包含随机客户的选择性旅行商问题建模及求解

针对快递配送过程中客户需求具有不确定性的特征,提出一种新的路径优化问题——包含随机客户的选择性旅行商问题,在该问题中客户每天是否具有配送需求存在一定概率,并且对客户进行配送可获取一定利润。同时考虑以上两种因素,建立该问题的数学模型, 目标为在满足行驶距离限制的条件下,找出一条经过部分客户的预优化路径,使得该路径的期望利润最大。其可用于模拟构建最后一公里快递配送的路径问题,提供更具有经济效益的配送路径。随后提出包含精细化局部搜索策略的改进遗传算法,算法根据问题特点构建初始可行解。最后通过多个计算比对结果表明,该算法具有较高的计算效率。     

Stochastic scheduling to minimize expected maximum lateness

This paper is concerned with the problems in scheduling a set of jobs associated with random due dates on a single machine so as to minimize the expected maximum lateness in stochastic environment. This is a difficult problem and few efforts have been reported on its solution in the literature. In this paper, we first derive a deterministic equivalent to the expected maximum lateness and then propose a dynamic programming algorithm to obtain the optimal solutions. The procedures to compute optimal solutions are initially developed in the case of deterministic processing times, and then extended to stochastic processing times following arbitrary probability distributions. Moreover, several heuristic rules are suggested to compute near-optimal solutions, which are shown to be highly efficient and accurate by computer-based experiments.     

A priori policy evaluation and cyclic-order-based simulated annealing for the multi-compartment vehicle routing problem with stochastic demands

We develop methods to estimate and exactly calculate the expected cost of a priori policies for the multi-compartment vehicle routing problem with stochastic demands, an extension of the classical vehicle routing problem where customer demands are uncertain and products must be transported in separate partitions. We incorporate our estimation procedure into a cyclic-order-based simulated annealing algorithm, significantly improving the best-known solution values for a set of benchmark problems. We also extend the updating procedure for a cyclic order’s candidate route set to duration-constrained a priori policies.     

A lot sizing model with queueing delays: The issue of safety time

The objective of this paper is to introduce the concept of safety time in a make-to-order production environment. The production facility is represented as a queueing model, explicitly including a non-zero setup time. A methodology is presented to quantify the safety time and to compute the associated service level based on the queueing delay. The main result is a convex relationship of the expected waiting time, the variance of the waiting time and the quoted lead time as a function of the lot size and a concave relationship of the service level as a function of the lot size. Most models in the literature assume batch arrivals. We relax that assumption so that an individual customer arrival process is allowed. We therefore have to derive a new closed form analytical expression for the expected waiting time. Both the deterministic and a stochastic case are studied.     

Local search for the probabilistic traveling salesman problem: Correction to the 2-p-opt and 1-shift algorithms

The probabilistic traveling salesman problem concerns the best way to visit a set of customers located in some metric space, where each customer requires a visit only with some known probability. A solution to this problem is an a priori tour which visits all customers, and the objective is to minimize the expected length of the a priori tour over all customer subsets, assuming that customers in any given subset must be visited in the same order as they appear in the a priori tour. This problem belongs to the class of stochastic vehicle routing problems, a class which has received increasing attention in recent years, and which is of major importance in real world applications.Several heuristics have been proposed and tested for the probabilistic traveling salesman problem, many of which are a straightforward adaptation of heuristics for the classical traveling salesman problem. In particular, two local search algorithms (2-p-opt and 1-shift) were introduced by Bertsimas.In a previous report we have shown that the expressions for the cost evaluation of 2-p-opt and 1-shift moves, as proposed by Bertsimas, are not correct. In this paper we derive the correct versions of these expressions, and we show that the local search algorithms based on these expressions perform significantly better than those exploiting the incorrect expressions.     

Product line selection and pricing under a share-of-surplus choice model

Product line selection and pricing decisions are critical to the profitability of many firms, particularly in today’s competitive business environment in which providers of goods and services are offering a broad array of products to satisfy customer needs.We address the problem of selecting a set of products to offer and their prices when customers select among the offered products according to a share-of-surplus choice model. A customer’s surplus is defined as the difference between his utility (willingness to pay) and the price of the product. Under the share-of-surplus model, the fraction of a customer segment that selects a product is defined as the ratio of the segment’s surplus from this particular product to the segment’s total surplus across all offered products with positive surplus for that segment.We develop a heuristic procedure for this non-concave, mixed-integer optimization problem. The procedure utilizes simulated annealing to handle the binary product selection variables, and a steepest-ascent-style procedure that relies on certain structural properties of the objective function to handle the non-concave, continuous portion of the problem involving the prices. We also develop a variant of our procedure to handle uncertainty in customer utilities. In computational studies, our basic procedures perform extremely well, producing solutions whose objective values are within about 5% of those obtained via enumerative methods. Our procedure to handle uncertain utilities also performs well, producing solutions with expected profit values that are roughly 10% higher than the corresponding expected profits from solutions obtained under the assumption of deterministic utilities.     

