Browsing by Subject "Valid inequalities"

Now showing 1 - 18 of 18

Open Access
A branch and cut algorithm for the inventory routing problem
(2019-07) Mahmutoğulları, Özlem
The inventory routing problem arises in vendor managed systems where products are distributed from a supplier to a set of retailers by a homogeneous eet of capacitated vehicles. The routes of the vehicles and the quantities of products sent to each retailer in each time period are determined in such a way that no stockouts occur and total costs arising from inventory holding and transportation are minimized. Different inventory replenishment policies can be used while managing the inventories at retailers. We consider the problem with the maximum level inventory replenishment policy. We present a mixed integer linear programming model and derive valid inequalities using several structured relaxations. We relate our valid inequalities to those in the previous studies. We also propose new valid inequalities, implement a branch and cut algorithm and present computational results on benchmark instances from the literature as well as new randomly generated instances.
Open Access
A branch-and-cut algorithm for the hub location and routing problem
(Elsevier, 2014) Rodríguez-Martín, I.; Salazar-González, J-J.; Yaman, H.
We study the hub location and routing problem where we decide on the location of hubs, the allocation of nodes to hubs, and the routing among the nodes allocated to the same hubs, with the aim of minimizing the total transportation cost. Each hub has one vehicle that visits all the nodes assigned to it on a cycle. We propose a mixed integer programming formulation for this problem and strengthen it with valid inequalities. We devise separation routines for these inequalities and develop a branch-and-cut algorithm which is tested on CAB and AP instances from the literature. The results show that the formulation is strong and the branch-and-cut algorithm is able to solve instances with up to 50 nodes.
Open Access
A branch-and-cut algorithm for two-level survivable network design problems
(Elsevier, 2016) Rodríguez-Martín, I.; Salazar-González, J-J.; Yaman, H.
This paper approaches the problem of designing a two-level network protected against single-edge failures. The problem simultaneously decides on the partition of the set of nodes into terminals and hubs, the connection of the hubs through a backbone network (first network level), and the assignment of terminals to hubs and their connection through access networks (second network level). We consider two survivable structures in both network levels. One structure is a two-edge connected network, and the other structure is a ring. There is a limit on the number of nodes in each access network, and there are fixed costs associated with the hubs and the access and backbone links. The aim of the problem is to minimize the total cost. We give integer programming formulations and valid inequalities for the different versions of the problem, solve them using a branch-and-cut algorithm, and discuss computational results. Some of the new inequalities can be used also to solve other problems in the literature, like the plant cycle location problem and the hub location routing problem.
Open Access
Comparison of the formulations for a hub-and-spoke network design problem under congestion
(Elsevier, 2016) Kian, Ramer; Kargar, Kamyar
In this paper, we study the hub location problem with a power-law congestion cost and propose an exact solution approach. We formulate this problem in a conic quadratic form and use a strengthening method which rests on valid inequalities of perspective cuts in mixed integer nonlinear programming. In a numerical study, we compare two well known types of mathematical modeling in the hub-location problems which are solved with different branch and cut strategies. The strength and weakness of the formulations are summarized based on an extensive numerical study over the CAB data set. © 2016 Elsevier Ltd
Open Access
Formulations and valid inequalities for the heterogeneous vehicle routing problem
(Springer, 2006) Yaman, H.
We consider the vehicle routing problem where one can choose among vehicles with different costs and capacities to serve the trips. We develop six different formulations: the first four based on Miller-Tucker-Zemlin constraints and the last two based on flows. We compare the linear programming bounds of these formulations. We derive valid inequalities and lift some of the constraints to improve the lower bounds. We generalize and strengthen subtour elimination and generalized large multistar inequalities.
Open Access
Hierarchical multimodal hub location problem with time-definite deliveries
(Elsevier, 2012) Alumur, S. A.; Yaman, H.; Kara, B. Y.
Hierarchical multimodal hub location problem is a cost-minimizing hub covering problem where two types of hubs and hub links, accounting for ground and air transportation, are to be established, while ensuring time-definite deliveries. We propose a mixed-integer programming formulation and perform a comprehensive sensitivity analysis on the Turkish network. We show that the locations of airport hubs are less sensitive to the cost parameters compared to the locations of ground hubs and it is possible to improve the service quality at not much additional cost in the resulting multimodal networks. Our methodology provides the means for a detailed trade-off analysis.
Open Access
The integer knapsack cover polyhedron
(Society for Industrial and Applied Mathematics, 2007) Yaman, H.
We study the integer knapsack cover polyhedron which is the convex hull of the set of vectors x ∈ ℤ+ n that satisfy C T x ≥ b, with C ∈ ℤ++ n and 6 ∈ ℤ++. We present some general results about the nontrivial facet-defining inequalities. Then we derive specific families of valid inequalities, namely, rounding, residual capacity, and lifted rounding inequalities, and identify cases where they define facets. We also study some known families of valid inequalities called 2-partition inequalities and improve them using sequence-independent lifting.
