# American Institute of Mathematical Sciences

ISSN:
1547-5816

eISSN:
1553-166X

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## Journal of Industrial & Management Optimization

July 2018 , Volume 14 , Issue 3

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2018, 14(3): 833-855 doi: 10.3934/jimo.2017078 +[Abstract](5072) +[HTML](1204) +[PDF](2338.72KB)
Abstract:

The alternating direction method of multipliers (ADMM) is an efficient approach for two-block separable convex programming, while it is not necessarily convergent when extending this method to multiple-block case directly. One appealing method is that converts the multiple-block variables into two groups firstly and then adopts the classic ADMM with inexact solving to the resulting model, which is so-called block-wise ADMM. However, solving the subproblems in block-wise ADMM are usually difficult when the linear mappings in the constraints are not diagonal or the proximal operator of the objective function is not easy to evaluate. Therefore, in this paper, we adopt the linearization technique to different terms presented in the block-wise ADMM subproblems, and obtain three kinds of linearized block-wise ADMM which make the subproblems easy to solve in general case. Moreover, under some mild conditions, we prove the global convergence of the three new methods and report some preliminary numerical results to indicate the feasibility and effectiveness of the linearization strategy.

2018, 14(3): 857-876 doi: 10.3934/jimo.2017079 +[Abstract](4623) +[HTML](952) +[PDF](466.54KB)
Abstract:

In this paper, an inventory control problem with a mean reverting inventory model is considered. The demand is assumed to follow a continuous diffusion process and a mean-reverting process which will take into account of the demand dependent of the inventory level. By choosing when and how much to stock, the objective is to minimize the long-run average cost, which consists of transaction cost for each replenishment, holding and shortage costs associated with the inventory level. An approach for deriving the average cost value of infinite time horizon is developed. By applying the theory of stochastic impulse control, we show that a unique (s, S) policy is indeed optimal. The main contribution of this work is to present a method to derive the (s, S) policy and hence the minimal long-run average cost.

2018, 14(3): 877-894 doi: 10.3934/jimo.2017080 +[Abstract](4635) +[HTML](889) +[PDF](491.76KB)
Abstract:

This paper considers an optimal production scheduling problem in a single-supplier-multi-manufacturer supply chain involving production and delivery time-delays, where the time-delays for the supplier and the manufacturers can have different values. The objective of both levels is to find an optimal production schedule so that their production rates and their inventory levels are close to the ideal values as much as possible in the whole planning horizon. Each manufacturer's problem, which involves one time-delayed argument, can be solved analytically by using the necessary condition of optimality. To tackle the supplier's problem involving $n+1$ different time-delayed arguments (where $n$ is the number of manufacturers) by the above approach, we need to introduce a model transformation technique which converts the original system of combined algebraic/differential equations with $n+1$ time-delayed arguments into a sum of $n$ sub-systems, each of which consists of only two time-delayed arguments. Thus, the supplier's problem can also be solved analytically. Numerical examples consisting of a single supplier and four manufacturers are solved to provide insight of the optimal strategies of both levels.

2018, 14(3): 895-912 doi: 10.3934/jimo.2017081 +[Abstract](4185) +[HTML](877) +[PDF](390.82KB)
Abstract:

Because interval-valued programming problem is used to tackle interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena, this paper considers a non-differentiable interval-valued optimization problem in which objective and all constraint functions are interval-valued functions, and the involved endpoint functions in interval-valued functions are locally Lipschitz and Clarke sub-differentiable. A necessary optimality condition is first established. Some sufficient optimality conditions of the considered problem are derived for a feasible solution to be an efficient solution under the $G-(F, ρ)$ convexity assumption. Weak, strong, and converse duality theorems for Wolfe and Mond-Weir type duals are also obtained in order to relate the efficient solution of primal and dual inter-valued programs.

2018, 14(3): 913-930 doi: 10.3934/jimo.2017082 +[Abstract](4432) +[HTML](891) +[PDF](435.61KB)
Abstract:

Uncertainty theory is a branch of axiomatic mathematics that deals with human uncertainty. Based on uncertainty theory, this paper discusses linear quadratic (LQ) optimal control with process state inequality constraints for discrete-time uncertain systems, where the weighting matrices in the cost function are assumed to be indefinite. By means of the maximum principle with mixed inequality constraints, we present a necessary condition for the existence of optimal state feedback control that involves a constrained difference equation. Moreover, the existence of a solution to the constrained difference equation is equivalent to the solvability of the indefinite LQ problem. Furthermore, the well-posedness of the indefinite LQ problem is proved. Finally, an example is provided to demonstrate the effectiveness of our theoretical results.