Analysis of seat allocation and overbooking decisions with hybrid information

We investigate a single-leg airline revenue management problem where an airline has limited demand information and uncensored no-show information. To use such hybrid information for simultaneous overbooking and booking control decisions, we combine expected overbooking cost with revenue. Then we take a robust optimization approach with a regret-based criterion. While the criterion is defined on a myriad of possible demand scenarios, we show that only a small number of them are necessary to compute the objective. We also prove that nested booking control policies are optimal among all deterministic ones. We further develop an effective computational method to find the optimal policy and compare our policy to others proposed in the literature.     

Cyclic-order neighborhoods with application to the vehicle routing problem with stochastic demand

We examine neighborhood structures for heuristic search applicable to a general class of vehicle routing problems (VRPs). Our methodology utilizes a cyclic-order solution encoding, which maps a permutation of the customer set to a collection of many possible VRP solutions. We identify the best VRP solution in this collection via a polynomial-time algorithm from the literature. We design neighborhoods to search the space of cyclic orders. Utilizing a simulated annealing framework, we demonstrate the potential of cyclic-order neighborhoods to facilitate the discovery of high quality a priori solutions for the vehicle routing problem with stochastic demand (VRPSD). Without tailoring our solution procedure to this specific routing problem, we are able to match 16 of 19 known optimal VRPSD solutions. We also propose an updating procedure to evaluate the neighbors of a current solution and demonstrate its ability to reduce the computational expense of our approach.     

New efficient numerical procedures for solving stochastic variational problems with a priori maximum pointwise error estimates

In a previous paper we gave a new formulation and derived the Euler equations and other necessary conditions to solve strong, pathwise, stochastic variational problems with trajectories driven by Brownian motion. Thus, unlike current methods which minimize the control over deterministic functionals (the expected value), we find the control which gives the critical point solution of random functionals of a Brownian path and then, if we choose, find the expected value.This increase in information is balanced by the fact that our methods are anticipative while current methods are not. However, our methods are more directly connected to the theory and meaningful examples of deterministic variational theory and provide better means of solution for free and constrained problems. In addition, examples indicate that there are methods to obtain nonanticipative solutions from our equations although the anticipative optimal cost function has smaller expected value.In this paper we give new, efficient numerical methods to find the solution of these problems in the quadratic case. Of interest is that our numerical solution has a maximal, a priori, pointwise error of O(h3/2) where h is the node size. We believe our results are unique for any theory of stochastic control and that our methods of proof involve new and sophisticated ideas for strong solutions which extend previous deterministic results by the first author where the error was O(h²).We note that, although our solutions are given in terms of stochastic differential equations, we are not using the now standard numerical methods for stochastic differential equations. Instead we find an approximation to the critical point solution of the variational problem using relations derived from setting to zero the directional derivative of the cost functional in the direction of simple test functions.Our results are even more significant than they first appear because we can reformulate stochastic control problems or constrained calculus of variations problems in the unconstrained, stochastic calculus of variations formulation of this paper. This will allow us to find efficient and accurate numerical solutions for general constrained, stochastic optimization problems. This is not yet being done, even in the deterministic case, except by the first author.     

A stochastic inventory problem with fuzzy shortage cost

Up to now, many inventory models have been considered in the literature. Some assume stochastic demands and others consider the deterministic case. Though they include a shortage cost due to lost sales, it is usually assumed to be known concretely and a priori. This paper introduces fuzziness of shortage cost explicitly into the classical newsboy problem. That is, we investigate the so-called fuzzy newsboy problem where its shortage cost is vague and given by an L shape fuzzy number. Then the total expected profit function also becomes a fuzzy number. Finally, we find an optimal ordering quantity realizing the fuzzy max order of the profit function (fuzzy min order considering the profit function) and compare it with the optimal ordering quantity of the non-fuzzy newsboy problem.     

