# American Institute of Mathematical Sciences

ISSN:
1547-5816

eISSN:
1553-166X

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## Journal of Industrial & Management Optimization

October 2016 , Volume 12 , Issue 4

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2016, 12(4): 1173-1186 doi: 10.3934/jimo.2016.12.1173 +[Abstract](1317) +[PDF](376.4KB)
Abstract:
$H$-tensor is a new developed concept in tensor analysis and it is an extension of $H$-matrix and $M$-tensor. Based on the spectral theory of nonnegative tensors, several equivalent conditions of nonsingular $H$-tensors are established in the literature. However, these conditions can not be used as a criteria to identify nonsingular $H$-tensors as they are hard to verify. In this paper, based on the diagonal product dominance and $S$ diagonal product dominance of a tensor, we establish some new implementable criteria in identifying nonsingular $H$-tensors. The positive definiteness of nonsingular $H$-tensors with positive diagonal entries is also discussed in this paper. The obtained results extend the corresponding conclusions for nonsingular $H$-matrices and improve the existing results for nonsingular $H$-tensors.
2016, 12(4): 1187-1197 doi: 10.3934/jimo.2016.12.1187 +[Abstract](916) +[PDF](312.2KB)
Abstract:
In this paper, we investigate semidefinite programming by using the image space analysis and present some equivalence between the (regular) linear separation and the saddle points of the Lagrangian functions related to semidefinite programming. Some necessary and sufficient optimality conditions for semidefinite programming are also given under some suitable assumptions. As an application, we obtain some equivalent characterizations for necessary and sufficient optimality conditions for linear semidefinite programming under Slater assumption.
2016, 12(4): 1199-1214 doi: 10.3934/jimo.2016.12.1199 +[Abstract](1026) +[PDF](237.3KB)
Abstract:
This paper analyzes a finite buffer multiple working vacations queue with balking, reneging and Bernoulli schedule vacation interruption. Arriving customers decide either to join the system or to balk. After joining the queue the customers may renege based on their desire for service or their unwillingness for waiting. At a service completion instant during working vacations, the server can decide either to continue the vacation with probability $q$ or interrupt the vacation and resume regular service period with probability $1-q$. The inter-arrival times of customers are assumed to be arbitrarily distributed. Service times during a regular service period, during a working vacation period and vacation times are assumed to be exponentially distributed. Using recursive technique, the steady state system length distributions at various epochs are obtained. Some performance measures of the model and cost analysis using ant colony optimization are presented. Finally, numerical results showing the effect of model parameters on key performance measures are presented.
2016, 12(4): 1215-1225 doi: 10.3934/jimo.2016.12.1215 +[Abstract](1179) +[PDF](672.6KB)
Abstract:
This study proposes a novel methodology towards using ant colony optimization ($ACO$) with stochastic demand. In particular, an optimization-simulation-optimization approach is used to solve the Stochastic uncapacitated location-allocation problem with an unknown number of facilities, and an objective of minimizing the fixed and transportation costs. $ACO$ is modeled using discrete event simulation to capture the randomness of customers' demand, and its objective is to optimize the costs. On the other hand, the simulated $ACO$'s parameters are also optimized to guarantee superior solutions. This approach's performance is evaluated by comparing its solutions to the ones obtained using deterministic data. The results show that simulation was able to identify better facility allocations where the deterministic solutions would have been inadequate due to the real randomness of customers' demands.
