Dantzig-Wolfe Decomposition Approach Research Articles

5th generation district heating and cooling network planning: A Dantzig–Wolfe decomposition approach

The adequate planning of district energy systems is a key process to provide communities with proper heating and cooling networks. To find cost-effective planning solutions, this decision-making process is usually formulated as mathematical model, which is optimized to determine where and how many assets should be installed in the network. However, this optimization problem is becoming more complex as district energy systems evolve and become more elaborate. For the 5th generation district heating and cooling (5GDHC) networks, this planning framework comprises multiple building energy systems, a thermal and electrical network as well as central heating and cooling units. As a result, the optimization problem associated with these circumstances can easily become intractable as the number of elements of the network increases. To alleviate this tractability problem, in this paper, a Dantzig–Wolfe approach is devised to decompose a mixed-integer linear program into multiple subproblems (for every building) and a master problem (thermal and electrical network and central units). A realistic case study based on a 5GDHC system in Germany is considered. For this case study, it is demonstrated that the proposed decomposition approach yields the same results attained by the original not decomposed problem while achieving significant gains in terms of scalability and computational times. More specifically, the decomposition approach reduces the computational time for districts with more than 10 buildings by in average 94%. This corroborates the potential of the proposed approach to improve the computational efficiency of models that will deliver cost-effective investment plans for 5GDHC networks.

Joint Routing and Scheduling of Mobile Charging Infrastructure for V2V Energy Transfer

An adequate charging infrastructure advocates to ameliorate the range anxiety to propel the disparaged electric vehicle (EV) market. But, the high initial installation cost, requirement of suitable places and the anticipated immense load on the grid during peak times hinder to elongate the charging station network, especially in urban areas. Fortunately, the bidirectional energy transferring capability between vehicles (i.e., V2V) may act as an auxiliary solution to charge an EV at any place and at any time without leaning on a stationary charging infrastructure. In this work, we assume a market where charging providers each has a number of charging trucks equipped with a larger battery and a fast charger to charge a number of EVs at some particular parking lots. A provider intends to maximize the served number of EVs using its limited number of charging trucks, when an EV should be considered as served only if it would be fully charged during its declared charging window. All charging requests are assumed to be received by an agent which provisions a route and schedule for each charging truck and all trucks should return to the depot after serving EVs. We formulate this combinatorially hard problem as an integer linear program (ILP) to maximize the number of served EVs by determining the optimal trajectory of each truck. Owing to its complexity, we present a solution methodology by decomposing the problem using Dantzig-Wolfe decomposition approach; we divide the problem into one master problem and a set of pricing problems (one for each EV) and achieve the solution iteratively. Though the solution achieved from the decomposition might not be optimal, it is faster to be applicable in practice. We also compare the performance with two heuristic algorithms and report on the collected results.

Pre-positioning and real-time disaster response operations: Optimization with mobile phone location data

Uncertainty of disaster information severely hinders efficiency of disaster relief operations. This paper explores how mobile phone location data are applied to capture rapidly accurate disaster information and proposes a preparedness-response two-stage scenario-based stochastic programming model with mobile phone location data for integrated pre-positioning and real-time response operation optimization. The Dantzig-Wolfe decomposition approach based on Lagrange duality and branch-and-bound algorithm are adopted to solve this model. The applicability of the model is proved via a real-world case study on the Haiti earthquake. Numerical experiments are performed to offer important managerial implications and insights in disaster management, especially for large-scale disasters.

Column Generation Algorithms for a Single Machine Problem with Deteriorating Jobs and Deterioration Maintenance Activities

This article addresses the single machine scheduling problem in which the actual processing time of jobs change depending on an elapsed time between its start time and a released time or completion time of a recent maintenance activity. The introduced problem reflects production industry of chemical and metallurgical processes. The objective of this problem is to determine a sequence of jobs and the number and positions of maintenance activities on a single machine to minimize the makespan. We introduce a mixed integer linear programming (MILP) model for the problem. We then propose column generation (CG) algorithms by using Dantzig-Wolfe decomposition approach. The performances of the CG algorithms are evaluated with randomly generated examples.