2018, 14(3): 931-951 doi: 10.3934/jimo.2017083 +[Abstract](4407) +[HTML](1212) +[PDF](528.9KB)
Abstract:

Existing papers on the Newsboy Problem that deal with price dependent demand and multiple discounts often analyze those two problems separately. This paper considers a setting where price dependence and multiple discounts are observed simultaneously, as is the case of the apparel industry. Henceforth, we analyze the optimal order quantity, initial selling price and discount scheme in the News-Vendor Problem context. The term of discount scheme is often used to specify the number of discounts as well as the discount percentages. We present a solution procedure of the problem with general demand distributions and two types of price-dependent demand: additive case and multiplicative case. We provide interesting insights based on a numerical study. An approximation method is proposed which confirms our numerical results.

2018, 14(3): 953-966 doi: 10.3934/jimo.2017084 +[Abstract](4494) +[HTML](956) +[PDF](367.36KB)
Abstract:

A Nash bargaining solution for Bayesian collective choice problem with general type and action spaces is built in this paper. Such solution generalizes the bargaining solution proposed by Myerson who uses finite sets to characterize the type and action spaces. However, in the real economics and industries, types and actions can hardly be characterized by a finite set in some circumstances. Hence our generalization expands the applications of bargaining theory in economic and industrial models.

2018, 14(3): 967-980 doi: 10.3934/jimo.2017085 +[Abstract](4925) +[HTML](1096) +[PDF](355.61KB)
Abstract:

Rescheduling in production planning means to schedule the sequenced jobs again together with a set of new arrived jobs so as to generate a new feasible schedule, which creates disruptions to any job between the original and adjusted position. In this paper, we study rescheduling problems with learning effect under disruption constraints to minimize several classical objectives, where learning effect means that the workers gain experience during the process of operation and make the actual processing time of jobs shorter than their normal processing time. The objectives are to find optimal sequences to minimize the makespan and the total completion time under a limit of the disruptions from the original schedule. For the considered objectives under a single disruption constraint or a disruption cost constraint, we propose polynomial-time algorithms and pseudo-polynomial time algorithms, respectively.

2018, 14(3): 981-1005 doi: 10.3934/jimo.2017086 +[Abstract](4285) +[HTML](1247) +[PDF](544.38KB)
Abstract:

This paper focuses on a class of mathematical programs with symmetric cone complementarity constraints (SCMPCC). The explicit expression of C-stationary condition and SCMPCC-linear independence constraint qualification (denoted by SCMPCC-LICQ) for SCMPCC are first presented. We analyze a parametric smoothing approach for solving this program in which SCMPCC is replaced by a smoothing problem \begin{document}$P_{\varepsilon}$\end{document} depending on a (small) parameter \begin{document}$\varepsilon$\end{document}. We are interested in the convergence behavior of the feasible set, stationary points, solution mapping and optimal value function of problem \begin{document}$P_{\varepsilon}$\end{document} when \begin{document}$\varepsilon \to 0$\end{document} under SCMPCC-LICQ. In particular, it is shown that the convergence rate of Hausdorff distance between feasible sets \begin{document}$\mathcal{F}_{\varepsilon}$\end{document} and \begin{document}$\mathcal{F}$\end{document} is of order \begin{document}$\mbox{O}(|\varepsilon|)$\end{document} and the solution mapping and optimal value of \begin{document}$P_{\varepsilon}$\end{document} are outer semicontinuous and locally Lipschitz continuous at \begin{document}$\varepsilon=0$\end{document} respectively. Moreover, any accumulation point of stationary points of \begin{document}$P_{\varepsilon}$\end{document} is a C-stationary point of SCMPCC under SCMPCC-LICQ.

