs have anti-agreeable parameters. Furthermore, we show that the single-machine scheduling is NP-hard in the strong sense if jobs have distinct release dates and distinct deteriorating dates.

2019, 15(4): 1965-1993 doi: 10.3934/jimo.2018132 +[Abstract](2436) +[HTML](1090) +[PDF](526.68KB)
Abstract:

This paper deals with the optimal liability and dividend strategies for an insurance company in Markov regime-switching models. The objective is to maximize the total expected discounted utility of dividend payment in the infinite time horizon in the logarithm and power utility cases, respectively. The switching process, which is interpreted by a hidden Markov chain, is not completely observable. By using the technique of the Wonham filter, the partially observed system is converted to a completely observed one and the necessary information is recovered. The upper-lower solution method is used to show the existence of classical solution of the associated second-order nonlinear Hamilton-Jacobi-Bellman equation in the two-regime case. The explicit solution of the value function is derived and the corresponding optimal dividend policies and liability ratios are obtained. In the multi-regime case, a general setting of the Wonham filter is presented, and the value function is proved to be a viscosity solution of the associated system of Hamilton-Jacobi-Bellman equations.

2019, 15(4): 1995-2008 doi: 10.3934/jimo.2018133 +[Abstract](1950) +[HTML](825) +[PDF](342.03KB)
Abstract:

Uncertain heat equation is a class of uncertain partial differential equations involving Liu processes. This paper first gives the uncertainty distributions and the inverse uncertainty distributions of extreme values of solutions for uncertain heat equations. Numerical methods are designed to gain the inverse uncertainty distributions of extreme values of solutions.

2019, 15(4): 2009-2021 doi: 10.3934/jimo.2018134 +[Abstract](2558) +[HTML](865) +[PDF](619.51KB)
Abstract:

It is a popular research topic in computer vision community to find a solution for the zero norm minimization problem via solving its non-convex relaxation problem. In fact, there are already many existing algorithms to solve the non-convex relaxation problem. However, most of them are computationally expensive due to the non-Lipschitz property of this problem and thus these existing algorithms are not suitable for many engineering problems with large dimensions.

In this paper, we first develop an efficient algorithm to solve the non-convex relaxation problem via solving a sequence of non-convex sub-problems based on our recent work. To this end, we reformulate the minimization problem into another non-convex one but with non-negative constraint. Then we can transform the non-Lipschitz continuous non-convex problem with the non-negative constraint into a Lipschitz continuous problem, which allows us to use some efficient existing algorithms for its solution. Based on the proposed algorithm, an important relation between the solutions of relaxation problem and the original zero norm minimization problem is established from a different point of view. The results in this paper reveal two important issues: ⅰ) The solution of non-convex relaxation minimization problem converges to the solution of the original problem; ⅱ) The general non-convex relaxation problem can be solved efficiently with another reformulated high dimension problem with nonnegative constraint. Finally, some numerical results are used to demonstrate effectiveness of the proposed algorithm.

2019, 15(4): 2023-2034 doi: 10.3934/jimo.2018135 +[Abstract](1953) +[HTML](819) +[PDF](369.4KB)
Abstract:

We propose two new double projection algorithms for solving the split feasibility problem (SFP). Different from the extragradient projection algorithms, the proposed algorithms do not require fixed stepsize and do not employ the same projection region at different projection steps. We adopt flexible rules for selecting the stepsize and the projection region. The proposed algorithms are shown to be convergent under certain assumptions. Numerical experiments show that the proposed methods appear to be more efficient than the relaxed- CQ algorithm.

2018  Impact Factor: 1.025