Connecting a population dynamic model with a multi-period location-allocation problem for post-disaster relief operations

In this study, we propose a mathematical model and heuristics for solving a multi-period location-allocation problem in post-disaster operations, which takes into account the impact of distribution over the population. Logistics restrictions such as human and financial resources are considered. In addition, a brief review on resilience system models is provided, as well as their connection with quantitative models for post-disaster relief operations. In particular, we highlight how one can improve resilience by means of OR/MS strategies. Then, a simpler resilience schema is proposed, which better reflects an active system for providing humanitarian aid in post-disaster operations, similar to the model focused in this work. The proposed model is non-linear and solved by a decomposition approach: the master level problem is addressed by a non-linear solver, while the slave subproblem is treated as a black-box coupling heuristics and a Variable Neighborhood Descent local search. Computational experiments have been done using several scenarios, and real data from Belo Horizonte city in Brazil.     

Formulation and numerical solution of finite-level quantum optimal control problems

Optimal control of finite-level quantum systems is investigated, and iterative solution schemes for the optimization of a control representing laser pulses are developed. The purpose of this external field is to channel the system's wavefunction between given states in its most efficient way. Physically motivated constraints, such as limited laser resources or population suppression of certain states, are accounted for through an appropriately chosen cost functional. First-order necessary optimality conditions and second-order sufficient optimality conditions are investigated. For solving the optimal control problems, a cascadic non-linear conjugate gradient scheme and a monotonic scheme are discussed. Results of numerical experiments with a representative finite-level quantum system demonstrate the effectiveness of the optimal control formulation and efficiency and robustness of the proposed approaches.     

Mathematical modeling and trajectory planning of mobile manipulators with flexible links and joints

This paper is concerned with mathematical modeling and optimal motion designing of flexible mobile manipulators. The system is composed of a multiple flexible links and flexible revolute joints manipulator mounted on a mobile platform. First, analyzing on kinematics and dynamics of the model is carried out then; open-loop optimal control approach is presented for optimal motion designing of the system. The problem is known to be complex since combined motion of the base and manipulator, non-holonomic constraint of the base and highly non-linear and complicated dynamic equations as a result of the flexible nature of both links and joints are taken into account. In the proposed method, the generalized coordinates and additional kinematic constraints are selected in such a way that the base motion coordination along the predefined path is guaranteed while the optimal motion trajectory of the end-effector is generated. This method by using Pontryagin’s minimum principle and deriving the optimality conditions converts the optimal control problem into a two point boundary value problem. A comparative assessment of the dynamic model is validated through computer simulations, and then additional simulations are done for trajectory planning of a two-link flexible mobile manipulator to demonstrate effectiveness and capability of the proposed approach.     

A decision tool for optimal irrigated crop planning and water resources sustainability

In the present study, the conjunctive use policies of surface and ground water resources are developed for minimizing water shortage in an irrigation district subject to constraints on groundwater withdrawals and crop planning capacities. An integrated soil water balance algorithm is coupled to a non-linear optimization model in order to carry out water allocation planning in complex deficit agricultural water resources systems based on an economic efficiency criterion. Various options of conjunctive use water resources along with current and proposed cropping patterns have been explored by Koohdasht Irrigation District (KID), a semi-arid region in I.R. Iran. The analysis provides various scenarios, which can help managers in decision-making for the optimum allocation plans of water resources within the irrigation area. The results reveal that the proposed model, as a decision tool for optimal irrigated crop planning and water resources sustainability, may be used for maximizing the overall net benefits and global water productivity of an irrigation district considering an allowable annual recharge of groundwater. Findings indicate the importance of the conjunctive water management modeling, which can be easily implemented and would enhance the overall benefits from cropping activities in the study area.     

A disruption recovery model for a single stage production-inventory system

This paper presents a newly developed disruption recovery model for a single stage production and inventory system, where the production is disrupted for a given period of time during the production up time. The model is categorized as a constrained non-linear optimization program which we have solved using an efficient heuristic developed in this paper. The model was also solved using an evolutionary algorithm and a comparison of the results from both methods was performed. The heuristic was able to accurately solve the model with significantly less time compared to the evolutionary algorithm. It can be shown that the optimal recovery schedule is dependent on the shortage cost parameters, as well as the extent of the disruption. The proposed model offers a potentially useful tool to help manufacturers decide on the optimal recovery plan in real time whenever the production system experiences a sudden disruption.     