Open Access
An intermodal multicommodity routing problem with scheduled services
(2012) Ayar, B.; Yaman, H.
We study a multicommodity routing problem faced by an intermodal service operator that uses ground and maritime transportation. Given a planning horizon, a set of commodities to be picked up at their release times and to be delivered not later than their duedates, the problem is to decide on routes for these commodities using trucks and scheduled and capacitated maritime services at minimum cost of transportation and stocking at the seaports. Two mixed integer programming formulations and valid inequalities are proposed for this problem. The results of a computational study to evaluate the strength of the linear programming relaxations and the solution times are reported.
Open Access
k-node-disjoint hop-constrained survivable networks: polyhedral analysis and branch and cut
(Springer-Verlag France, 2018) Diarrassouba, I.; Mahjoub, M.; Mahjoub, A. R.; Yaman, Hande
Given a graph with weights on the edges, a set of origin and destination pairs of nodes, and two integers L ≥ 2 and k ≥ 2, the k-node-disjoint hop-constrained network design problem is to find a minimum weight subgraph of G such that between every origin and destination there exist at least k node-disjoint paths of length at most L. In this paper, we consider this problem from a polyhedral point of view. We propose an integer linear programming formulation for the problem for L ∈{2,3} and arbitrary k, and investigate the associated polytope. We introduce new valid inequalities for the problem for L ∈{2,3,4}, and give necessary and sufficient conditions for these inequalities to be facet defining. We also devise separation algorithms for these inequalities. Using these results, we propose a branch-and-cut algorithm for solving the problem for both L = 3 and L = 4 along with some computational results.
Open Access
The latest arrival hub location problem for cargo delivery systems with stopovers
(Elsevier, 2007) Yaman, H.; Kara, B. Y.; Tansel, B. C.
In this paper, we concentrate on the service structure of ground-transportation based cargo delivery companies. The transient times that arise from nonsimultaneous arrivals at hubs (typically spent for unloading, loading, and sorting operations) can constitute a significant portion of the total delivery time for cargo delivery systems. The latest arrival hub location problem is a new minimax model that focuses on the minimization of the arrival time of the last item to arrive, taking into account journey times as well as the transient times at hubs. We first focus on a typical cargo delivery firm operating in Turkey and observe that stopovers are essential components of a ground-based cargo delivery system. The existing formulations of the hub location problem in the literature do not allow stopovers since they assume direct connections between demand centers and hubs. In this paper, we propose a generic mathematical model, which allows stopovers for the latest arrival hub location problem. We improve the model using valid inequalities and lifting. We present computational results using data from the US and Turkey.
Open Access
Lot sizing with perishable items
(2019-07) Arslan, Nazlıcan
We address the uncapacitated lot sizing problem for a perishable item that has a deterministic and fixed lifetime. In the first part of the study, we assume that the demand is also deterministic. We conduct a polyhedral analysis and derive valid inequalities to strengthen the LP relaxation. We develop a separation algorithm for the valid inequalities and propose a branch and cut algorithm to solve the problem. We conduct a computational study to test the effiectiveness of the valid inequalities. In the second part, we study the multistage stochastic version of the problem where the demand is uncertain. We use the valid inequalities we found for the deterministic problem to strengthen the LP relaxation of the stochastic problem and test their effiectiveness. As the size of the stochastic model grows exponentially in the number of periods, we also implement a decomposition method based on scenario grouping to obtain lower and upper bounds.
Open Access
Manufacturer's mixed pallet design problem
(Elsevier, 2008) Yaman, H.; Şen, A.
We study a problem faced by a major beverage producer. The company produces and distributes several brands to various customers from its regional distributors. For some of these brands, most customers do not have enough demand to justify full pallet shipments. Therefore, the company decided to design a number of mixed or "rainbow" pallets so that its customers can order these unpopular brands without deviating too much from what they initially need. We formally state the company's problem as determining the contents of a pre-determined number of mixed pallets so as to minimize the total inventory holding and backlogging costs of its customers over a finite horizon. We first show that the problem is NP-hard. We then formulate the problem as a mixed integer linear program, and incorporate valid inequalities to strengthen the formulation. Finally, we use company data to conduct a computational study to investigate the efficiency of the formulation and the impact of mixed pallets on customers' total costs.
Open Access
New exact solution approaches for the split delivery vehicle routing problem
(Springer Verlag, 2018) Özbaygın, Gizem; Karasan, Oya Ekin; Yaman, Hande
In this study, we propose exact solution methods for the split delivery vehicle routing problem (SDVRP). We first give a new vehicle-indexed flow formulation for the problem and then a relaxation obtained by aggregating the vehicle-indexed variables over all vehicles. This relaxation may have optimal solutions where several vehicles exchange loads at some customers. We cut off such solutions, in a nontraditional way, either by extending the formulation locally with vehicle-indexed variables or by node splitting. We compare these approaches using instances from the literature and new randomly generated instances. Additionally, we introduce two new extensions of the SDVRP by restricting the number of splits and by relaxing the depot return requirement and modify our algorithms to handle these extensions.
