Eskişehir İktisadi ve Ticari İlimler Akademisi Dergisi, Dec 6, 1973
Gerek bireysel gerek toplumsal ya~antImlzda, bireyler veya gruplar herhangi bir amaca ula~abilmek... more Gerek bireysel gerek toplumsal ya~antImlzda, bireyler veya gruplar herhangi bir amaca ula~abilmek i<;:in varacaklan kararda bin;ok eylem (faaliyet) alternatifleri ile ka:r~l kar~lya gelmektedirler. iyi bir karar, eylem alternatiflerinden amaca en uygun olam, ama<;: i<;:in en learh olam, amaca gore en <;:lkarh veya ge<;:erli olam se<;:ebilmektir 0 Ku~kusuz, bu eylem alternatifleri kiimesinin elemanlannm sa-yIlabilme veya sayIlamamalan, kantitatif olup olmamalan, eylem alternatifleri klimesi ile ama<;: arasmdaki ili~kinin, ba~ka bir deyi~le, bu kiimenin herbir e1emamnm amaca etkilerinin ol<;:iiliip ol<;:iilememesi, ama<;: i<,:in konu olan eylem alternatiflerinin tiimiiniin i<;:erilmesi veya i<;:erilmemesi son karan etkileyen degi~kenlerdiro A~~lk<;:a da gorlilecegi gibi, ama<;: bir dizi degi~kenin etkisi aItmdadlr. byle ki, bu degi~kenlerin herbir kan~lml alternatif eylemleri olu~tururo Degi~kenlerin karakterleri ve aralarmdaki ili~kilerin bi<;:'imi, en iyi, en ge<;:erli, en uygun veya kIvamh terimleriyle belirtilen Optimum eylem i<;:in en etkin bir eleman olmaktadlr.
Central European Journal of Operations Research, 2017
Coaches of sports clubs aim to form the team that optimally determines the roles of positions bef... more Coaches of sports clubs aim to form the team that optimally determines the roles of positions before the match. These types of decisions are referred to as the team formation problem, and they are critical for the sports industry in the financial sense. Finding the optimal solution to the team formation problem is more difficult without the use of systematical approaches, as the number of players and their past performance records have increased substantially in recent years. In this paper, we discuss previous studies on the team formation problems of sports clubs and outline the deficiencies of their results in real-life decision processes. Then, we propose two new formulations that address coaches' preferences in the decision-making process. A real-life application of the proposed models is displayed for a volleyball team that participates in the first division of the Turkish Volleyball League.
Order acceptance and scheduling problem consist of simultaneously deciding which orders to be sel... more Order acceptance and scheduling problem consist of simultaneously deciding which orders to be selected and how to schedule these selected orders. An extension of the sequence-dependent setup times and release dates was introduced in 2010 and a mathematical formulation was presented. Since then, a few mathematical formulations have appeared in the literature by addressing this problem on the basis of sequence-dependent setup times. However, some of the mathematical formulations are nonlinear or lack usability. Therefore, the model presented in 2010 is still being considered in recent studies. In this paper, we investigated the case in which there are sequence-dependent setup times with no release dates for all orders. We developed a new mathematical formulation with O(n) binary variables and O(n) constraints. In order to see the performance of our formulation, we conducted a computational analysis with CPLEX 12.4 by solving benchmark instances available in the literature. To manage t...
In this study, a new model has been developed for the order selection problem for a bakery firm l... more In this study, a new model has been developed for the order selection problem for a bakery firm located in Turkey. We consider a combined order selection and process planning problem where a make-to-order bakery goods producer firm has to determine a set of orders to process so as to maximize the total profit. The developed model combines setup costs, sales price, lot size, demand and other important daily data of the products. After collecting the related data, fuzzy TOPSIS method is used to obtain the rankings of the orders (bread types). Then, the ranking scores are incorporated in the knapsack problem to determine the lot size and which orders to select. A computer application is also provided in the paper. With the help of MS Excel Visual Basic program, the computer application updates data daily, ranks the products, determines the lot size, and helps the decision maker with the selection of orders. The computational results show that the proposed method presented better results than the current method applied in the firm. Comparing the two methods in terms of cost reduction, the proposed method gives better results than the current method with 27%. Furthermore, the proposed computer program does not need extra special software or a commercial computer program, which is favorable for small-size enterprises.