Solving the Probabilistic TSP with Ant Colony Optimization

In this paper, we describe new ways to apply Ant Colony Optimization (ACO) to the Probabilistic Traveling Salesperson Problem (PTSP). PTSP is a stochastic extension of the well known Traveling Salesperson Problem (TSP), where each customer will require a visit only with a certain probability. The goal is to find an a priori tour visiting all customers with minimum expected length, customers not requiring a visit simply being skipped in the tour.We show that ACO works well even when only an approximative evaluation function is used, which speeds up the algorithm, leaving more time for the actual construction. As we demonstrate, this idea can also be applied successfully to other state-of-the-art heuristics. Furthermore, we present new heuristic guidance schemes for ACO, better adapted to the PTSP than what has been used previously. We show that these modifications lead to significant improvements over the standard ACO algorithm, and that the resulting ACO is at least competitive to other state-of-the-art heuristics.     

Asymptotically optimal staffing of service systems with joint QoS constraints

We consider the problem of staffing large-scale service systems with multiple customer classes and multiple dedicated server pools under joint quality-of-service (QoS) constraints. We first analyze the case in which arrival rates are deterministic and the QoS metric is the probability a customer is queued, given by the Erlang-C formula. We use the Janssen–Van Leeuwaarden–Zwart bounds to obtain asymptotically optimal solutions to this problem. The second model considered is one in which the arrival rates are not completely known in advance (before the server staffing levels are chosen), but rather are known via a probability distribution. In this case, we provide asymptotically optimal solutions to the resulting stochastic integer program, leveraging results obtained for the case of deterministic arrivals.     

Selective and periodic inventory routing problem for waste vegetable oil collection

We consider a biodiesel production company that collects waste vegetable oil from source points that generate waste in large amounts. The company uses the collected waste as raw material for biodiesel production. The manager of this company needs to decide which of the present source points to include in the collection program, which of them to visit on each day, which periodic routing schedule to repeat over an infinite horizon and how many vehicles to operate such that the total collection, inventory and purchasing costs are minimized while the production requirements and operational constraints are met. For this selective and periodic inventory routing problem, we propose two different formulations, compare them and apply the better performing one on a real-world problem with 36 scenarios. We generate lower bounds using a partial linear relaxation model, and observe that the solutions obtained through our model are within 3.28% of optimality on the average. Several insights regarding the customer selection, routing and purchasing decisions are acquired with sensitivity analysis.     

Modelling imperfect advance demand information and analysis of optimal inventory policies

We consider an inventory control problem where it is possible to collect some imperfect information on future demand. We refer to such information as imperfect Advance Demand Information (ADI), which may occur in different forms of applications. A simple example is a company that uses sales representatives to market its products, in which case the collection of sales representatives’ information as to the number of customers interested in a product can generate an indication about the future sales of that product, hence it constitutes imperfect ADI. Other applications include internet retailing, Vendor Managed Inventory (VMI) applications and Collaborative Planning, Forecasting, and Replenishment (CPFR) environments. We develop a model that incorporates imperfect ADI with ordering decisions. Under our system settings, we show that the optimal policy is of order-up-to type, where the order level is a function of imperfect ADI. We also provide some characterizations of the optimal solution. We develop an expression for the expected cost benefits of imperfect ADI for the myopic problem. Our analytical and empirical findings reveal the conditions under which imperfect ADI is more valuable.     

A location–allocation heuristic for the capacitated multi-facility Weber problem with probabilistic customer locations

The capacitated multi-facility Weber problem is concerned with locating m facilities in the Euclidean plane, and allocating their capacities to n customers at minimum total cost. The deterministic version of the problem, which assumes that customer locations and demands are known with certainty, is a non-convex optimization problem and difficult to solve. In this work, we focus on a probabilistic extension and consider the situation where the customer locations are randomly distributed according to a bivariate distribution. We first present a mathematical programming formulation, which is even more difficult than its deterministic version. We then propose an alternate location–allocation local search heuristic generalizing the ideas used originally for the deterministic problem. In its original form, the applicability of the heuristic depends on the calculation of the expected distances between the facilities and customers, which can be done for only very few distance and probability density function combinations. We therefore propose approximation methods which make the method applicable for any distance function and bivariate location distribution.     

A memetic algorithm for the team orienteering problem

The team orienteering problem (TOP) is a generalization of the orienteering problem. A limited number of vehicles is available to visit customers from a potential set. Each vehicle has a predefined running-time limit, and each customer has a fixed associated profit. The aim of the TOP is to maximize the total collected profit. In this paper we propose a simple hybrid genetic algorithm using new algorithms dedicated to the specific scope of the TOP: an Optimal Split procedure for chromosome evaluation and local search techniques for mutation. We have called this hybrid method a memetic algorithm for the TOP. Computational experiments conducted on standard benchmark instances clearly show our method to be highly competitive with existing ones, yielding new improved solutions in at least 5 instances.     