Open Access
The periodic vehicle routing problem with driver consistency
(Elsevier, 2019) Rodríguez-Martín, I.; Salazar-González, J. -J.; Yaman, Hande
The Periodic Vehicle Routing Problem is a generalization of the classical capacitated vehicle routing problem in which routes are determined for a planning horizon of several days. Each customer has an associated set of allowable visit schedules, and the objective of the problem is to design a set of minimum cost routes that give service to all the customers respecting their visit requirements. In this paper we study a new variant of this problem in which we impose that each customer should be served by the same vehicle/driver at all visits. We call this problem the Periodic Vehicle Routing Problem with Driver Consistency. We present an integer linear programming formulation for the problem and derive several families of valid inequalities. We solve it using an exact branch-and-cut algorithm, and show computational results on a wide range of randomly generated instances.
Open Access
The ring/κ-rings network design problem: model and branch-and-cut algorithm
(John Wiley & Sons, 2016) Rodríguez-Martín, I.; Salazar-González, J-J.; Yaman, H.
This article considers the problem of designing a two-level network where the upper level consists of a backbone ring network connecting the so-called hub nodes, and the lower level is formed by access ring networks that connect the non-hub nodes to the hub nodes. There is a fixed cost for each type of link, and a facility opening cost associated to each hub. The number of nodes in each access ring is bounded, and the number of access rings connected to a hub is limited to κ, thus resulting in a ring/κ-rings topology. The aim is to decide the hubs to open and to design the backbone and access rings to minimize the installation cost. We propose a mathematical model, give valid inequalities, and describe a branch-and-cut algorithm to solve the problem. Computational results show the algorithm is able to find optimal solutions on instances involving up to 40 nodes within a reasonable time.
Open Access
Time constrained maximal covering salesman problem with weighted demands and partial coverage
(Elsevier Ltd, 2016) Ozbaygin, G.; Yaman, H.; Karasan, O. E.
In a routing framework, it may not be viable to visit every single customer separately due to resource limitations or efficiency concerns. In such cases, utilizing the notion of coverage; i.e., satisfying the demand of multiple customers by visiting a single customer location, may be advantageous. With this motivation, we study the time constrained maximal covering salesman problem (TCMCSP) in which the aim is to find a tour visiting a subset of customers so that the amount of demand covered within a limited time is maximized. We provide flow and cut formulations and derive valid inequalities. Since the connectivity constraints and the proposed valid inequalities are exponential in the size of the problem, we devise different branch-and-cut schemes. Computational experiments performed on a set of problem instances demonstrate the effectiveness of the proposed valid inequalities in terms of strengthening the linear relaxation bounds as well as speeding up the solution procedure. Moreover, the results indicate the superiority of using a branch-and-cut methodology over a flow-based formulation. Finally, we discuss the relation between the problem parameters and the structure of optimal solutions based on the results of our experiments. © 2016 Elsevier Ltd
Open Access
Toz deterjan için üretim planlama ve çizelgeleme sistemi tasarımı
(TMMOB Makina Mühendisleri Odası, 2009) Sepin, Tardu Selim; Yatkın, Mehmet Diyar; Eralp, Merve Nazlı; Memişoğlu, Gökhan; Taner, Mehmet Rüştü; Özcan, Mehmet; Akdemir, Deniz
Unilever Gebze Fabrikasının toz deterjan üretimi planlama sürecinde çizelgeleme işlemi için karar destek sisteminin eksikliği, ürün değişikliklerinin neden olduğu kurulum sayısının ve fırsat maliyetlerinin artmasına sebep olmaktadır. Projenin amacı, sürekli imalat yapısına sahip olan toz deterjan üretimine hızlı ve tutarlı sonuçlar veren, toplam kurulum sayı ve süresini en aza indirecek bir çizelgeleme sistemi tasarlanmasıdır. Problem dört aşamada incelenmiş; sırasıyla bütünleşik, bölünmüş, kısıtlı bölünmüş matematiksel modeller ve sezgisel metot ile çözülmüştür. Sonuçların karşılaştırılmasıyla sezgisel metodun kısa zamanda tutarlı çözümler verdiği görülmüş ve oluşturulan arayüzle sisteme entegre edilmiştir.
Open Access
The vendor location problem
(Elsevier, 2011) Çınar, Y.; Yaman, H.
The vendor location problem is the problem of locating a given number of vendors and determining the number of vehicles and the service zones necessary for each vendor to achieve at least a given profit. We consider two versions of the problem with different objectives: maximizing the total profit and maximizing the demand covered. The demand and profit generated by a demand point are functions of the distance to the vendor. We propose integer programming models for both versions of the vendor location problem. We then prove that both are strongly NP-hard and we derive several families of valid inequalities to strengthen our formulations. We report the outcomes of a computational study where we investigate the effect of valid inequalities in reducing the duality gaps and the solution times for the vendor location problem.