The Frobenius number of a set of relatively prime positive integers α1,α2,…,αn such that α1< α... more The Frobenius number of a set of relatively prime positive integers α1,α2,…,αn such that α1< α2< …< αn, is the largest integer that can not be written as a nonnegative integer linear combination of the given set. Finding the Frobenius number is known as the Frobenius problem, which is also named as the coin exchange problem or the postage stamp problem. This problem is closely related with the equality constrained integer knapsack problem. It is known that this problem is NP-hard. Extensive research has been conducted for finding the Frobenius number of a given set of positive integers. An exact formula exists for the case n=2 and various formulas have been derived for all special cases of n = 3. Many algorithms have been proposed for n≥4. As far as we are aware, there does not exist any integer programming approach for this problem which is the main motivation of this paper. We present four integer linear programming formulations about the Frobenius number of a given set o...
Proportional symbol maps are a cartographic tool that employs scaled symbols to represent data as... more Proportional symbol maps are a cartographic tool that employs scaled symbols to represent data associated with specific locations. The symbols we consider are opaque disks, which may be partially covered by other overlapping disks. We address the problem of creating a suitable drawing of the disks that maximizes one of two quality metrics: the total and the minimum visible length of disk boundaries. We study three variants of this problem, two of which are known to be NP-hard and another whose complexity is open. We propose novel integer programming formulations for each problem variant and test them on real-world instances with a branch-and-cut algorithm. When compared with state-of-the-art models from the literature, our models significantly reduce computation times for most instances.
Gazi Üniversitesi Mühendislik-Mimarlık Fakültesi Dergisi, 2015
Lojistik yönetiminde en önemli operasyonel kararlardan birisi müşterilere hizmet verecek araç rot... more Lojistik yönetiminde en önemli operasyonel kararlardan birisi müşterilere hizmet verecek araç rotalarının belirlenmesidir. Araç Rotalama Problemi (ARP), bir depodan müşterilerin dağıtım (toplama) taleplerini karşılayacak en uygun rotaların belirlenmesi olarak tanımlanabilir. Gerçek hayat lojistik uygulamalarında, filoda bulunan araçlar farklı özelliklerde olabilirler. Ayrıca müşterilerden/taşınanlardan kaynaklı gereklilikler de farklı özellikte araç kullanımı zorunluluğunu ortaya çıkarabilir. Bunun yanısıra firmalar, mamul, yarı mamul ve hammaddelerin tersine akışının yönetimini de hem ekonomik getirisi hem de yasal ve çevresel yükümlülüklerinden dolayı daha fazla önemsemektedirler. Bu makalede, heterojen araç filosunun bulunduğu ve müşterilerin dağıtım ve toplama taleplerinin eşzamanlı gerçekleştiği durumların birlikte dikkate alındığı bir ARP türü üzerinde çalışılmıştır. Bu problem Heterojen Eşzamanlı Topla-Dağıt Araç Rotalama Problemi (HETD-ARP) olarak adlandırılmıştır. HETD-ARP, toplam maliyeti enküçükleyen araç rotalarının ve herbir rotada kullanılan araç tipinin belirlenmesi olarak tanımlanabilir. Problem için polinom sayıda kısıta sahip akış tabanlı bir matematiksel model önerilmiştir. HETD-ARP, NP-zor problemler sınıfında olduğundan dolayı makul sürelerde orta boyutlu problemlere bile en iyi çözümü bulmak zordur. Bu nedenle bu makalede orta ve büyük boyutlu HETD-ARP'nin çözümü için basit bir kurucu sezgisel algoritma önerilmiştir. Bu algoritma, kaynaklarda ARP için önerilen Clarke-Wright Tasarruf (CWT) algoritmasının HETD-ARP için uyarlanmış halidir. Önerilen matematiksel modelin ve sezgisel algoritmanın etkinliği test problemleri üzerinde incelenmiştir.
