Time and multiple objectives in scheduling and routing problems

Many optimization problems encountered in practice are multi-objective by nature, i.e., different objectives are conflicting and equally important. Many times, it is not desirable to drop some of them or to optimize them in a composite single objective or hierarchical manner. Furthermore, cost parameters change over time which makes optimization problems harder. For instance, in the transport sector, travel costs are a function of travel time which changes depending on the time of the day a vehicle is travelling (e.g., due to road congestion). Road congestion results in tremendous delays which lead to a decrease in the service quality and the responsiveness of logistic service providers. In Chapter 2, we develop a generic approach to deal with Multi-Objective Scheduling Problems (MOSPs) with State-Dependent Cost Parameters. The aim is to determine the set of Pareto solutions that capture the trade offs between the different conflicting objectives. Due to the complexity of MOSPs, an efficient approximation based on dynamic programming is developed. The approximation has a provable worse case performance guarantee. Even though the generated approximate Pareto front consist of fewer solutions, it still represents a good coverage of the true Pareto front. Furthermore, considerable gains in computation times are achieved. In Chapter 3, the developed methodology is validated on the multi-objective timedependent knapsack problem. In the classical knapsack problem, the input consists of a knapsack with a finite capacity and a set of items, each with a certain weight and a cost. A feasible solution to the knapsack problem is a selection of items such that their total weight does not exceed the knapsack capacity. The goal is to maximize the single objective function consisting of the total pro t of the selected items. We extend the classical knapsack problem in two ways. First, we consider time-dependent profits (e.g., in a retail environment profit depends on whether it is Christmas or not)

A stochastic inventory policy with limited transportation capacity

In this paper we consider a stochastic single-item inventory problem. A retailer keeps a single product on stock to satisfy customers stochastic demand. The retailer is replenished periodically from a supplier with ample stock. For the delivery of the product, trucks with finite capacity are available and a fixed shipping cost is charged whenever a truck is dispatched regardless of its load. Furthermore, linear holding and backorder costs are considered at the end of a review period. A replenishment policy is proposed to determine order quantities taking into account transportation capacity and aiming at minimizing total average cost. Every period an order quantity is determined based on an order-up-to logic. If the order quantity is smaller than a given threshold then the shipment is delayed. On the other hand, if the order quantity is larger than a second threshold then the initial order size is enlarged and a full truckload is shipped. An order size between these two thresholds results in no adaption of the order quantity and the order is shipped as it is. We illustrate that this proposed policy is close to the optimal policy and much better than an order-up-to policy without adaptations. Moreover, we show how to compute the cost optimal policy parameters exactly and how to compute them by relying on approximations. In a detailed numerical study, we compare the results obtained by the heuristics with those given by the exact analysis. A very good cost performance of the proposed heuristics can be observed

Single item inventory control under periodic review and a minimum order quantity

In this paper we study a periodic review single item single stage inventory system with stochastic demand. In each time period the system must order none or at least as much as a minimum order quantity Qmin. Since the optimal structure of an ordering policy with a minimum order quantity is complicated, we propose an easy-to-use policy, which we call (R, S,Qmin) policy. Assuming linear holding and backorder costs we determine the optimal numerical value of the level S using a Markov Chain approach. In addition, we derive simple news-vendor-type inequalities for near-optimal policy parameters, which can easily be implemented within spreadsheet applications. In a numerical study we compare our policy with others and test the performance of the approximation for three different demand distributions: Poisson, negative binomial, and a discretized version of the gamma distribution. Given the simplicity of the policy and its cost performance as well as the excellent performance of the approximation we advocate the application of the (R, S,Qmin) policy in practice

A dynamic programming approach to multi-objective time-dependent capacitated single vehicle routing problems with time windows

A single vehicle performs several tours to serve a set of geographically dis- persed customers. The vehicle has a finite capacity and is only available for a limited amount of time. Moreover, tours' duration is restricted (e.g. due to quality or security issues). Because of road congestion, travel times are time-dependent: depending on the departure time at a customer, a different travel time is incurred. Furthermore, all customers need to get delivered in their specicified time windows. Contrary to most of the literature, we con- sider a multi-objective cost function: simultaneously minimizing the total time traveled including waiting times at customers due to time windows, and maximizing the total demand fulfilled. Efficient dynamic programming algorithms are developed to compute the Pareto set of routes, assuming a specific structure for time windows and travel time profiles

Approximating multi-objective time-dependent optimization problems

In many practical situations, decisions are multi-objective in nature. Furthermore, costs and profits are time-dependent, i.e. depending upon the time a decision is taken, different costs and profits are incurred. In this paper, we propose a generic approach to deal with multi-objective time-dependent optimization problems (MOTDP). The aim is to determine the set of Pareto solutions that capture the interactions between the different objectives. Due, to the complexity of MOTDP, an efficient approximation based on dynamic programming is developed. The approximation has a provable worst case performance guarantee. Even though the approximate Pareto set consists of less solutions, it represents a good coverage of the true set of Pareto solutions. Numerical results are presented showing the value of the approximation

The Time Window Assignment Vehicle Routing Problem with Time Dependent Travel Times

In this paper, we introduce the time window assignment vehicle routing problem (TWAVRP) with time-dependent travel times. It is the problem of assigning time windows to customers before their demand is known and creating vehicle routes adhering to these time windows after demand becomes known. The goal is to assign the time windows in such a way that the expected transportation costs are minimized. We develop a branch-price-and-cut algorithm to solve this problem to optimality. The pricing problem that has to be solved is a new variant of the shortest path problem, which includes a capacity constraint, time-dependent travel times, time window constraints on both the nodes and on the arcs, and linear node costs. For solving the pricing problem, we develop an exact labeling algorithm and a tabu search heuristic. Furthermore, we present new valid inequalities, which are specifically designed for the TWAVRP with time-dependent travel times. Finally, we present results of numerical experiments to illustrate the performance of the algorithm

Cover inequalities for a vehicle routing problem with time windows and shifts

This paper introduces the vehicle routing problem with time windows and shifts (VRPTWS). At the depot, several shifts with nonoverlapping operating periods are available to load the planned trucks. Each shift has a limited loading capacity. We solve the VRPTWS exactly by a branch-and-cut-and-price algorithm. The master problem is a set partitioning with an additional constraint for every shift. Each constraint requires the total quantity loaded in a shift to be less than its loading capacity. For every shift, a pricing subproblem is solved by a label-setting algorithm. Shift capacity constraints define knapsack inequalities; hence we use valid inequalities inspired from knapsack inequalities to strengthen the linear programming relaxation of the master problem when solved by column generation. In particular, we use a family of tailored robust cover inequalities and a family of new nonrobust cover inequalities. Numerical results show that nonrobust cover inequalities significantly improve the algorithm

Cover inequalities for a vehicle routing problem with time windows and shifts

This paper introduces the vehicle routing problem with time windows and shifts (VRPTWS). At the depot, several shifts with nonoverlapping operating periods are available to load the planned trucks. Each shift has a limited loading capacity. We solve the VRPTWS exactly by a branch-and-cut-and-price algorithm. The master problem is a set partitioning with an additional constraint for every shift. Each constraint requires the total quantity loaded in a shift to be less than its loading capacity. For every shift, a pricing subproblem is solved by a label-setting algorithm. Shift capacity constraints define knapsack inequalities; hence we use valid inequalities inspired from knapsack inequalities to strengthen the linear programming relaxation of the master problem when solved by column generation. In particular, we use a family of tailored robust cover inequalities and a family of new nonrobust cover inequalities. Numerical results show that nonrobust cover inequalities significantly improve the algorithm