Dantzig–Wolfe decomposition approach to the vehicle assignment problem with demand uncertainty in a hybrid hub-and-spoke network

In this article, we investigate the vehicle assignment problem with demand uncertainty in a hybrid hub-and-spoke network with a single hub. The problem is deciding both the transportation routes and the number and types of vehicles to be deployed to minimize the sum of costs to transport all quantities in a hybrid hub-and-spoke network which allows direct transportation between spokes. Daily changes in quantities are reflected with a finite number of scenarios. Regularly scheduled vehicles and temporarily scheduled vehicles are considered to meet the demand variation. We propose a Dantzig–Wolfe decomposition approach which yields a strong LP relaxation bound by introducing a set of feasible direct route patterns. We develop an algorithm which incorporates a column generation procedure at the root node and repeats iteratively a variable fixing and column generation procedure at the non-root nodes until an integral solution is found. Finally, we present computational results using the well-known CAB data sets and real-life data from the Korea Post. The results show that our algorithm can find near optimal solutions very efficiently.

A coalitional game approach for fractional cooperative caching in content-oriented networks

In content-oriented networks, popular contents are replicated at the intermediate nodes to enhance content delivery performance. Under cooperative caching, the caching nodes collaborate to leverage one another’s cache capability and to reduce the amount of traffic transferring inside the network. This study considers the cooperation among service providers (SPs). The transferable-payoff coalitional game model is applied for analysis. We investigate the stability of the grand coalition and show that the dual-based cost allocation is in the core. A linear program (LP) minimizing the network bandwidth-expense is used for the characteristic function of the game model. However, solving the LP is a challenge because of a large amount of contents in the network. The Dantzig–Wolfe decomposition approach is further applied to decompose the large-scale problem into many subproblems, which can be solved in parallel. The analysis provides not only a deeper insight into the cooperative cache among SPs but also content placement and distribution strategies as a solution to the LP.

Optimal allocation of aviation security screening devices

Recent advances in aviation security screening technologies have presented a new challenge in optimally allocating these devices across a set of airports. Following check-in at the ticket counter, self-service kiosk, or airline website application, each passenger is assigned to a security screening class through an automated passenger prescreening system based on their measured perceived risk level. The class to which a passenger is assigned can be used to determine how this passenger’s baggage will be screened by a set of security devices and procedures. In this paper, an explosive screening device allocation model is formulated as a nonlinear integer program to assign both the type of and number of devices to each class at each airport such that the total security is maximized, given a set of budget, resource, and passenger throughput constraints. A Dantzig-Wolfe decomposition approach is used to transform the nonlinear program into a binary integer program by redefining the binding constraint associated with the number of individual devices allocated across all airports. Computational results are provided for several randomly generated problems to demonstrate that the resulting binary integer program can be quickly solved to optimality.

Robust vehicle routing problem with deadlines and travel time/demand uncertainty

In this article, we investigate the vehicle routing problem with deadlines, whose goal is to satisfy the requirements of a given number of customers with minimum travel distances while respecting both of the deadlines of the customers and vehicle capacity. It is assumed that the travel time between any two customers and the demands of the customer are uncertain. Two types of uncertainty sets with adjustable parameters are considered for the possible realizations of travel time and demand. The robustness of a solution against the uncertain data can be achieved by making the solution feasible for any travel time and demand defined in the uncertainty sets. We propose a Dantzig-Wolfe decomposition approach, which enables the uncertainty of the data to be encapsulated in the column generation subproblem. A dynamic programming algorithm is proposed to solve the subproblem with data uncertainty. The results of computational experiments involving two well-known test problems show that the robustness of the solution can be greatly improved.