Ba Baş şkent kent Ü Üniversitesi niversitesi End Endü üstri M stri Mü ühendisli hendisliğ ği B i ... more Ba Baş şkent kent Ü Üniversitesi niversitesi End Endü üstri M stri Mü ühendisli hendisliğ ği B i Bö öl lü üm mü ü
Gazi Üniversitesi Mühendislik-Mimarlık Fakültesi Dergisi, 2021
Homogeneous and heterogeneous multiple traveler minimum latency problem with time windows New... more Homogeneous and heterogeneous multiple traveler minimum latency problem with time windows New mathematical models with polynomial number of decision variables and constraints Analysis of mathematical models in terms of CPU times and linear programming relaxation values Purpose: The aim of this study is to develop new mathematical formulations for homogeneous and heterogeneous multiple traveler minimum latency problem with time windows. Theory and Methods: Based on the mixed integer linear programming, mathematical models with polynomial number of decision variables and constraints are developed. Benchmark instances from the literature are solved with existing formulations and proposed new formulations by using CPLEX 12.5.0.1. CPU times and percentage deviation from linear programming relaxation values are considered as performance criteria. Results: We solved 125 problems with varying number of nodes and time windows. In all the problem solved proposed formulations are better than the existing formulations in terms of both of the performance criteria. Conclusion: The proposed formulations for homogeneous and heterogeneous multiple traveler minimum latency problem with time windows are superior than the existing formulations and able to solve the problems up to 100 nodes with narrow time windows. Proposed formulations may be used to solve small and moderate real-life problems very easily. They may also be used for testing the performance of the heuristics constructed for kLPTW.
The Generalized Vehicle Routing Problem (GVRP) is an extension of the Vehicle Routing Problem (VR... more The Generalized Vehicle Routing Problem (GVRP) is an extension of the Vehicle Routing Problem (VRP) defined on a graph in which the nodes (customers, vertices) are grouped into a given number of mutually exclusive and exhaustive clusters (nodesets). In this paper, an integer linear programming formulation of the GVRP with O(n 2) binary variables and O(n 2) constraints is presented. It is shown that, under specific circumstances, the proposed model reduces to the well-known routing problems. The computational performance of the models solved using a commercial code on test problems are also presented.
The Generalized Vehicle Routing Problem (GVRP) is an extension of the Vehicle Routing Problem (VR... more The Generalized Vehicle Routing Problem (GVRP) is an extension of the Vehicle Routing Problem (VRP) defined on a graph in which the nodes (customers, vertices) are grouped into a given number of mutually exclusive and exhaustive clusters (nodesets). In this paper, an integer linear programming formulation of the GVRP with O(n 2 ) binary variables and O(n 2 ) constraints is presented. It is shown that, under specific circumstances, the proposed model reduces to the well-known routing problems. The computational performance of the models solved using a commercial code on test problems are also presented.
Corrected Miller–Tucker–Zemlin type subtour elimination constraints for the capacitated vehicle r... more Corrected Miller–Tucker–Zemlin type subtour elimination constraints for the capacitated vehicle routing problem are presented. 2003 Elsevier B.V. All rights reserved.
The Generalized Traveling Salesman Problem (GTSP) is an extension of the well known Traveling Sal... more The Generalized Traveling Salesman Problem (GTSP) is an extension of the well known Traveling Salesman Problem (TSP). The GTSP is defined on a graph in which the nodes (customers or vertices) are grouped into a given number of clusters (node sets). Solution procedures for the GTSP are generally focused on transforming the problem to the TSP and applying the exact or heuristic solution methods developed for the TSP. There exist a few integer programming formulations for the GTSP some of which are exponential size with respect to number of the nodes. In this paper, we propose two new formulations for the GTSP with polynomial size with respect to number of the nodes. For preliminary computational analysis, GTSP instances from TSPLIB are solved by proposed formulations and also by the previously existing formulations in the literature. Performances of the formulations in terms of linear programming relaxations and CPU times are analyzed. We observe that, performances of the proposed for...