Profitability ratio maximization in an inventory model with stock-dependent demand rate and non-linear holding cost

This paper studies a deterministic inventory model with a stock-dependent demand pattern where the cumulative holding cost is a non-linear function of both time and stock level. When the monetary resources are limited and the inventory manager can invest his/her money in buying different products, it seems reasonable to select the ones that provide a higher profitability. Thus, a new approach with the aim of maximizing the profitability ratio (defined as the profit/cost quotient) is considered in this paper. We prove that the profitability ratio maximization is equivalent to minimizing the inventory cost per unit of an item. The optimal policy is obtained in a closed form, whose general expression is a generalization of the classical EOQ formula for inventory models with a stock-dependent demand rate and a non-linear holding cost. This optimal solution is different from the other policies proposed for the problems of minimum cost or maximum profit per unit time. A complete sensitivity analysis of the optimal solution with respect to all the parameters of the model is developed. Finally, numerical examples are solved to illustrate the theoretical results and the solution methodology.     

A multi-objective optimal allocation model for irrigation water resources under multiple uncertainties

This paper proposed a multi-objective optimal water resources allocation model under multiple uncertainties. The proposed model integrated the chance-constrained programming, semi-infinite programming and integer programming into an interval linear programming. Then, the developed model is applied to irrigation water resources optimal allocation system in Minqin’s irrigation areas, Gansu Province, China. In this study, the irrigation areas’ economic benefits, social benefits and ecological benefits are regarded as the optimal objective functions. As a result, the optimal irrigation water resources allocation plans of different water types (surface water and groundwater) under different hydrological years (wet year, normal year and dry year) and probabilities are obtained. The proposed multi-objective model is unique by considering water-saving measures, irrigation water quality impact factors and the dynamic changes of groundwater exploitable quantity in the irrigation water resources optimal allocation system under uncertain environment. The obtained results are valuable for supporting the adjustment of the existing irrigation patterns and identify a desired water-allocation plan for irrigation under multiple uncertainties.     

A zero-one model for project portfolio selection and scheduling

A zero-one integer linear programming model is proposed for selecting and scheduling an optimal project portfolio, based on the organisation's objectives and constraints such as resource limitations and interdependence among projects. The model handles some of the issues that frequently arise in real world applications but are not addressed by previously suggested models, such as situations in which the amount of available and consumed resources varies in different periods. It also allows for interactive adjustment following the optimisation process, to provide decision makers a method for controlling portfolio selection, based on criteria that may be difficult to elicit directly. It is critical for such a system to provide fast evaluation of alternatives the decision makers may want to examine, and this requirement is addressed. The proposed model not only suggests projects that should be incorporated in the optimal portfolio, but it also determines the starting period for each project. Scheduling considerations can have a major impact on the combination of projects that can be incorporated in the portfolio, and may allow the addition of certain projects to the portfolio that could not have been selected otherwise. An example problem is described and solved with the proposed model, and some areas for future research are discussed.     

Determining optimal resource recycling boundary at regional level: A case study on Tokyo Metropolitan Area in Japan

Conventionally, material inputs to industries come from either natural resources or intermediate products. With the increasing attention on greening the industrial process, municipal solid waste (MSW) can be separated and utilized as inputs to large-scale industries. As such, waste flows need to be managed to meet the requirements of industries by considering both quality and scale. Small scales of municipal recycling centers (MRCs) for pre-treating recyclable wastes often lead to higher pre-treatment cost and lower recycling efficiencies. Theoretically, appropriate facility scale and recycling boundary should be based upon a balanced consideration between economies of scale and transportation costs. However, only a few empirical studies on recycling boundaries have been conducted. Existing modeling studies are mostly case studies for planning purposes and do not theoretically examine the mechanisms and factors that determine recycling boundary. Under such a circumstance, this paper fills such a gap by quantitatively seeking the determinants for recycling boundaries and the related mechanism. An optimization model applying on the recycling of waste plastics is developed and tested in the Tokyo Metropolitan Region (TMR) in Japan. The results indicate two determinants for recycling boundary at the regional level: spatial density of separated wastes and the ratio of unit transportation cost to unit treatment cost. Separated wastes that are collected in low spatial density and have a low ratio of unit transportation cost to unit treatment cost should be better recycled in a large region. The finding implies that regional recycling networks should have multiple layers designed for different types of wastes due to the differences in their spatial density, unit transportation cost and unit treatment cost.     

A reducing of a chaotic movement to a periodic orbit,of a micro-electro-mechanical system,by using an optimal linear control design

This paper analyzes the non-linear dynamics, with a chaotic behavior of a particular micro-electro-mechanical system. We used a technique of the optimal linear control for reducing the irregular (chaotic) oscillatory movement of the non-linear systems to a periodic orbit. We use the mathematical model of a (MEMS) proposed by Luo and Wang.     