2016, 12(4): 1227-1247 doi: 10.3934/jimo.2016.12.1227 +[Abstract](1192) +[PDF](439.6KB)
Abstract:
Circulant tensors naturally arise from stochastic process and spectral hypergraph theory. The joint moments of stochastic processes are symmetric circulant tensors. The adjacency, Laplacian and signless Laplacian tensors of circulant hypergraphs are also symmetric circulant tensors. The adjacency, Laplacian and signless Laplacian tensors of directed circulant hypergraphs are circulant tensors, but they are not symmetric in general. In this paper, we study spectral properties of circulant tensors and their applications in spectral hypergraph theory and stochastic process. We show that in certain cases, the largest H-eigenvalue of a circulant tensor can be explicitly identified. In particular, the largest H-eigenvalue of a nonnegative circulant tensor can be explicitly identified. This confirms the results in circulant hypergraphs and directed circulant hypergraphs. We prove that an even order circulant B$_0$ tensor is always positive semi-definite. This shows that the Laplacian tensor and the signless Laplacian tensor of a directed circulant even-uniform hypergraph are positive semi-definite. If a stochastic process is $m$th order stationary, where $m$ is even, then its $m$th order moment, which is a circulant tensor, must be positive semi-definite. In this paper, we give various conditions for an even order circulant tensor to be positive semi-definite.
2016, 12(4): 1249-1265 doi: 10.3934/jimo.2016.12.1249 +[Abstract](968) +[PDF](528.3KB)
Abstract:
$2$-CatSP is a fundamental NP-Complete optimization problem, which is formulated as given a ground set $U$ of $n$ items, $m$ subsets $S_1,S_2,\cdots,$ $S_m$ of $U$ and an integer $1\leq t\leq n$, find two subsets $A_1$ and $A_2$ of $U$ such that $|A_1|=|A_2|\leq t$ and $\sum_{i=1}^{m}\max\left(|S_i\cap A_1|,|S_i\cap A_2|\right)$ is maximized. In this paper, we reconsider randomized approximation algorithms for $2$-CatSP without and with triangle inequalities in terms of a new positive semidefinite matrix reflecting more information on unbalanced properties. The performance ratios of our algorithm are much better than the current best ones of Xu et al in (Optim. Method. Softw., 18(6): 705-719, 2003) for general cases under unbalanced parameters $\sigma=t/n\geq 0.50$ by our rigorous analysis. In particular, we get 0.6700 and 0.7250 without and with triangle inequalities for the most important subcase $\sigma=0.50$, improving previously best ratios 0.6430 and 0.6736 in corresponding environment. Indeed recently Wu et al (J. Ind. Manag. Optim. 8: 117-126, 2012) and Xu et al in (J. Comb. Optim., 27: 315-327, 2014) also considered approximate strategies for its disjoint subcase $A_1\cap A_2=\emptyset$ as $\sigma=0.50$ with ratios 0.7317 and 0.7469, which can be reduced to max bisection problems of graphs.
2016, 12(4): 1267-1285 doi: 10.3934/jimo.2016.12.1267 +[Abstract](1500) +[PDF](485.0KB)
Abstract:
In this paper, we formulate an alcohol quitting model in which we consider the impact of distributed time delay between contact and infection process by characterizing dynamic nature of alcoholism behaviours, and we generalize the infection rate to the general case, simultaneously, we consider two different control strategies. Next, we discuss the qualities on the model, the existence and boundedness as well as positivity of the equilibrium are involved. Then, under certain proper conditions, we construct appropriate Lyapunov functionals to prove the global stability of alcohol free equilibrium point $E_{0}$ and alcoholism equilibrium $E^{*}$ respectively. Furthermore, the optimal control strategies are derived by proposing an objective functional and using classic Pontryagin's Maximum Principle. Numerical simulations are conducted to support our theoretical results derived in optimal control.
2016, 12(4): 1287-1301 doi: 10.3934/jimo.2016.12.1287 +[Abstract](1392) +[PDF](413.5KB)
Abstract:
This paper is concerned with a mean-field type optimal control problem, whose new features are that the state $x^v_t$ is partially observed by a noisy process $y(t)$, and the control problem is time inconsistent in the sense that Bellman optimality principle does not work. A necessary condition for optimality is derived by convex variation, dual technique and backward stochastic differential equations (BSDEs). A linear-quadratic (LQ) optimal control example is studied, and the optimal solution is obtained by the optimal filtering for BSDEs and the necessary condition.