One of the exact solution methods for the traveling salesman problem (TSP) is the mixed0 integer ... more One of the exact solution methods for the traveling salesman problem (TSP) is the mixed0 integer formulation, based on the subtour elimination constraints. Dantzig, Fulkerson and Johnson have first proposed subtour elimination constraints in 1959. Later on, Miller, Tucker and Zemlin proposed an alternative formulation by introducing new variables. Their approach was only based on experiments, ie developing several models and selecting the most e cient one in terms of generality, number of variables and number of constraints. ...
The Home Health Care routing problem (HHCRP) is an extension of the vehicle routing problem (VRP)... more The Home Health Care routing problem (HHCRP) is an extension of the vehicle routing problem (VRP) considering scheduling and routing simultaneously to perform home health care (HHC) services. These services may be examining the patients, preparing a diet plan, applying physical therapy, and picking up the blood samples. In this paper, HHCRP has been addressed for a public hospital in Ankara, Turkey, that provides HHC services with limited resources. There are teams that consist of a doctor, a nurse, a dietitian, a physiotherapist and a vehicle driver. The multiple tours of those teams are searched by considering time windows to meet patient demands on time. This problem is defined as the multi-trip home health care routing problem with time-windows (MT-HHCRP-TW). In this study, a mixed integer linear programming model to minimize the total travelling time is proposed. Test problems that are generated based on real data for computational analysis and results are discussed. Main contr...
In classical scheduling problem, it is assumed that all orders must be processed. In the order ac... more In classical scheduling problem, it is assumed that all orders must be processed. In the order acceptance and scheduling (OAS) problem, some orders are rejected due to limited capacity. In make-to-order production environment, in which the OAS problem occurs, accepting all orders may cause overloads, delay in deliveries and unsatisfied customers. Oguz et al. (2010) introduced the OAS problem with sequence-dependent setup times and release dates. In this paper, we propose a new mixed integer programming formulation with O(n2) decision variables and O(n2) constraints for the same problem. We conduct a computational analysis comparing the performance of our formulation with Oguz et al. (2010) formulation. We use the benchmark instances, which are available in the literature. We observe that our formulation can solve all the instances up to 50 orders in a reasonable time, while Oguz et al. (2010) formulation can solve only the instances with 10 orders in the same time limit.
Abstract The Traveling Repairman Problem (TRP) is one of the most important variants of the Trave... more Abstract The Traveling Repairman Problem (TRP) is one of the most important variants of the Traveling Salesman Problem (TSP). The objective function of TRP is to find a Hamiltonian path or tour starting from the origin while minimizing the total latency (waiting or delay time) for all customers. The latency of a customer is defined as the time passed from the beginning of a tour (or path) until a customer’s service is completed. TRP with time windows (TRPTW) is the case where the earliest and latest times for visiting each customer are restricted by prescribed time windows. The literature on TRPTW is scarce. We only found one formulation for TRPTW and one formulation for its variant. In this paper, we propose four new mathematical models for TRPTW with O(n2) binary variables and O(n2) constraints. We computationally analyze the performance of existing and new formulations by solving symmetric and asymmetric benchmark instances with CPLEX 12.5.0.1 and compare the results in terms of CPU times and optimality gap. We observed that our two formulations were extremely faster than existing formulations, and they could optimally solve symmetric instances up to 150 nodes and asymmetric instances up to 131 nodes within seconds.
Eskişehir İktisadi ve Ticari İlimler Akademisi Dergisi, Dec 6, 1973
Gerek bireysel gerek toplumsal ya~antImlzda, bireyler veya gruplar herhangi bir amaca ula~abilmek... more Gerek bireysel gerek toplumsal ya~antImlzda, bireyler veya gruplar herhangi bir amaca ula~abilmek i<;:in varacaklan kararda bin;ok eylem (faaliyet) alternatifleri ile ka:r~l kar~lya gelmektedirler. iyi bir karar, eylem alternatiflerinden amaca en uygun olam, ama<;: i<;:in en learh olam, amaca gore en <;:lkarh veya ge<;:erli olam se<;:ebilmektir 0 Ku~kusuz, bu eylem alternatifleri kiimesinin elemanlannm sa-yIlabilme veya sayIlamamalan, kantitatif olup olmamalan, eylem alternatifleri klimesi ile ama<;: arasmdaki ili~kinin, ba~ka bir deyi~le, bu kiimenin herbir e1emamnm amaca etkilerinin ol<;:iiliip ol<;:iilememesi, ama<;: i<,:in konu olan eylem alternatiflerinin tiimiiniin i<;:erilmesi veya i<;:erilmemesi son karan etkileyen degi~kenlerdiro A~~lk<;:a da gorlilecegi gibi, ama<;: bir dizi degi~kenin etkisi aItmdadlr. byle ki, bu degi~kenlerin herbir kan~lml alternatif eylemleri olu~tururo Degi~kenlerin karakterleri ve aralarmdaki ili~kilerin bi<;:'imi, en iyi, en ge<;:erli, en uygun veya kIvamh terimleriyle belirtilen Optimum eylem i<;:in en etkin bir eleman olmaktadlr.
