Search

Production planning must account for uncertainty in a production system, arising from fluctuating demand forecasts. Therefore, this article focuses on the integration of updated customer demand into the rolling horizon planning cycle. We use scenario-based stochastic programming to solve capacitated lot sizing problems under stochastic demand in a rolling horizon environment. This environment is replicated using a discrete event simulation-optimization framework, where the optimization problem is periodically solved, leveraging the latest demand information to continually adjust the production plan. We evaluate the stochastic optimization approach and compare its performance to solving a deterministic lot sizing model, using expected demand figures as input, as well as to standard Material Requirements Planning (MRP). In the simulation study, we analyze three different customer behaviors related to forecasting, along with four levels of shop load, within a multi-item and multi-stage production system. We test a range of significant parameter values for the three planning methods and compute the overall costs to benchmark them. The results show that the production plans obtained by MRP are outperformed by deterministic and stochastic optimization. Particularly, when facing tight resource restrictions and rising uncertainty in customer demand, the use of stochastic optimization becomes preferable compared to deterministic optimization.

Branch and bound methods which are based on the principle "divide and conquer" are a well established solution approach in single-objective integer programming. In multi-objective optimization branch and bound algorithms are increasingly attracting interest. However, the larger number of objectives raises additional difficulties for implicit enumeration approaches like branch and bound. Since bounding and pruning is considerably weaker in multiple objectives, many branches have to be (partially) searched and may not be pruned directly. The adaptive use of objective space information can guide the search in promising directions to determine a good approximation of the Pareto front already in early stages of the algorithm. In particular we focus in this article on improving the branching and queuing of subproblems and the handling of lower bound sets. In our numerical test we evaluate the impact of the proposed methods in comparison to a standard implementation of multiobjective branch and bound on knapsack problems, generalized assignment problems and (un)capacitated facility location problems.

For many real-world decision-making problems subject to uncertainty, it may be essential to deal with multiple and often conflicting objectives while taking the decision-makers' risk preferences into account. Conditional value-at-risk (CVaR) is a widely applied risk measure to address risk-averseness of the decision-makers. In this paper, we use the subset-based polyhedral representation of the CVaR to reformulate the bi-objective two-stage stochastic facility location problem presented in Nazemi et al. (2021). We propose an approximate cutting-plane method to deal with this more computationally challenging subset-based formulation. Then, the cutting plane method is embedded into the epsilon-constraint method, the balanced-box method, and a recently developed matheuristic method to address the bi-objective nature of the problem. Our computational results show the effectiveness of the proposed method. Finally, we discuss how incorporating an approximation of the subset-based polyhedral formulation affects the obtained solutions., Comment: arXiv admin note: substantial text overlap with arXiv:2007.07767

Motivated by a real-world application, we model and solve a complex staff scheduling problem. Tasks are to be assigned to workers for supervision. Multiple tasks can be covered in parallel by a single worker, with worker shifts being flexible within availabilities. Each worker has a different skill set, enabling them to cover different tasks. Tasks require assignment according to priority and skill requirements. The objective is to maximize the number of assigned tasks weighted by their priorities, while minimizing assignment penalties. We develop an adaptive large neighborhood search (ALNS) algorithm, relying on tailored destroy and repair operators. It is tested on benchmark instances derived from real-world data and compared to optimal results obtained by means of a commercial MIP-solver. Furthermore, we analyze the impact of considering three additional alternative objective functions. When applied to large-scale company data, the developed ALNS outperforms the previously applied solution approach.