2016, 12(4): 1303-1309 doi: 10.3934/jimo.2016.12.1303 +[Abstract](1111) +[PDF](308.4KB)
Abstract:
In this paper, the lower semicontinuity of the approximate solution mapping to generalized strong vector equilibrium problems is established by using a new proof method which is different from the ones used in the literature. Simultaneously, we also obtain the upper semicontinuity of the approximate solution mapping without the assumptions about monotonicity and approximate solution mappings. Some examples are given to illustrate our results.
2016, 12(4): 1311-1322 doi: 10.3934/jimo.2016.12.1311 +[Abstract](1413) +[PDF](376.0KB)
Abstract:
Congestion is generally used in the economics and indicates a situation where a decrease (increase) in one or more inputs can increase (decrease) one or more outputs. In this paper, we introduce a new concept using data envelopment analysis, and call it revenue congestion. The new concept implies a situation where reduction in some inputs may result in an increase in revenue. This improvement in revenue is rather possible by a simultaneous increase and decrease in outputs due to a reduction in inputs. Then, we try to propose a method to distinguish the revenue congestion and identify its sources and amounts. To illustrate the use of the proposed method, an empirical application corresponding to 30 Iranian bank branches is provided. 16 branches evidence revenue congestion via the proposed approach. This identification is very significant because these branches can increase the revenue of their outputs by eliminating the amounts of revenue congestion in each of their inputs. Moreover, it is found that an increase in all outputs is not always profitable, but rather in some cases a decrease in some outputs and an increase in some other outputs can help the firms to make more profits.
2016, 12(4): 1323-1331 doi: 10.3934/jimo.2016.12.1323 +[Abstract](958) +[PDF](348.7KB)
Abstract:
We investigate the problem of optimal investment and consumption of Merton in the case of discrete markets in an infinite horizon. We suppose that there is frictions in the markets due to loss in trading. These frictions are modeled through nonlinear penalty functions and the classical transaction cost and liquidity models are included in this formulation. In this context, the solvency region is defined taking into account this penalty function and every investigator have to maximize his utility, that is derived from consumption, in this region. We give the dynamic programming of the model and we prove the existence and uniqueness of the value function.
2016, 12(4): 1333-1347 doi: 10.3934/jimo.2016.12.1333 +[Abstract](1180) +[PDF](371.6KB)
Abstract:
In this paper we develop a new inventory model for items with imperfect quality and quantity discounts under adjusted screening rate and earned interest. Three highlights are included in this new model: (1) interest is earned by depositing the sales revenue from the perfect and imperfect items into an interest-bearing account (2) the screening rate may not be given but is a decision variable (3) the supplier offers quantity discounts to trigger the retailer into ordering greater lot sizes. This scenario has not been discussed in previous EOQ models with imperfect quality. Our model could determine two decision variables, order quantity and screening rate, to maximize retailer profit. The expected total profit function is derived with two special cases explored to validate the proposed model. An algorithm is developed to help the manager determine the optimal order quantity and screening rate. A numerical example is given to illustrate the proposed model and algorithm. Sensitivity analyses are carried out to investigate the model parameters effects on the optimal solution. Managerial insights are also included.
2016, 12(4): 1349-1366 doi: 10.3934/jimo.2016.12.1349 +[Abstract](1237) +[PDF](442.4KB)
Abstract:
We consider an optimization problem that minimizes a function of the form $f=f_0+f_1-f_2$ with the constraint $g-h\leq 0$, where $f_0$ is continuous differentiable, $f_1,f_2$ are convex and $g,h$ are lower semicontinuous convex. We propose to solve the problem by an inexact subgradient-based convex approximations method. Under mild assumptions, we show that the method is guaranteed to converge to a stationary point. Finally, some preliminary numerical results are given.