Modeling cadmium accumulation in radish,carrot, spinach and cabbage

Heavy metals like cadmium and arsenic have serious health consequences and ecosystem impacts. Due to various factors including the disposal of municipal and industrial wastes, application of fertilizers, atmospheric deposition and discharge of wastewater on land, has resulted in increase in the concentration of heavy metals in the soil. Crops and vegetables grown on such soil accumulate heavy metals, which leads to phyto-toxicity. For understanding and managing precious natural resources, mathematical models are increasingly being used. This paper describes a dynamic macroscopic numerical model for heavy metal transport and its uptake by vegetables in the root zone. The model is applied for simulating cadmium uptake by radish (Raphanus sativus), carrot (Daucos carota), spinach (Spinacia oleracea) and cabbage (Brassica oleracea) by using measured field data. The governing non-linear partial differential equations are solved numerically by an implicit finite difference method using Picard’s iterative technique and the source code is written in MATLAB.     

A conic relaxation model for searching for the global optimum of network data envelopment analysis

Network data envelopment analysis (DEA) models the internal structures of decision-making units (DMUs). Unlike the standard DEA model, multiplier-based network DEA models are often highly non-linear and cannot be converted into linear programs. As such, obtaining a non-linear network DEA's global optimal solution is a challenge because it corresponds to a nonconvex optimization problem. In this paper, we introduce a conic relaxation model that searches for the global optimum to the general multiplier-based network DEA model. We reformulate the general network DEA models and relax the new models into second order cone programming (SOCP) problems. In comparison with linear relaxation models, which is potentially applicable to general network DEA structures, the conic relaxation model guarantees applicability in general network DEA, since McCormick envelopes involved are ensured to be finite. Furthermore, the conic relaxation model avoids unnecessary linear relaxations of some nonlinear constraints. It generates, in a more convenient manner, feasible approximations and tighter upper bounds on the global optimal overall efficiency. Compared with a line-parameter search method that has been applied to solve non-linear network DEA models, the conic relaxation model keeps track of the distances between the optimal overall efficiency and its approximations. As a result, it is able to determine whether a qualified approximation has been achieved or not, with the help of a branch and bound algorithm. Hence, our proposed approach can substantially reduce the computations involved.     

Tracking Demands in Optimal Control of Managerial Systems with Continuously-Divisible, Doubly Constrained Resources

The paper addresses problems of allocating continuously divisible resources among multiple production activities. The resources are allowed to be doubly constrained, so that both usage at every point of time and cumulative consumption over a planning horizon are limited as it is often the case in project and production scheduling. The objective is to track changing in time demands for the activities as closely as possible. We propose a general continuous-time model that states the problem in a form of the optimal control problem with non-linear speed-resource usage functions. With the aid of the maximum principle, properties of the solutions are derived to characterize optimal resource usage policies. On the basis of this analytical investigation, numerical scheduling methods are suggested and computationally studied. This revised version was published online in July 2006 with corrections to the Cover Date.     

A numerical computation of the non-dimensional form of a non-linear hydrodynamic model in a uniform reservoir

A mathematical model is used to simulate the water current and the elevation in a uniform reservoir. A non-linear hydrodynamic model that provides the velocity field and elevation of the water flow is considered. In the simulating process, the Lax–Wendroff technique is used to approximate the solutions. The numerical solution can be the input data for a water-quality model that is applicable for the optimal control of water treatment in the system to achieve minimum cost.     

Optimal selection of retailers for a manufacturing vendor in a vendor managed inventory system

A Vendor Managed Inventory (VMI) system consists of a manufacturing vendor and a number of retailers. In such a system, it is essential for the vendor to optimally determine retailer selection and other related decisions, such as the product’s replenishment cycle time and the wholesale price, in order to maximize his profit. Meanwhile, each retailer’s decisions on her willingness to enter the system and retail price are simultaneously considered in the retailer selection process. However, the above interactive decision making is complex and the available studies on interactive retailer selection are scarce. In this study, we formulate the retailer selection problem as a Stackelberg game model to help the manufacturer, as a vendor, optimally select his retailers to form a VMI system. This model is non-linear, mixed-integer, game-theoretic, and analytically intractable. Therefore, we further develop a hybrid algorithm for effectively and efficiently solving the developed model. The hybrid algorithm combines dynamic programming (DP), genetic algorithm (GA) and analytical methods. As demonstrated by our numerical studies, the optimal retailer selection can increase the manufacturer’s profit by up to 90% and the selected retailers’ profits significantly compared to non-selection strategy. The proposed hybrid algorithm can solve the model within a minute for a problem with 100 candidate retailers, whereas a pure GA has to take more than 1 h to solve a small sized problem of 20 candidate retailers achieving an objective value no worse than that obtained by the hybrid algorithm.     