2018, 14(3): 1007-1022 doi: 10.3934/jimo.2017087 +[Abstract](4192) +[HTML](821) +[PDF](371.85KB)
Abstract:

The problem of solving a fuzzy relational equation plays an important role in fuzzy systems. In this paper, we investigate the uniqueness of solutions of fuzzy relational equations regarding Max-av composition through the relationship between minimal solutions and minimal coverage. A method for verifying the strong regularity of matrices in fuzzy Max-av algebra is proposed in the paper.

2018, 14(3): 1023-1040 doi: 10.3934/jimo.2017088 +[Abstract](6094) +[HTML](1653) +[PDF](559.23KB)
Abstract:

We discuss the optimal pricing and production decisions in a channel supply chain under symmetric and asymmetric information cases. We compare the optimum policies between the asymmetric information and the full information cases. We analyze the effect of the reseller's cost information on the profits of the partners. We find that information asymmetry is beneficial to the reseller, but is inefficient to the manufacturer and the whole supply chain. The information value increases with uncertainties arising from the reseller's cost structure. The dual-channel supply chain can share information and achieve coordination, if the lump-sum side payment from the manufacturer can make up the loss of the reseller due to sharing information. Finally, the effectiveness of the proposed models is verified by numerical examples.

2018, 14(3): 1041-1054 doi: 10.3934/jimo.2017089 +[Abstract](4287) +[HTML](923) +[PDF](424.89KB)
Abstract:

The aim of this paper is to develop second-order necessary and second-order sufficient optimality conditions for cone constrained multi-objective optimization. First of all, we derive, for an abstract constrained multi-objective optimization problem, two basic necessary optimality theorems for weak efficient solutions and a second-order sufficient optimality theorem for efficient solutions. Secondly, basing on the optimality results for the abstract problem, we demonstrate, for cone constrained multi-objective optimization problems, the first-order and second-order necessary optimality conditions under Robinson constraint qualification as well as the second-order sufficient optimality conditions under upper second-order regularity for the conic constraint. Finally, using the optimality conditions for cone constrained multi-objective optimization obtained, we establish optimality conditions for polyhedral cone, second-order cone and semi-definite cone constrained multi-objective optimization problems.

2018, 14(3): 1055-1083 doi: 10.3934/jimo.2017090 +[Abstract](4662) +[HTML](972) +[PDF](557.56KB)
Abstract:

This paper assumes that an insurer can control the dividend, refinancing and reinsurance strategies dynamically. Particularly, the reinsurance is provided by two reinsurers and the variance premium principle is applied in pricing insurance contracts. Using the optimal control method, we identify the optimal strategies for maximizing the insurance company's value. Meanwhile, the effects of transaction costs and terminal value at bankruptcy are investigated. The results turn out that the insurer should consider refinancing when and only when the transaction costs and terminal value are relatively low. Also, it should buy less reinsurance when the surplus increases, while the proportion of risk allocation between two reinsurers remains constant. When the dividend rate is unbounded, dividends should be paid according to the barrier strategy. When the dividend rate is restricted, dividends should be distributed according to the threshold strategy. Some examples are provided to illustrate the implementation of our results.

2018, 14(3): 1085-1104 doi: 10.3934/jimo.2017091 +[Abstract](4185) +[HTML](1141) +[PDF](482.28KB)
Abstract:

The classical multifacility Weber problem (MFWP) is one of the most important models in facility location. This paper considers more general and practical case of MFWP called constrained multifacility Weber problem (CMFWP), in which the gauge is used to measure distances and locational constraints are imposed to facilities. In particular, we develop a variational inequality approach for solving it. The CMFWP is reformulated into a linear variational inequality, whose special structures lead to new projection-type methods. Global convergence of the projection-type methods is proved under mild assumptions. Some preliminary numerical results are reported which verify the effectiveness of proposed methods.

2018, 14(3): 1105-1122 doi: 10.3934/jimo.2018001 +[Abstract](4550) +[HTML](967) +[PDF](678.95KB)
Abstract:

This study investigates a budget-constrained retailer's optimal financing and portfolio order policies in a supply chain with option contracts. To this end, we develop two analytical models: a basic model with wholesale price contracts as the benchmark and a model with option contracts. Each model considers both the financing scenario and the no-financing scenario. Our analyses show that the retailer uses wholesale price contracts for procurement, instead of option contracts, when its budget is extremely tight. The retailer starts to use a combination of these two types of contracts when the budget constraint is relieved. As the budget increases, the retailer adjusts the procurement ratio through both types until it can implement the optimal ordering policy with an adequate budget. In addition, the condition for seeking external financing is determined by the retailer's initial budget, financing cost, and profit margin.