2016, 12(4): 1367-1389 doi: 10.3934/jimo.2016.12.1367 +[Abstract](1131) +[PDF](545.5KB)
Abstract:
2016, 12(4): 1391-1415 doi: 10.3934/jimo.2016.12.1391 +[Abstract](1905) +[PDF](1310.5KB)
Abstract:
In this study, a genetic algorithm (GA) with priority-based representation is proposed for a flexible job shop scheduling problem (FJSP) which is one of the hardest operations research problems. Investigating the effect of the proposed representation schema on FJSP is the main contribution to the literature. The priority of each operation is represented by a gene on the chromosome which is used by a constructive algorithm performed for decoding. All active schedules, which constitute a subset of feasible schedules including the optimal, can be generated by the constructive algorithm. To obtain improved solutions, iterated local search (ILS) is applied to the chromosomes at the end of each reproduction process. The most widely used FJSP data sets generated in the literature are used for benchmarking and evaluating the performance of the proposed GA methodology. The computational results show that the proposed GA performed at the same level or better with respect to the makespan for some data sets when compared to the results from the literature.
2016, 12(4): 1417-1433 doi: 10.3934/jimo.2016.12.1417 +[Abstract](1327) +[PDF](444.8KB)
Abstract:
In this paper, an algorithm of the branch and bound type in outcome space is proposed for solving a global optimization problem that includes, as a special case, generalized multiplicative problems. As an application, we solve the problem of optimizing over the efficient set of a bicriteria concave maximization problem. Preliminary computational experiments show that this algorithm works well for problems where the dimensions of the decision space can be fairly large.
2016, 12(4): 1435-1464 doi: 10.3934/jimo.2016.12.1435 +[Abstract](1185) +[PDF](799.8KB)
Abstract:
In this paper, we deal with a discrete-time $Geo/G/1$ queueing system under the control of Min($N, V$)-policy in which the server takes single vacation whenever the system becomes empty. The Min($N, V$)-policy means that the server commences its service once the number of waiting customers reaches threshold $N$ or when its vacation time ends with at least one but less than $N$ customers waiting for processing, whichever occurs first. Otherwise, if no customer is presenting at the end of the server vacation, the server remains idle until the first arrival occurs. Under these assumptions, the $z$-transform expressions for the transient queue size distribution at time epoch $n^+$ are obtained by employing the renewal process theory and the total probability decomposition technique. Based on the transient analysis, the explicit recursive formulas of the steady-state queue length distribution at time epochs $n^+$, $n$, $n^-$ and outside observer's time epoch are derived, respectively. Additionally, the stochastic decomposition structure is presented and some other performance measures are also discussed. Furthermore, some computational experiments are implemented to demonstrate the significant application value of the recursive formulas for the steady-state queue size in designing system capacity. Finally, the optimal threshold of $N$ for economizing the system cost is numerically determined.
2016, 12(4): 1465-1493 doi: 10.3934/jimo.2016.12.1465 +[Abstract](1331) +[PDF](436.3KB)
Abstract:
We provide an overview of the history, the methods and the people who researched on minimizing the (weighted) number of tardy jobs as a performance measure. The review presents cases on multiple machines: parallel machines (including the identical, uniform and unrelated machines, flow shop, job shop and the open shop). The literature is divided into various sections for proper categorization. This includes setup time, preemption, batching, on-line and off-line scheduling, and other classifications. The complexity status of the various classifications is enumerated with its results and methods. Possible extension for future work is also highlighted.
2016, 12(4): 1495-1505 doi: 10.3934/jimo.2016.12.1495 +[Abstract](946) +[PDF](391.3KB)
Abstract:
In this paper, a class of optimization problems coupled with differential equations in finite dimensional spaces are introduced and studied. An existence theorem of a Carathéodory weak solution of the differential optimization problem is established. Furthermore, when both the mapping and the constraint set in the optimization problem are perturbed by two different parameters, the stability analysis of the differential optimization problem is considered. Finally, an algorithm for solving the differential optimization problem is established.
