A hybrid genetic algorithm for the multi-depot open vehicle routing problem

This paper studies the multi-depot open vehicle routing problem (MDOVRP), a variant of the vehicle routing problem (VRP), in which vehicles start from several depots and are not required to return to the depot. Despite the vast amount of literature about VRPs, the MDOVRP has received very little attention from researchers. In this paper, a new hybrid genetic algorithm is presented for finding the routes that minimize the traveling cost of the vehicles. Computational results on a number of test instances indicate the proposed algorithm dominates the CPLEX solver and the existing approach in the literature. Meanwhile, experiments are conducted on multi-depot VRP benchmarks, and the results are compared with a sophisticated tabu search approach and an exact method.     

Scenario-based approach for flexible resource loading under uncertainty

Order acceptance decisions in manufacture-to-order environments are often made based on incomplete or uncertain information. To quote reliable due dates in order processing, manage resource capacity adequately and take into account uncertainty, the paper presents and analyses models and tools for more robust resource loading. We refer to the problem as flexible resource loading under uncertainty. We propose a scenario-based solution approach that can deal with a wide range of uncertainty types. The approach is based on an MILP to find a plan with minimum expected costs over all relevant scenarios. To solve this MILP, we propose an exact branch-and-price algorithm. Further, we propose a much faster improvement heuristic based on an LP (linear programming) approximation. A disadvantage of the scenario-based MILP, is that a large number of scenarios may make the model intractable. We therefore propose an approximate approach that uses the aforementioned solution techniques and only a sample of all scenarios. Computational experiments show that, especially for instances with sufficient slack, solutions obtained with deterministic techniques that only use the expected scenario can be significantly improved with respect to their expected costs (i.e. robustness). We also show that for large instances, our heuristics outperform the exact approach given a maximum computation time as a stopping criterion. Moreover, it turns out that using a small sample of selected scenarios generally yields better results than using all scenarios.     

A branch and cut algorithm for the VRP with satellite facilities

An important aspect of the vehicle routing problem (VRP) that has been largely overlooked is the use of satellite facilities to replenish vehicles during a route. When possible, satellite replenishment allows the drivers to continue making deliveries until the close of their shift without necessarily returning to the central depot. This situation arises primarily in the distribution of fuels and certain retail items. When demand is random, optimizing customer routes a priori may result in significant additional costs for a particular realization of demand. Satellite facilities are one way of safeguarding against unexpected demand. This paper presents a branch and cut methodology for solving the VRP with satellite facilities subject to capacity and route time constraints. We begin with a mixed-integer linear programming formulation and then describe a series of valid inequalities that can be used to cut off solutions to the linear programming relaxation. Several separation heuristics are then outlined that are used to generate the cuts. Embedded in the methodology is a VRP heuristic for finding good feasible solutions at each stage of the computations. Results are presented for a set of problems derived from our experience with a leading propane distributor.     

Solving the heterogeneous fixed fleet open vehicle routing problem by a combined metaheuristic algorithm

The vehicle routing problem (VRP) is a well-known combinatorial optimisation problem and holds a central place in logistics management. Many exact, heuristic and metaheuristic approaches have been proposed to solve VRP. An important variant of the VRP arises when a ?eet of vehicles is fixed and characterised by different capacities for distribution activities. The problem is known as the heterogeneous fixed fleet VRP (HFFVRP). The HFFVRP is a natural generalisation of the VRP with several vehicle types, each type being defined by a capacity, a fixed cost and a cost per distance unit, and can cover more practical situations in transportation. This problem consists of determining a set of vehicle trips of minimum total length in which a set of customers is to be satisfied in the demand constraints using identical vehicles with limited capacity. If open routes instead of closed ones are considered in the HFFVRP, the problem becomes a heterogeneous fixed fleet Open VRP (HFFOVRP) which has numerous applications in industrial and service problems. In this paper, a bone route algorithm which uses the tabu search as an improved procedure is utilised to solve the HFFOVRP. The proposed algorithm was tested empirically on a 24 of generated VRPs, and compared with elite ant system and ant colony system. In all cases, the proposed algorithm finds the best-known solutions within a reasonable time.     

Capacity and material requirement planning modelling by comparing deterministic and fuzzy models

A model for the capacity and material requirement planning problem with uncertainty in a multi-product, multi-level and multi-period manufacturing environment is proposed. An optimization model is formulated which takes into account the uncertainty that exists in both the market demand and capacity data, and the uncertain costs for backlog. This work uses the concept of possibilistic programming by comparing trapezoidal fuzzy numbers. Such an approach makes it possible to model the ambiguity in market demand, capacity data, cost information, etc. that could be present in production planning systems. The main goal is to determine the master production schedule, stock levels, backlog, and capacity usage levels over a given planning horizon in such a way as to hedge against the uncertainty. Finally, the fuzzy model and the deterministic model adopted as the basis of this work are compared using real data from an automobile seat manufacturer. The paper concludes that fuzzy numbers could improve the solution of production planning problems.     