Solving multi-stage stochastic in-house production and outsourcing planning by two-level decomposition

In-house production and outsourcing are important strategic decisions for planning production and capacity in business organisations. Outsourcing to overseas suppliers is often associated with risk with respect to the quality of the products. Hence, we developed a multi-stage stochastic programming model that takes into account the uncertainty involved in the production of the quality of outsourced products in the face of stochastic demand. The goal is to find an optimal way to choose between in-house capacity expansion and buying from local suppliers with assured quality versus buying from overseas suppliers. Moreover, we propose three alternative algorithms for solving the problem. These three approaches are: a two-level column generation by using the analytic centre cutting plane method (ACCPM), a two-level Benders' decomposition by using the ACCPM and a two-level decomposition where the first level is solved by using the classical Dantzig-Wolfe decomposition approach and the second level is solved by using the ACCPM.

Knapsack problems with setups

Knapsack problems with setups find their application in many concrete industrial and financial problems. Moreover, they also arise as subproblems in a Dantzig-Wolfe decomposition approach to more complex combinatorial optimization problems, where they need to be solved repeatedly and therefore efficiently. Here, we consider the multiple-class integer knapsack problem with setups. Items are partitioned into classes whose use implies a setup cost and associated capacity consumption. Item weights are assumed to be a multiple of their class weight. The total weight of selected items and setups is bounded. The objective is to maximize the difference between the profits of selected items and the fixed costs incurred for setting-up classes. A special case is the bounded integer knapsack problem with setups where each class holds a single item and its continuous version where a fraction of an item can be selected while incurring a full setup. The paper shows the extent to which classical results for the knapsack problem can be generalized to these variants with setups. In particular, an extension of the branch-and-bound algorithm of Horowitz and Sahni is developed for problems with positive setup costs. Our direct approach is compared experimentally with the approach proposed in the literature consisting in converting the problem into a multiple choice knapsack with pseudo-polynomial size.

Long-Term Security-Constrained Unit Commitment: Hybrid Dantzig–Wolfe Decomposition and Subgradient Approach

The solution of a long-term security-constrained unit commitment (SCUC) problem, which often spans several months to a year, may consider multiple long-term fuel and emission constraints in addition to operating constraints embedded in short-term SCUC. The size and the complexity of long-term SCUC are often beyond reasonable computing time and resources. Hence, Lagrangian relaxation is applied in this paper to manage coupling constraints over the entire period. Based on dual relaxation, the large-scale optimization problem is decomposed into many tractable short-term SCUC subproblems without long-term fuel and emission constraints. The resource penalty prices are linking signals for the coordination of subproblems. The short-term SCUC may be solved by any numerical optimization methods, including mixed integer programming and Lagrangian relaxation. A hybrid subgradient and Dantzig-Wolfe decomposition approach is presented for managing Lagrangian multipliers in the large-scale dual optimization of long-term SCUC problem. The proposed hybrid approach is a tradeoff between calculation speed and accuracy of the long-term SCUC solution. A modified IEEE 118-bus system is analyzed to exhibit the effectiveness of the proposed approach.

Solution Algorithm for Combined Interregional Commodity Flow and Transportation Network Model with Link Capacity Constraints

A solution algorithm for the multicommodity combined freight distribution and assignment problem that computes the system optimum link flow pattern is introduced here. The distribution component of the model is based on an input-output materials balance formulation that satisfies each region’s intermediate and final demands for each commodity. An entropy term is included to capture cross-hauling and dispersion effects. The network links are assumed to be capacitated. This problem is formulated as a nonlinear program, with the nonlinear entropy term appearing in the objective function. This problem tends to be intractable even for small networks, so a Dantzig–Wolfe decomposition approach is introduced to make the problem computationally feasible. Wilson’s balancing technique is used to solve the nonlinear distribution subproblem. Test results demonstrate that realistic networks can be solved in a reasonable amount of time. The model is used to obtain insights into the trade-offs between dispersion coefficients and transportation costs on both capacitated and uncapacitated networks.