In the bi-objective branch-and-bound literature, a key ingredient is objective branching, i.e. to create smaller and disjoint sub-problems in the objective space, obtained from the partial dominance of the lower bound set by the upper bound set. When considering three or more objective functions, however, applying objective branching becomes more complex, and its benefit has so far been unclear. In this paper, we investigate several ingredients which allow to better exploit objective branching in a multi-objective setting. We extend the idea of probing to multiple objectives, enhance it in several ways, and show that when coupled with objective branching, it results in significant speed-ups in terms of CPU times. We also investigate cut generation based on the objective branching constraints. Besides, we generalize the best-bound idea for node selection to multiple objectives and we show that the proposed rules outperform the, in the multi-objective literature, commonly employed depth-first and breadth-first strategies. We also analyze problem specific branching rules. We test the proposed ideas on available benchmark instances for three problem classes with three and four objectives, namely the capacitated facility location problem, the uncapacitated facility location problem, and the knapsack problem. Our enhanced multi-objective branch-and-bound algorithm outperforms the best existing branch-and-bound based approach and is the first to obtain competitive and even slightly better results than a state-of-the-art objective space search method on a subset of the problem classes.

The Electric Autonomous Dial-A-Ride Problem (E-ADARP) consists in scheduling a fleet of electric autonomous vehicles to provide ride-sharing services for customers that specify their origins and destinations. The E-ADARP differs from the classical DARP in two aspects: (i) a weighted-sum objective that minimizes both total travel time and total excess user ride time; (ii) the employment of electric autonomous vehicles and a partial recharging policy. This paper presents a highly-efficient labeling algorithm, which is integrated into Branch-and-Price (B&P) algorithms to solve the E-ADARP. To handle (i), we introduce a fragment-based representation of paths. A novel approach is invoked to abstract fragments to arcs while ensuring excess-user-ride-time optimality. We then construct a new graph that preserves all feasible routes of the original graph by enumerating all feasible fragments, abstracting them to arcs, and connecting them with each other, depots, and recharging stations in a feasible way. On the new graph, partial recharging (ii) is tackled exactly by tailored Resource Extension Functions (REFs). We apply strong dominance rules and constant-time feasibility checks to compute the shortest paths efficiently. These methods construct the first labeling algorithm that can deal with minimizing (excess) user ride time. In the computational experiments, the B&P algorithm achieves optimality in 71 out of 84 instances. Remarkably, among these instances, 50 were solved optimally at the root node without branching. We identify 26 new best solutions, improve 30 previously reported lower bounds, and provide 17 new lower bounds for large-scale instances with up to 8 vehicles and 96 requests. In total 42 new best solutions are generated on previously solved and unsolved instances.

Many real-world optimisation problems involve multiple objectives. When considered concurrently, they give rise to a set of optimal trade-off solutions, also known as efficient solutions. These solutions have the property that neither objective can be improved without deteriorating another objective. Motivated by the success of matheuristics in the single-objective domain, we propose a linear programming-based matheuristic for tri-objective binary integer programming. To achieve a high-quality approximation of the optimal set of trade-off solutions, a lower bound set is first obtained using the vector linear programming solver Bensolve. Then, feasibility pump-based ideas in combination with path relinking are applied in novel ways so as to obtain a high quality upper bound set. Our matheuristic is compared to a recently-suggested algorithm that is, to the best of our knowledge, the only existing matheuristic method for tri-objective integer programming. In an extensive computational study, we show that our method generates a better approximation of the true Pareto front than the benchmark method on a large set of tri-objective benchmark instances. Since the developed approach starts from a potentially fractional lower bound set, it may also be used as a primal heuristic in the context of linear relaxation-based multi-objective branch-and-bound algorithms.

In this paper, we present the first outer approximation algorithm for multi-objective mixed-integer linear programming problems with any number of objectives. The algorithm also works for certain classes of non-linear programming problems. It produces the non-dominated extreme points as well as the facets of the convex hull of these points. The algorithm relies on an oracle which solves single-objective weighted-sum problems and we show that the required number of oracle calls is polynomial in the number of facets of the convex hull of the non-dominated extreme points in the case of multiobjective mixed-integer programming (MOMILP). Thus, for MOMILP problems for which the weighted-sum problem is solvable in polynomial time, the facets can be computed with incremental-polynomial delay. From a practical perspective, the algorithm starts from a valid lower bound set for the non-dominated extreme points and iteratively improves it. Therefore it can be used in multi-objective branch-and-bound algorithms and still provide a valid bound set at any stage, even if interrupted before converging. Moreover, the oracle produces Pareto optimal solutions, which makes the algorithm also attractive from the primal side in a multi-objective branch-and-bound context. Finally, the oracle can also be called with any relaxation of the primal problem, and the obtained points and facets still provide a valid lower bound set. A computational study on a set of benchmark instances from the literature and new non-linear multi-objective instances is provided., Comment: 21 pages