A multiperiod stochastic production planning and sourcing problem with service level constraints

We study a stochastic multiperiod production planning and sourcing problem of a manufacturer with a number of plants and/or subcontractors. Each source, i.e. each plant and subcontractor, has a different production cost, capacity, and lead time. The manufacturer has to meet the demand for different products according to the service level requirements set by its customers. The demand for each product in each period is random. We present a methodology that a manufacturer can utilize to make its production and sourcing decisions, i.e., to decide how much to produce, when to produce, where to produce, how much inventory to carry, etc. This methodology is based on a mathematical programming approach. The randomness in demand and related probabilistic service level constraints are integrated in a deterministic mathematical program by adding a number of additional linear constraints. Using a rolling horizon approach that solves the deterministic equivalent problem based on the available data at each time period yields an approximate solution to the original dynamic problem. We show that this approach yields the same result as the base stock policy for a single plant with stationary demand. For a system with dual sources, we show that the results obtained from solving the deterministic equivalent model on a rolling horizon gives similar results to a threshold subcontracting policy. Correspondence to: Fikri KaraesmenThe authors are grateful to Yves Dallery for his ideas, comments and suggestions on the earlier versions of this paper.     

Robust material requirement planning with cumulative demand under uncertainty

In this paper, we deal with the problem of tactical capacitated production planning with the demand under uncertainty modelled by closed intervals. We propose a single-item with backordering model under small uncertainty in the cumulative demand for the Master Production Scheduling (MPS) problem with different rules, namely the Lot For Lot rule and the Periodic Order Quantity rule. Then we study a general multilevel, multi-item, multi-resource model with backordering and the external demand on components for the Material Requirement Planning (MRP) problem under uncertainty in the cumulative demand. In order to choose robust production plans for the above problems that hedge against uncertainty, we adopt the well-known minmax criterion. We propose polynomial methods for evaluating the impact of uncertainty on a given production plan in terms of its cost and for computing optimal robust production plans for both problems (MPS/MRP) under the assumed interval uncertainty representation. We show in this way that the robust problems (MPS/MRP) under this uncertainty representation are not much computationally harder than their deterministic counterparts.     

Multi-item capacitated lot-sizing with demand uncertainty

We consider a stochastic version of the classical multi-item Capacitated Lot-Sizing Problem (CLSP). Demand uncertainty is explicitly modeled through a scenario tree, resulting in a multi-stage mixed-integer stochastic programming model with recourse. We propose a plant-location-based model formulation and a heuristic solution approach based on a fix-and-relax strategy. We report computational experiments to assess not only the viability of the heuristic, but also the advantage (if any) of the stochastic programming model with respect to the considerably simpler deterministic model based on expected value of demand. To this aim we use a simulation architecture, whereby the production plan obtained from the optimization models is applied in a realistic rolling horizon framework, allowing for out-of-sample scenarios and errors in the model of demand uncertainty. We also experiment with different approaches to generate the scenario tree. The results suggest that there is an interplay between different managerial levers to hedge demand uncertainty, i.e. reactive capacity buffers and safety stocks. When there is enough reactive capacity, the ability of the stochastic model to build safety stocks is of little value. When capacity is tightly constrained and the impact of setup times is large, remarkable advantages are obtained by modeling uncertainty explicitly.     

An approach for robust PDE-constrained optimization with application to shape optimization of electrical engines and of dynamic elastic structures under uncertainty

We present a robust optimization framework that is applicable to general nonlinear programs (NLP) with uncertain parameters. We focus on design problems with partial differential equations (PDE), which involve high computational cost. Our framework addresses the uncertainty with a deterministic worst-case approach. Since the resulting min–max problem is computationally intractable, we propose an approximate robust formulation that employs quadratic models of the involved functions that can be handled efficiently with standard NLP solvers. We outline numerical methods to build the quadratic models, compute their derivatives, and deal with high-dimensional uncertainties. We apply the presented approach to the parametrized shape optimization of systems that are governed by different kinds of PDE and present numerical results.     