A Lagrangean relax-and-cut approach for the two-stage capacitated facility location problem

The two-stage capacitated facility location problem is to find the optimal locations of depots to serve customers with a given demand, the optimal assignment of customers to depots and the optimal product flow from plants to depots. In order to solve this problem, a Lagrangean heuristic is proposed which is based on the relaxation of the capacity constraints. The resulting Lagrangean subproblem is an uncapacitated facility location problem with an aggregate capacity constraint and can be solved efficiently by branch-and-bound methods. The best bound available from this relaxation is computed by means of a “weighted” Dantzig–Wolfe decomposition approach. Feasible solutions are constructed from those of the Lagrangean subproblems by applying simple reassignment procedures. Furthermore, the relaxation is strengthened by adding and dualizing valid inequalities which are violated by the fractional primal solution of the dual master program. Computational results and comparisons with some other bounds and heuristics for this problem are presented.

Decomposition of a Combined Inventory and Time Constrained Ship Routing Problem

In contrast to vehicle routing problems, little work has been done in ship routing and scheduling, although large benefits may be expected from improving this scheduling process. We will present a real ship planning problem, which is a combined inventory management problem and a routing problem with time windows. A fleet of ships transports a single product (ammonia) between production and consumption harbors. The quantities loaded and discharged are determined by the production rates of the harbors, possible stock levels, and the actual ship visiting the harbor. We describe the real problem and the underlying mathematical model. To decompose this model, we discuss some model adjustments. Then, the problem can be solved by a Dantzig–Wolfe decomposition approach including both ship routing subproblems and inventory management subproblems. The overall problem is solved by branch-and-bound. Our computational results indicate that the proposed method works for the real planning problem.

2-Path Cuts for the Vehicle Routing Problem with Time Windows

This paper introduces a strong valid inequality, the 2-path cut, to produce better lower bounds for the vehicle routing problem with time windows. It also develops an effective separation algorithm to find such inequalities. We next incorporate them as needed in the master problem of a Dantzig–Wolfe decomposition approach. In this enhanced optimization algorithm, the coupling constraints require that each customer be serviced. The subproblem is a shortest path problem with time window and capacity constraints. We apply branch and bound to obtain integer solutions. We first branch on the number of vehicles if this is fractional, and then on the flow variables. The algorithm has been implemented and tested on problems of up to 100 customers from the Solomon datasets. It has succeeded in solving to optimality several previously unsolved problems and a new 150-customer problem. In addition, the algorithm proved faster than algorithms previously considered in the literature. These computational results indicate the effectiveness of the valid inequalities we have developed.

Cost–benefit analysis of generation additions in system planning

Selection of the optimum generation reserve depends on both economic and reliability considerations. A new approach to select the optimum generator additions in a bulk power system by minimising the total costs is presented in the paper. This includes investment, operational costs and unreliability costs in the form of outage costs. The proposed approach uses the Dantzig–Wolfe decomposition approach to minimise the total costs.

Daily Aircraft Routing and Scheduling

In this paper we consider the daily aircraft routing and scheduling problem (DARSP). It consists of determining daily schedules which maximize the anticipated profits derived from the aircraft of a heterogeneous fleet. This fleet must cover a set of operational flight legs with known departure time windows, durations and profits according to the aircraft type. We present two models for this problem: a Set Partitioning type formulation and a time constrained multicommodity network flow formulation. We describe the network structure of the subproblem when a column generation technique is applied to solve the linear relaxation of the first model and when a Dantzig-Wolfe decomposition approach is used to solve the linear relaxation of the second model. The linear relaxation of the first model provides upper bounds. Integer solutions to the overall problem are derived through branch-and-bound. By exploiting the equivalence between the two formulations, we propose various optimal branching strategies compatible with the column generation technique. Finally we report computational results obtained on data provided by two different airlines. These results show that significant profit improvement can be generated by solving the DARSP using our approach and that this can be obtained in a reasonable amount of CPU time.