2018, 14(3): 1123-1141 doi: 10.3934/jimo.2018002 +[Abstract](4463) +[HTML](978) +[PDF](473.12KB)
Abstract:

Supply chain network (SCN) is a complex nonlinear system and may have a chaotic behavior. This network involves multiple entities that cooperate to satisfy customers demand and control network inventory. The policy of each entity in demand forecast and inventory control, and constraints and uncertainties of demand and supply (or production) significantly affects the complexity of its behavior. In this paper, a supply chain network is investigated that has two ordering policies: smooth ordering policy and a new policy that is designed based on proportional-derivative controller. Two forecast methods are used in the network: moving average (MA) forecast and exponential smoothing (ES) forecast. The supply capacity of each entity is constrained. The effect of demand elasticity, which is the result of marketing activities, is involved in the SCN. The inventory adjustment parameter and demand elasticity are the most important decision parameters in the SCN. Overall, four scenarios are designed for modeling and analyzing the chaotic behavior of the network and in each scenario the maximum Lyapunov exponent is calculated and drawn. Finally, the best scenario for decision-making is obtained.

2018, 14(3): 1143-1155 doi: 10.3934/jimo.2018003 +[Abstract](3837) +[HTML](910) +[PDF](435.3KB)
Abstract:

Motivated by the proximal-like bundle method [K. C. Kiwiel, Journal of Optimization Theory and Applications, 104(3) (2000), 589-603.], we establish a new proximal Chebychev center cutting plane algorithm for a type of nonsmooth optimization problems. At each step of the algorithm, a new optimality measure is investigated instead of the classical optimality measure. The convergence analysis shows that an \begin{document}$\varepsilon$\end{document}-optimal solution can be obtained within \begin{document}$O(1/\varepsilon^3)$\end{document} iterations. The numerical result is presented to show the validity of the conclusion and it shows that the method is competitive to the classical proximal-like bundle method.

2018, 14(3): 1157-1178 doi: 10.3934/jimo.2018004 +[Abstract](4195) +[HTML](889) +[PDF](379.56KB)
Abstract:

We introduce the concept of null set in the space of all bounded closed intervals. Based on this concept, we can define two partial orderings according to the substraction and Hukuhara difference between any two bounded closed intervals, which will be used to define the solution concepts of interval-valued optimization problems. On the other hand, we transform the interval-valued optimization problems into the conventional vector optimization problem. Under these settings, we can apply the technique of scalarization to solve this transformed vector optimization problem. Finally, we show that the optimal solution of the scalarized problem is also the optimal solution of the original interval-valued optimization problem.

2018, 14(3): 1179-1201 doi: 10.3934/jimo.2018005 +[Abstract](3946) +[HTML](829) +[PDF](668.34KB)
Abstract:

In this paper, a modified dividend strategy is proposed by delaying dividend payments until the insurer's surplus level remains at or above a threshold level b for a predetermined period of time h. We consider two cases depending on whether the period of time sustained at or above level b is counted either consecutively or accumulatively (referred to as standard or cumulative waiting period). In both cases, we develop a recursive computational procedure to calculate the expected total discounted dividend payments made prior to ruin for a discrete-time Sparre Andersen renewal risk process. By varying the values of b and h, a numerical study of the trade-off effects between finite-time ruin probabilities and expected total discounted dividend payments is investigated under a variety of scenarios. Finally, a generalized threshold-based strategy with a delayed dividend payment rule is studied under the compound binomial model.