A Benders decomposition algorithm for a multi-area,multi-stage integrated resource planning in power systems

The aim of this paper is to propose an integrated model for resource planning in power systems by taking into account both supply and demand sides options simultaneously. At supply-side, investment in generation capacity and transmission lines is considered. Demand side management (DSM) technologies are also incorporated to correct the shape of the load duration curve in terms of peak clipping and load shifting programmes. A mixed integer non-linear programming model is developed to find the optimal location and timing of electricity generation/transmission as well as DSM options. To solve the resulting complex model, nonlinearity caused by transmission loss terms are first eliminated using the piecewise linearization technique. Then, a Benders decomposition (BD) algorithm is developed to solve the linearized model. The performance of the proposed BD algorithm is validated via applying it to the 6-bus Garver test system and a modified 21-bus IEEE reliability test system.     

Robust scheduling of parallel machines with sequence-dependent set-up costs

In this paper we propose a robust approach for solving the scheduling problem of parallel machines with sequence-dependent set-up costs. In the literature, several mathematical models and solution methods have been proposed to solve such scheduling problems, but most of which are based on the strong assumption that input data are known in a deterministic way. In this paper, a fuzzy mathematical programming model is formulated by taking into account the uncertainty in processing times to provide the optimal solution as a trade-off between total set-up cost and robustness in demand satisfaction. The proposed approach requires the solution of a non-linear mixed integer programming (NLMIP), that can be formulated as an equivalent mixed integer linear programming (MILP) model. The resulting MILP model in real applications could be intractable due to its NP-hardness. Therefore, we propose a solution method technique, based on the solution of an approximated model, whose dimension is remarkably reduced with respect to the original counterpart. Numerical experiments conducted on the basis of data taken from a real application show that the average deviation of the reduced model solution over the optimum is less than 1.5%.     

Period doubling cascades of prey–predator model with nonlinear harvesting and control of over exploitation through taxation

The present study investigates a prey predator type model for conservation of ecological resources through taxation with nonlinear harvesting. The model uses the harvesting function as proposed by Agnew (1979) [1] which accounts for the handling time of the catch and also the competition between standard vessels being utilized for harvesting of resources. In this paper we consider a three dimensional dynamic effort prey–predator model with Holling type-II functional response. The conditions for uniform persistence of the model have been derived. The existence and stability of bifurcating periodic solution through Hopf bifurcation have been examined for a particular set of parameter value. Using numerical examples it is shown that the system admits periodic, quasi-periodic and chaotic solutions. It is observed that the system exhibits periodic doubling route to chaos with respect to tax. Many forms of complexities such as chaotic bands (including periodic windows, period-doubling bifurcations, period-halving bifurcations and attractor crisis) and chaotic attractors have been observed. Sensitivity analysis is carried out and it is observed that the solutions are highly dependent to the initial conditions. Pontryagin’s Maximum Principle has been used to obtain optimal tax policy to maximize the monetary social benefit as well as conservation of the ecosystem.     

A continuous model of the dynamical systems capable to memorise multiple shapes

This paper proposes the novel approach to the mathematical synthesis of continuous self-organising systems capable to memorise and restore own multiple shapes defined by means of functions of single spatial variable or parametric models in two-dimensional space. The model is based on the certain universal form of the integral operator with the kernel representing the system memory. The technique for memorising shapes uses the composition of singular kernels of integral operators. The whole system is described by the potential function, whose minimisation leads to the non-linear dynamics of shape reconstruction by integro-differential non-linear equations with partial derivatives. The corresponding models are proposed and analysed for both parametric and non-parametric shape definitions. Main features of the proposed model are considered, and the results of numerical simulation are shown in case of three shapes memorising and retrieval. The proposed model can be used in theory of smart materials, artificial intelligence and some other branches of non-linear sciences where the effect of multiple shapes memorising and retrieval appears as the core feature.     

A resource portfolio model for equipment investment and allocation of semiconductor testing industry

Profitable but risky semiconductor testing market has led companies in the industry to carefully seek to maximize their profits by developing a proper resource portfolio plan for simultaneously deploying resources and selecting the most profitable orders. Various important factors, such as resource investment alternatives, trade-offs between the price and speed of equipment and capital time value, further increase the complexity of the simultaneous resource portfolio problem. This study develops a simultaneous resource portfolio decision model as a non-linear integer programming, and proposes a genetic algorithm to solve it efficiently. The proposed method is employed in the context of semiconductor testing industry to support decisions regarding equipment investment alternatives (including new equipment procurement, rent and transfer by outsourcing, and phasing outing) for simultaneous resources (such as testers and handlers) and task allocation. Experiments have showed that our approach, in contrast to an optimal solution tool, obtains a near-optimal solution in a relatively short computing time.