A VNS-LP algorithm for the robust dynamic maximal covering location problem

This study introduces a robust variant of the well-known dynamic maximal covering location problem (DMCLP) and proposes an integer linear programming formulation of the robust DMCLP. A hybrid approach for solving both deterministic and robust variant of the DMCLP is developed, which is based on hybridization of a Variable neighborhood search and a linear programming technique. The main idea is to split the problem into subproblems and to combine optimal solutions of the obtained subproblems in order to construct solution of the initial problem. The results of the proposed hybrid approach on instances of the deterministic DMCLP are presented and compared with the results of the state-of-the-art approach from the literature and with the results of commercial CPLEX solver. The presented computational analysis shows that the proposed hybrid algorithm outperforms other approaches for the DMCLP. In addition, the algorithm was tested on the instances of the robust variant of DMCLP, and obtained results are discussed in detail.

     

Polymorphic uncertain nonlinear programming approach for maximizing the capacity of V-belt driving

In this paper, a polymorphic uncertain nonlinear programming (PUNP) approach is developed to formulate the problem of maximizing the capacity in a system of V-belt driving with uncertainties. The constructed optimization model is found to consist of a nonlinear objective function and some nonlinear constraints with some parameters which are of uncertain nature. These uncertain parameters are interval parameters, random interval parameters, fuzzy parameters or fuzzy interval parameters. To find a robust solution of the problem, a deterministic equivalent formulation (DEF) is established for the polymorphic uncertain nonlinear programming model. For a given satisfaction level, this DEF turns out to be a nonlinear programming involving only interval parameters. A solution method, called a sampling based interactive method, is developed such that a robust solution of the original model with polymorphic uncertainties is obtained by using standard smooth optimization techniques. The proposed method is applied into a real-world design of V-belt driving, and the results indicate that both the PUNP approach and the developed algorithm are useful to the optimization problem with polymorphic uncertainty.     

Distributed robust optimization (DRO), part I: framework and example

Robustness of optimization models for network problems in communication networks has been an under-explored topic. Most existing algorithms for solving robust optimization problems are centralized, thus not suitable for networking problems that demand distributed solutions. This paper represents a first step towards a systematic theory for designing distributed and robust optimization models and algorithms. We first discuss several models for describing parameter uncertainty sets that can lead to decomposable problem structures and thus distributed solutions. These models include ellipsoid, polyhedron, and D-norm uncertainty sets. We then apply these models in solving a robust rate control problem in wireline networks. Three-way tradeoffs among performance, robustness, and distributiveness are illustrated both analytically and through simulations. In Part II of this two-part paper, we will present applications to wireless power control using the framework of distributed robust optimization.     

Optimization of laminated composites subject to uncertain buckling loads

Optimal design of composite laminates under buckling load uncertainty is presented. The laminates are subjected to biaxial compressive loads and the buckling load is maximized under worst case in-plane loading which is computed using an anti-optimization approach. The magnitudes of the in-plane loads are not known a priori resulting in load uncertainty subject to the only constraint that the loads belong to a given uncertainty domain. Results are given for continuous and discrete fibre orientations which constitute the optimization problem coupled to load anti-optimization problem leading to a nested solution method. It is observed that the stacking sequence of a laminate designed for a deterministic load case only differs considerably from that of a robust laminate designed taking load uncertainties into account. Consequently the buckling load carried by a deterministic design is considerably less than the one carried by a robust design when both are subjected to uncertain loads.     

Comparing deterministic,robust and online scheduling using entropy

We conjecture that when the uncertainty of scheduling information increases, one should change from deterministic, through robust to, finally, online scheduling techniques. Previously, extensive mathematical investigations have been carried out on the stability of a deterministic schedule for uncertain operation processing times. In this paper, we will use an empirical approach and an entropy measure to justify the transition between deterministic, robust and online scheduling. The use of an entropy measure in our context can be perceived, in a broader sense, as a pro-active approach to deal with changes in the level of information uncertainty and relative importance of each term in the total schedule execution cost. The level of information uncertainty may change due to the performance deterioration of processors (machines or human) and the replacement of old machines with new ones; and the changes in relative importance of cost elements may be due to changes in shop floor priorities and pressure from partners in the supply chain network. One can decide upon the scheduling strategies to be employed based on the latest entropy value of the information considered and the relative importance of each cost term.     

Fast numerical methods for robust optimal design

A fast numerical approach for robust design optimization is presented. The core of the method is based on the state-of-the-art fast numerical methods for stochastic computations with parametric uncertainty. These methods employ generalized polynomial chaos (gPC) as a high-order representation for random quantities and a stochastic Galerkin (SG) or stochastic collocation (SC) approach to transform the original stochastic governing equations to a set of deterministic equations. The gPC-based SG and SC algorithms are able to produce highly accurate stochastic solutions with (much) reduced computational cost. It is demonstrated that they can serve as efficient forward problem solvers in robust design problems. Possible alternative definitions for robustness are also discussed. Traditional robust optimization seeks to minimize the variance (or standard deviation) of the response function while optimizing its mean. It can be shown that although variance can be used as a measure of uncertainty, it is a weak measure and may not fully reflect the output variability. Subsequently a strong measure in terms of the sensitivity derivatives of the response function is proposed as an alternative robust optimization definition. Numerical examples are provided to demonstrate the efficiency of the gPC-based algorithms, in both the traditional weak measure and the newly proposed strong measure.     