Central European Journal of Operations Research, 2017
Coaches of sports clubs aim to form the team that optimally determines the roles of positions bef... more Coaches of sports clubs aim to form the team that optimally determines the roles of positions before the match. These types of decisions are referred to as the team formation problem, and they are critical for the sports industry in the financial sense. Finding the optimal solution to the team formation problem is more difficult without the use of systematical approaches, as the number of players and their past performance records have increased substantially in recent years. In this paper, we discuss previous studies on the team formation problems of sports clubs and outline the deficiencies of their results in real-life decision processes. Then, we propose two new formulations that address coaches' preferences in the decision-making process. A real-life application of the proposed models is displayed for a volleyball team that participates in the first division of the Turkish Volleyball League.
Order acceptance and scheduling problem consist of simultaneously deciding which orders to be sel... more Order acceptance and scheduling problem consist of simultaneously deciding which orders to be selected and how to schedule these selected orders. An extension of the sequence-dependent setup times and release dates was introduced in 2010 and a mathematical formulation was presented. Since then, a few mathematical formulations have appeared in the literature by addressing this problem on the basis of sequence-dependent setup times. However, some of the mathematical formulations are nonlinear or lack usability. Therefore, the model presented in 2010 is still being considered in recent studies. In this paper, we investigated the case in which there are sequence-dependent setup times with no release dates for all orders. We developed a new mathematical formulation with O(n) binary variables and O(n) constraints. In order to see the performance of our formulation, we conducted a computational analysis with CPLEX 12.4 by solving benchmark instances available in the literature. To manage t...
In this study, a new model has been developed for the order selection problem for a bakery firm l... more In this study, a new model has been developed for the order selection problem for a bakery firm located in Turkey. We consider a combined order selection and process planning problem where a make-to-order bakery goods producer firm has to determine a set of orders to process so as to maximize the total profit. The developed model combines setup costs, sales price, lot size, demand and other important daily data of the products. After collecting the related data, fuzzy TOPSIS method is used to obtain the rankings of the orders (bread types). Then, the ranking scores are incorporated in the knapsack problem to determine the lot size and which orders to select. A computer application is also provided in the paper. With the help of MS Excel Visual Basic program, the computer application updates data daily, ranks the products, determines the lot size, and helps the decision maker with the selection of orders. The computational results show that the proposed method presented better results than the current method applied in the firm. Comparing the two methods in terms of cost reduction, the proposed method gives better results than the current method with 27%. Furthermore, the proposed computer program does not need extra special software or a commercial computer program, which is favorable for small-size enterprises.
The Frobenius number of a set of relatively prime positive integers α1,α2,…,αn such that α1< α... more The Frobenius number of a set of relatively prime positive integers α1,α2,…,αn such that α1< α2< …< αn, is the largest integer that can not be written as a nonnegative integer linear combination of the given set. Finding the Frobenius number is known as the Frobenius problem, which is also named as the coin exchange problem or the postage stamp problem. This problem is closely related with the equality constrained integer knapsack problem. It is known that this problem is NP-hard. Extensive research has been conducted for finding the Frobenius number of a given set of positive integers. An exact formula exists for the case n=2 and various formulas have been derived for all special cases of n = 3. Many algorithms have been proposed for n≥4. As far as we are aware, there does not exist any integer programming approach for this problem which is the main motivation of this paper. We present four integer linear programming formulations about the Frobenius number of a given set o...