Motivated by their success in the single-objective domain, we propose a very simple linear programming-based matheuristic for tri-objective binary integer programming. To tackle the problem, we obtain lower bound sets by means of the vector linear programming solver Bensolve. Then, simple heuristic approaches, such as rounding and path relinking, are applied to this lower bound set to obtain high-quality approximations of the optimal set of trade-off solutions. The proposed algorithm is compared to a recently suggested algorithm which is, to the best of our knowledge, the only existing matheuristic method for tri-objective integer programming. Computational experiments show that our method produces a better approximation of the true Pareto front using significantly less time than the benchmark method on standard benchmark instances for the three-objective knapsack problem.

The aim of the bi-objective multimodal car-sharing problem (BiO-MMCP) is to determine the optimal mode of transport assignment for trips and to schedule the routes of available cars and users whilst minimizing cost and maximizing user satisfaction. We investigate the BiO-MMCP from a user-centred point of view. As user satisfaction is a crucial aspect in shared mobility systems, we consider user preferences in a second objective. Users may choose and rank their preferred modes of transport for different times of the day. In this way we account for, e.g., different traffic conditions throughout the planning horizon. We study different variants of the problem. In the base problem, the sequence of tasks a user has to fulfill is fixed in advance and travel times as well as preferences are constant over the planning horizon. In variant 2, time-dependent travel times and preferences are introduced. In variant 3, we examine the challenges when allowing additional routing decisions. Variant 4 integrates variants 2 and 3. For this last variant, we develop a branch-and-cut algorithm which is embedded in two bi-objective frameworks, namely the $\epsilon$-constraint method and a weighting binary search method. Computational experiments show that the branch-and cut algorithm outperforms the MIP formulation and we discuss changing solutions along the Pareto frontier.

Multiple and usually conflicting objectives subject to data uncertainty are main features in many real-world problems. Consequently, in practice, decision-makers need to understand the trade-off between the objectives, considering different levels of uncertainty in order to choose a suitable solution. In this paper, we consider a two-stage bi-objective single source capacitated model as a base formulation for designing a last-mile network in disaster relief where one of the objectives is subject to demand uncertainty. We analyze scenario-based two-stage risk-neutral stochastic programming, adaptive (two-stage) robust optimization, and a two-stage risk-averse stochastic approach using conditional value-at-risk (CVaR). To cope with the bi-objective nature of the problem, we embed these concepts into two criterion space search frameworks, the $\epsilon$-constraint method and the balanced box method, to determine the Pareto frontier. Additionally, a matheuristic technique is developed to obtain high-quality approximations of the Pareto frontier for large-size instances. In an extensive computational experiment, we evaluate and compare the performance of the applied approaches based on real-world data from a Thies drought case, Senegal.

In many real-world optimization problems, more than one objective plays a role and input parameters are subject to uncertainty. In this paper, motivated by applications in disaster relief and public facility location, we model and solve a bi-objective stochastic facility location problem. The considered objectives are cost and uncovered demand, whereas the demand at the different population centers is uncertain but its probability distribution is known. The latter information is used to produce a set of scenarios. In order to solve the underlying optimization problem, we apply a Benders' type decomposition approach which is known as the L-shaped method for stochastic programming and we embed it into a recently developed branch-and-bound framework for bi-objective integer optimization. We analyze and compare different cut generation schemes and we show how they affect lower bound set computations, so as to identify the best performing approach. Finally, we compare the branch-and-Benders-cut approach to a straight-forward branch-and-bound implementation based on the deterministic equivalent formulation.