2018, 14(3): 1203-1218 doi: 10.3934/jimo.2018006 +[Abstract](4092) +[HTML](1120) +[PDF](586.4KB)
Abstract:

This paper develops a multi-period product pricing and service investment model to discuss the optimal decisions of the participants in a supplier-dominant supply chain under uncertainty. The supply chain consists of a risk-neutral supplier and two risk-averse manufacturers, of which one manufacturer can provide real-time customer service based on the Internet of Things (IoT). In each period of the Stackelberg game, the supplier decides its wholesale price to maximize the profit while the manufacturers make pricing and service investment decisions to maximize their respective utility. Using the backward induction, we first investigate the effects of risk-averse coefficients and price sensitive coefficients on the optimal decisions of the manufacturers. We find that the decisions of one manufacturer are inversely proportional to both risk-averse coefficients and its own price sensitive coefficient, while proportional to the price sensitive coefficient of its rival. Then, we derive the first-best wholesale price of the supplier and analyze how relevant factors affect the results. A numerical example is conducted to verify our conclusions and demonstrate the advantages of the IoT technology in long-term competition. Finally, we summarize the main contributions of this paper and put forward some advices for further study.

2018, 14(3): 1219-1249 doi: 10.3934/jimo.2018007 +[Abstract](5257) +[HTML](2404) +[PDF](1181.83KB)
Abstract:

This paper focuses on the batch scheduling problem in multi-hybrid cell manufacturing systems (MHCMS) in a dual-resource constrained (DRC) setting, considering skilled workforce assignment (SWA). This problem consists of finding the sequence of batches on each cell, the starting time of each batch, and assigning employees to the operations of batches in accordance with the desired objective. Because handling both the scheduling and assignment decisions simultaneously presents a challenging optimization problem, it is difficult to solve the formulated model, even for small-sized problem instances. Three metaheuristics are proposed to solve the batch scheduling problem, namely the genetic algorithm (GA), simulated annealing (SA) algorithm, and artificial bee colony (ABC) algorithm. A serial scheduling scheme (SSS) is introduced and employed in all metaheuristics to obtain a feasible schedule for each individual. The main aim of this study is to identify an effective metaheuristic for determining the scheduling and assignment decisions that minimize the average cell response time. Detailed computational experiments were conducted, based on real production data, to evaluate the performance of the metaheuristics. The experimental results show that the performance of the proposed ABC algorithm is superior to other metaheuristics for different levels of experimental factors determined for the number of batches and the employee flexibility.

2018, 14(3): 1251-1269 doi: 10.3934/jimo.2018052 +[Abstract](4078) +[HTML](898) +[PDF](661.75KB)
Abstract:

Diffusiophoresis is a common phenomenon that occurs when colloids are placed in the non-uniform solute concentration. It generates solute gradients which force the colloids to transfer toward or away from the higher solute concentration side. In this paper, we consider the input sequence control of the colloid transport in a dead-end micro-channel with a boundary solute concentration being manipulated, which has a wide range of applications such as drug delivery, biology transport, oil recovery system and so on. We model this process by a coupled system, which involves the solute diffusion equation and the colloid transport model. Then an optimal control problem is formulated, in which the goal is to minimize colloid density distribution deviation between the computational one and the target at a pre-specified terminal time. To solve this partial differential equation (PDE) optimal control problem, we first apply the control parameterization method to discretize the boundary control and transfer it into an optimal parameter selection problem. Then, using the variational method, the gradient of the objective function with respect to the decision parameters can be derived, which depends on the solution of the coupled system and the costate system. Based on this, we propose an effective computational method and a gradient-based optimization algorithm to solve the optimal control problem numerically. Finally, we give the simulation results to demonstrate that the objective function based on the proposed method is less nearly two orders of magnitude than that of a constant value control strategy, which well illustrates the effectiveness of the proposed method.

2018, 14(3): 1271-1295 doi: 10.3934/jimo.2018083 +[Abstract](3946) +[HTML](1709) +[PDF](2909.41KB)
Abstract:

In the present paper, a robust approach is used to locate hub facilities considering network risks. An additional objective function, minimax regret, is added to the classical objective function in the hub location problem. In the proposed model, risk factors such as availability, security, delay time, environmental guidelines and regional air pollution are considered using triangular fuzzy-stochastic numbers. Then an equivalent crisp single objective model is proposed and solved by the Benders decomposition method. Finally, the results of both Benders decomposition and commercial optimization software are compared for different instances. Numerical instances were developed based on the well-known Civil Aeronautics Board (CAB) data set, considering different levels of uncertainty in parameters. The results show that the proposed model is capable of selecting nodes as sustainable hubs. Also, the results confirm that using Benders decomposition is more efficient than using classical solution methods for large-scale problems.

2019  Impact Factor: 1.366