Proportional symbol maps are a cartographic tool that employs scaled symbols to represent data as... more Proportional symbol maps are a cartographic tool that employs scaled symbols to represent data associated with specific locations. The symbols we consider are opaque disks, which may be partially covered by other overlapping disks. We address the problem of creating a suitable drawing of the disks that maximizes one of two quality metrics: the total and the minimum visible length of disk boundaries. We study three variants of this problem, two of which are known to be NP-hard and another whose complexity is open. We propose novel integer programming formulations for each problem variant and test them on real-world instances with a branch-and-cut algorithm. When compared with state-of-the-art models from the literature, our models significantly reduce computation times for most instances.
Gazi Üniversitesi Mühendislik-Mimarlık Fakültesi Dergisi, 2015
Lojistik yönetiminde en önemli operasyonel kararlardan birisi müşterilere hizmet verecek araç rot... more Lojistik yönetiminde en önemli operasyonel kararlardan birisi müşterilere hizmet verecek araç rotalarının belirlenmesidir. Araç Rotalama Problemi (ARP), bir depodan müşterilerin dağıtım (toplama) taleplerini karşılayacak en uygun rotaların belirlenmesi olarak tanımlanabilir. Gerçek hayat lojistik uygulamalarında, filoda bulunan araçlar farklı özelliklerde olabilirler. Ayrıca müşterilerden/taşınanlardan kaynaklı gereklilikler de farklı özellikte araç kullanımı zorunluluğunu ortaya çıkarabilir. Bunun yanısıra firmalar, mamul, yarı mamul ve hammaddelerin tersine akışının yönetimini de hem ekonomik getirisi hem de yasal ve çevresel yükümlülüklerinden dolayı daha fazla önemsemektedirler. Bu makalede, heterojen araç filosunun bulunduğu ve müşterilerin dağıtım ve toplama taleplerinin eşzamanlı gerçekleştiği durumların birlikte dikkate alındığı bir ARP türü üzerinde çalışılmıştır. Bu problem Heterojen Eşzamanlı Topla-Dağıt Araç Rotalama Problemi (HETD-ARP) olarak adlandırılmıştır. HETD-ARP, toplam maliyeti enküçükleyen araç rotalarının ve herbir rotada kullanılan araç tipinin belirlenmesi olarak tanımlanabilir. Problem için polinom sayıda kısıta sahip akış tabanlı bir matematiksel model önerilmiştir. HETD-ARP, NP-zor problemler sınıfında olduğundan dolayı makul sürelerde orta boyutlu problemlere bile en iyi çözümü bulmak zordur. Bu nedenle bu makalede orta ve büyük boyutlu HETD-ARP'nin çözümü için basit bir kurucu sezgisel algoritma önerilmiştir. Bu algoritma, kaynaklarda ARP için önerilen Clarke-Wright Tasarruf (CWT) algoritmasının HETD-ARP için uyarlanmış halidir. Önerilen matematiksel modelin ve sezgisel algoritmanın etkinliği test problemleri üzerinde incelenmiştir.
Ba Baş şkent kent Ü Üniversitesi niversitesi End Endü üstri M stri Mü ühendisli hendisliğ ği B i ... more Ba Baş şkent kent Ü Üniversitesi niversitesi End Endü üstri M stri Mü ühendisli hendisliğ ği B i Bö öl lü üm mü ü
Gazi Üniversitesi Mühendislik-Mimarlık Fakültesi Dergisi, 2021
Homogeneous and heterogeneous multiple traveler minimum latency problem with time windows New... more Homogeneous and heterogeneous multiple traveler minimum latency problem with time windows New mathematical models with polynomial number of decision variables and constraints Analysis of mathematical models in terms of CPU times and linear programming relaxation values Purpose: The aim of this study is to develop new mathematical formulations for homogeneous and heterogeneous multiple traveler minimum latency problem with time windows. Theory and Methods: Based on the mixed integer linear programming, mathematical models with polynomial number of decision variables and constraints are developed. Benchmark instances from the literature are solved with existing formulations and proposed new formulations by using CPLEX 12.5.0.1. CPU times and percentage deviation from linear programming relaxation values are considered as performance criteria. Results: We solved 125 problems with varying number of nodes and time windows. In all the problem solved proposed formulations are better than the existing formulations in terms of both of the performance criteria. Conclusion: The proposed formulations for homogeneous and heterogeneous multiple traveler minimum latency problem with time windows are superior than the existing formulations and able to solve the problems up to 100 nodes with narrow time windows. Proposed formulations may be used to solve small and moderate real-life problems very easily. They may also be used for testing the performance of the heuristics constructed for kLPTW.