Motivated by the change in mobility patterns, we present a scheduling approach for a vehicle-sharing problem, considering several alternative modes of transport, from a company viewpoint with centralized planning. We consider vehicle-sharing in a company having one or more depots and a fixed number of users, i.e. employees. The users have appointments with a fixed location and fixed start and end times. A vehicle must be used for a full trip of a user from depot to depot. We aim at assigning vehicles to user trips so as to maximize savings compared to other modes of transport. We first consider that only one type of vehicle is used, and second that multiple vehicle types can be used. For the first case, we show that the vehicle-sharing problem can be formulated as a minimum-cost flow problem. Secondly, if multiple types of vehicles are available the problem can be formulated as a multi-commodity flow problem. These formulations make the problem applicable in daily operations due to efficient solution methods. We provide a comprehensive computational study for both cases on instances based on demographic, spatial, and economic data of Vienna. We show that our formulations for this problem solve these instances in a few seconds, which makes them usable in an online booking system. In the analysis we discuss different potential settings. We study the optimal composition of a shared fleet, restricted sets of modes of transport, and variations of the objective function.

We introduce the multimodal car- and ride-sharing problem (MMCRP), in which a pool of cars is used to cover a set of ride requests while uncovered requests are assigned to other modes of transport (MOT). A car's route consists of one or more trips. Each trip must have a specific but non-predetermined driver, start in a depot and finish in a (possibly different) depot. Ride-sharing between users is allowed, even when two rides do not have the same origin and/or destination. A user has always the option of using other modes of transport according to an individual list of preferences. The problem can be formulated as a vehicle scheduling problem. In order to solve the problem, an auxiliary graph is constructed in which each trip starting and ending in a depot, and covering possible ride-shares, is modeled as an arc in a time-space graph. We propose a two-layer decomposition algorithm based on column generation, where the master problem ensures that each request can only be covered at most once, and the pricing problem generates new promising routes by solving a kind of shortest-path problem in a time-space network. Computational experiments based on realistic instances are reported. The benchmark instances are based on demographic, spatial, and economic data of Vienna, Austria. We solve large instances with the column generation based approach to near optimality in reasonable time, and we further investigate various exact and heuristic pricing schemes.

Project scheduling in manufacturing environments often requires flexibility in terms of the selection and the exact length of alternative production activities. Moreover, the simultaneous scheduling of multiple lots is mandatory in many production planning applications. To meet these requirements, a new resource-constrained project scheduling problem (RCPSP) is introduced where both decisions (activity flexibility and time flexibility) are integrated. Besides the minimization of makespan, two new alternative objectives are presented: maximization of balanced length of selected activities (time balance) and maximization of balanced resource utilization (resource balance). New mixed integer and constraint programming (CP) models are proposed for the developed integrated flexible project scheduling problem. Benchmark instances on an already existing flexible RCPSP and the newly developed problem are solved to optimality. The real-world applicability of the suggested CP models is shown by additionally solving a large industry case.

The unrestricted block relocation problem is an important optimization problem encountered at terminals, where containers are stored in stacks. It consists in determining the minimum number of container moves so as to empty the considered bay following a certain retrieval sequence. A container move can be either the retrieval of a container or the relocation of a certain container on top of a stack to another stack. The latter types of moves are necessary so as to provide access to containers which are currently not on top of a stack. They might also be useful to prepare future removals. In this paper, we propose the first local search type improvement heuristic for the block relocation problem. It relies on a clever definition of the state space which is explored by means of a dynamic programming algorithm so as to identify the locally optimal sequence of moves of a given container. Our results on large benchmark instance reveal unexpectedly high improvement potentials (up to 50%) compared to results obtained by state-of-the-art constructive heuristics.

In bi-objective integer optimization the optimal result corresponds to a set of non-dominated solutions. We propose a generic bi-objective branch-and-bound algorithm that uses a problem-independent branching rule exploiting available integer solutions and takes advantage of integer objective coefficients. The developed algorithm is applied to bi-objective facility location problems, to the bi-objective set covering problem, as well as to the bi-objective team orienteering problem with time windows. In the latter case, lower bound sets are computed by means of column generation. Comparison to state-of-the-art exact algorithms shows the effectiveness of the proposed branch-and-bound algorithm.