Ensuring service levels in routing problems with time windows and stochastic travel times

In the stochastic variant of the vehicle routing problem with time windows, known as the SVRPTW, travel times are assumed to be stochastic. In our chance-constrained approach to the problem, restrictions are placed on the probability that individual time window constraints are violated, while the objective remains based on traditional routing costs. In this paper, we propose a way to offer this probability, or service level, for all customers. Our approach carefully considers how to compute the start-service time and arrival time distributions for each customer. These distributions are used to create a feasibility check that can be “plugged” into any algorithm for the VRPTW and thus be used to solve large problems fairly quickly. Our computational experiments show how the solutions change for some well-known data sets across different levels of customer service, two travel time distributions, and several parameter settings.     

Upper and lower bounds for the sales force deployment problem with explicit contiguity constraints

The sales force deployment problem arises in many selling organizations. This complex planning problem involves the concurrent resolution of four interrelated subproblems: sizing of the sales force, sales representatives locations, sales territory alignment, and sales resource allocation. The objective is to maximize the total profit. For this, a well-known and accepted concave sales response function is used. Unfortunately, literature is lacking approaches that provide valid upper bounds. Therefore, we propose a model formulation with an infinite number of binary variables. The linear relaxation is solved by column generation where the variables with maximum reduced costs are obtained analytically. For the optimal objective function value of the linear relaxation an upper bound is provided. To obtain a very tight gap for the objective function value of the optimal integer solution we introduce a Branch-and-Price approach. Moreover, we propose explicit contiguity constraints based on flow variables. In a series of computational studies we consider instances which may occur in the pharmaceutical industry. The largest instance comprises 50 potential locations and more than 500 sales coverage units. We are able to solve this instance in 1273 seconds with a gap of less than 0.01%. A comparison with Drexl and Haase (1999) shows that we are able to halve the solution gap due to tight upper bounds provided by the column generation procedure.     

Efficient and effective heuristics for the coordinated capacitated lot-size problem

Coordinating procurement decisions for a family of products that share a constrained resource, such as an ocean shipping container, is an important managerial problem. However due to the problem’s difficult mathematical properties, efficient and effective solution procedures for the problem have eluded researchers. This paper proposes two heuristics, for the capacitated, coordinated dynamic demand lot-size problem with deterministic but time-varying demand. In addition to inventory holding costs, the problem assumes a joint setup cost each time any member of the product family is replenished and an individual item setup cost for each item type replenished. The objective is to meet all customer demand without backorders at minimum total cost. We propose a six-phase heuristic (SPH) and a simulated annealing meta-heuristic (SAM). The SPH begins by assuming that each customer demand is met by a unique replenishment and then it seeks to iteratively maximize the net savings associated with order consolidation. Using SPH to find a starting solution, the SAM orchestrates escaping local solutions and exploring other areas of the solution state space that are randomly generated in an annealing search process. The results of extensive computational experiments document the effectiveness and efficiency of the heuristics. Over a wide range of problem parameter values, the SPH and SAM find solutions with an average optimality gap of 1.53% and 0.47% in an average time of 0.023 CPU seconds and 0.32 CPU seconds, respectively. The heuristics are strong candidates for application as stand alone solvers or as an upper bounding procedure within an optimization based algorithm. The procedures are currently being tested as a solver in the procurement software suite of a nationally recognized procurement software provider.     

The efficient solution of the (quietly constrained) noisy, linear regulator problem

In a previous paper we gave a new, natural extension of the calculus of variations/optimal control theory to a (strong) stochastic setting. We now extend the theory of this most fundamental chapter of optimal control in several directions. Most importantly we present a new method of stochastic control, adding Brownian motion which makes the problem “noisy.” Secondly, we show how to obtain efficient solutions: direct stochastic integration for simpler problems and/or efficient and accurate numerical methods with a global a priori error of O(h3/2) for more complex problems. Finally, we include “quiet” constraints, i.e. deterministic relationships between the state and control variables. Our theory and results can be immediately restricted to the non “noisy” (deterministic) case yielding efficient, numerical solution techniques and an a priori error of O(h²). In this event we obtain the most efficient method of solving the (constrained) classical Linear Regulator Problem. Our methods are different from the standard theory of stochastic control. In some cases the solutions coincide or at least are closely related. However, our methods have many advantages including those mentioned above. In addition, our methods more directly follow the motivation and theory of classical (deterministic) optimization which is perhaps the most important area of physical and engineering science. Our results follow from related ideas in the deterministic theory. Thus, our approximation methods follow by guessing at an algorithm, but the proof of global convergence uses stochastic techniques because our trajectories are not differentiable. Along these lines, a general drift term in the trajectory equation is properly viewed as an added constraint and extends ideas given in the deterministic case by the first author.