2016, 12(4): 1507-1519 doi: 10.3934/jimo.2016.12.1507 +[Abstract](1122) +[PDF](743.6KB)
Abstract:
The affine rank minimization problem is to find a low-rank matrix satisfying a set of linear equations, which includes the well-known matrix completion problem as a special case and draws much attention in recent years. In this paper, a new model for affine rank minimization problem is proposed. The new model not only enhances the robustness of affine rank minimization problem, but also leads to high nonconvexity. We show that if the classical projected gradient method is applied to solve our new model, the linear convergence rate can be established under some conditions. Some preliminary experiments have been conducted to show the efficiency and effectiveness of our method.
2016, 12(4): 1521-1533 doi: 10.3934/jimo.2016.12.1521 +[Abstract](1205) +[PDF](403.6KB)
Abstract:
This paper studies the portfolio optimization of mean-variance utility with state-dependent risk aversion, where the stock asset is driven by a stochastic process. The sub-game perfect Nash equilibrium strategies and the extended Hamilton-Jacobi-Bellman equations have been used to derive the system of non-linear partial differential equations. From the economic point of view, we demonstrate the numerical evaluation of the suggested solution for a special case where the risk aversion rate is proportional to the wealth value. Our results show that the asset driven by the stochastic volatility process is more general and reasonable than the process with a constant volatility.
2016, 12(4): 1535-1556 doi: 10.3934/jimo.2016.12.1535 +[Abstract](1031) +[PDF](444.3KB)
Abstract:
This paper investigates piecewise observer design for rectangular discrete fuzzy descriptor systems with multiple time-varying delays. Via a series of simple transformations, the considered rectangular descriptor plants are converted into standard ones with multiple time-varying delays. Then, two sufficient delay-dependent conditions for existence of piecewise fuzzy observers are derived based on piecewise Lyapunov functions. Finally, two numerical examples are presented to show the effectiveness of the theoretical results.
2016, 12(4): 1557-1585 doi: 10.3934/jimo.2016.12.1557 +[Abstract](1172) +[PDF](866.9KB)
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In this paper, we investigate the consumption-investment problem for a member of the defined contribution pension plan with non-constant time preferences. The aim of the member is to maximize the discounted utility of the consumption. It leads to a time-inconsistent control problem in the sense that the Bellman optimality principle does no longer hold. In our model, the contribution rate is assumed to be a fixed proportion of the scheme member's salary, and the pension fund can be invested in a risk-free asset, an index bond and a stock whose return follows a geometric Brownian motion. Two utility functions are considered: the power utility and the logarithmic utility. We characterize the time-consistent equilibrium consumption-investment strategies and the value function in terms of a solution of an integral equation in both situations. The existence and uniqueness of the solution is verified and the approximation of the solution is obtained. We present some numerical results of the equilibrium consumption rate and equilibrium investment policy with three types of discount functions.
2016, 12(4): 1587-1611 doi: 10.3934/jimo.2016.12.1587 +[Abstract](1019) +[PDF](609.1KB)
Abstract:
We examine the pure rebate strategies in a two-echelon supply chain under stochastic demand with multiplicative error. Given exogenous wholesale price and retail price, we characterize the unique Nash equilibrium when both manufacturer and retailer provide rebate policy to consumers under nonlinear and linear price-dependent demand functions, including iso-elastic multiplicative demand function (EMDF) and linear multiplicative demand function (LMDF). Based on a game theoretical framework, we prove that there still exists a unique equilibrium when the price elasticity is rather small with constraint conditions in the former case. We also find that in this case the retailer(manufacturer) may increase its rebate value in reaction to the manufacturer's(retailer's) rebate value in order to stimulate sales, which is contrary to the conventional wisdom that the retailer(manufacturer) will shrink its rebate value to gain an extra advantage" unfairly. As a result, both parties share the same profit at equilibrium. Further, we compare the expected profit outcomes at equilibrium among joint-rebate game, single-party rebate game and no-rebate game by using numerical examples. It is shown that the joint-rebate policy is not always dominates the others unless the price elasticity is sufficiently flexible.

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