鲁棒线性最优化的若干扩展

本文基于凸锥理论对鲁棒线性最优化作了若干拓展。本文的拓展分为三部分。首先我们放松了对不确定集的限制,把鲁棒线性最优化拓展到凸锥和子空间平移的交的不确定集的情形。其次我们考虑了由凸不等式定义的不确定集的鲁棒线性最优化。再次,我们把鲁棒线性最优化拓展到了包含系数不确定性和解的实现误差的情形。对某些特殊的情形,我们导出了鲁棒线性最优化的确定性等价问题。     

Reliability reallocation for cost uncertainty of fuel cell vehicles via improved differential evolution algorithm

A fuel cell vehicle (FCV) is composed of many subsystems; it is necessary to reallocate subsystem reliability to improve FCV system reliability. A comprehensive evaluation method considering uncertainty is proposed to obtain the interval value of feasibility factor. The FCV cost uncertainty model is established on the basic of parameters such as feasibility factor, initial reliability, limit reliability, and subsystem cost. To solve the optimization problem for cost uncertainty, the cost interval model is transformed into a deterministic model by interval order relation. An improved differential evolution (DE) algorithm is proposed to reallocate subsystem reliability for minimum cost.     

Adaptive gradient-assisted robust design optimization under interval uncertainty

Robust optimization techniques attempt to find a solution that is both optimum and relatively insensitive to input uncertainty. In general, these techniques are computationally more expensive than their deterministic counterparts. In this article two new robust optimization methods are presented. The first method is called gradient-assisted robust optimization (GARO). In GARO, a robust optimization problem is first converted to a deterministic one by using a gradient-based approximation technique. After solving this deterministic problem, the solution robustness and the accuracy of the approximation are checked. If the accuracy meets a threshold, a robust optimum solution is found; otherwise, the approximation is adaptively modified until the threshold is met and a solution, if it exists, is obtained. The second method is a faster version of GARO called quasi-concave gradient-assisted robust optimization (QC-GARO). QC-GARO is for problems with quasi-concave objective and constraint functions. The difference between GARO and QC-GARO is in the way that they check the approximation accuracy. Both GARO and QC-GARO methods are applied to a set of six engineering design test problems and the results are compared with a few related previous methods. It was found that, compared to the methods considered, GARO could solve all test problems but with a higher computational effort compared to QC-GARO. However, QC-GARO was computationally much faster when it was able to solve the problems.     

ROBUST DESIGN FOR IMPROVED VEHICLE HANDLING UNDER A RANGE OF MANEUVER CONDITIONS

In this article, a robust design procedure is applied to achieve improved vehicle handling performance as an integral part of simulation-based vehicle design. Recent developments in the field of robust design optimization and the techniques for creating global approximations of design behaviors are applied to improve the computational efficiency of robust vehicle design built upon sophisticated vehicle dynamic simulations. The approach is applied to the design of a M916A1 6-wheel tractor/M870A2 3-axle semi-trailer. The results illustrate that the proposed procedure is effective for preventing the rollover of ground vehicles as well as for identifying a design that is not only optimal against the worst maneuver condition but is also robust with respect to a range of maneuver inputs. Furthermore, a comparison is made between a statistical approach and a bi-level optimization approach in terms of their effectiveness in solving robust design problems     

A multi-objective robust stochastic programming model for disaster relief logistics under uncertainty

Humanitarian relief logistics is one of the most important elements of a relief operation in disaster management. The present work develops a multi-objective robust stochastic programming approach for disaster relief logistics under uncertainty. In our approach, not only demands but also supplies and the cost of procurement and transportation are considered as the uncertain parameters. Furthermore, the model considers uncertainty for the locations where those demands might arise and the possibility that some of the pre-positioned supplies in the relief distribution center or supplier might be partially destroyed by the disaster. Our multi-objective model attempts to minimize the sum of the expected value and the variance of the total cost of the relief chain while penalizing the solution’s infeasibility due to parameter uncertainty; at the same time the model aims to maximize the affected areas’ satisfaction levels through minimizing the sum of the maximum shortages in the affected areas. Considering the global evaluation of two objectives, a compromise programming model is formulated and solved to obtain a non-dominating compromise solution. We present a case study of our robust stochastic optimization approach for disaster planning for earthquake scenarios in a region of Iran. Our findings show that the proposed model can help in making decisions on both facility location and resource allocation in cases of disaster relief efforts.