The Generalized Vehicle Routing Problem (GVRP) is an extension of the Vehicle Routing Problem (VR... more The Generalized Vehicle Routing Problem (GVRP) is an extension of the Vehicle Routing Problem (VRP) defined on a graph in which the nodes (customers, vertices) are grouped into a given number of mutually exclusive and exhaustive clusters (nodesets). In this paper, an integer linear programming formulation of the GVRP with O(n 2) binary variables and O(n 2) constraints is presented. It is shown that, under specific circumstances, the proposed model reduces to the well-known routing problems. The computational performance of the models solved using a commercial code on test problems are also presented.
The Generalized Vehicle Routing Problem (GVRP) is an extension of the Vehicle Routing Problem (VR... more The Generalized Vehicle Routing Problem (GVRP) is an extension of the Vehicle Routing Problem (VRP) defined on a graph in which the nodes (customers, vertices) are grouped into a given number of mutually exclusive and exhaustive clusters (nodesets). In this paper, an integer linear programming formulation of the GVRP with O(n 2 ) binary variables and O(n 2 ) constraints is presented. It is shown that, under specific circumstances, the proposed model reduces to the well-known routing problems. The computational performance of the models solved using a commercial code on test problems are also presented.
Corrected Miller–Tucker–Zemlin type subtour elimination constraints for the capacitated vehicle r... more Corrected Miller–Tucker–Zemlin type subtour elimination constraints for the capacitated vehicle routing problem are presented. 2003 Elsevier B.V. All rights reserved.
The Generalized Traveling Salesman Problem (GTSP) is an extension of the well known Traveling Sal... more The Generalized Traveling Salesman Problem (GTSP) is an extension of the well known Traveling Salesman Problem (TSP). The GTSP is defined on a graph in which the nodes (customers or vertices) are grouped into a given number of clusters (node sets). Solution procedures for the GTSP are generally focused on transforming the problem to the TSP and applying the exact or heuristic solution methods developed for the TSP. There exist a few integer programming formulations for the GTSP some of which are exponential size with respect to number of the nodes. In this paper, we propose two new formulations for the GTSP with polynomial size with respect to number of the nodes. For preliminary computational analysis, GTSP instances from TSPLIB are solved by proposed formulations and also by the previously existing formulations in the literature. Performances of the formulations in terms of linear programming relaxations and CPU times are analyzed. We observe that, performances of the proposed for...
One of the exact solution methods for the traveling salesman problem (TSP) is the mixed0 integer ... more One of the exact solution methods for the traveling salesman problem (TSP) is the mixed0 integer formulation, based on the subtour elimination constraints. Dantzig, Fulkerson and Johnson have first proposed subtour elimination constraints in 1959. Later on, Miller, Tucker and Zemlin proposed an alternative formulation by introducing new variables. Their approach was only based on experiments, ie developing several models and selecting the most e cient one in terms of generality, number of variables and number of constraints. ...
The Home Health Care routing problem (HHCRP) is an extension of the vehicle routing problem (VRP)... more The Home Health Care routing problem (HHCRP) is an extension of the vehicle routing problem (VRP) considering scheduling and routing simultaneously to perform home health care (HHC) services. These services may be examining the patients, preparing a diet plan, applying physical therapy, and picking up the blood samples. In this paper, HHCRP has been addressed for a public hospital in Ankara, Turkey, that provides HHC services with limited resources. There are teams that consist of a doctor, a nurse, a dietitian, a physiotherapist and a vehicle driver. The multiple tours of those teams are searched by considering time windows to meet patient demands on time. This problem is defined as the multi-trip home health care routing problem with time-windows (MT-HHCRP-TW). In this study, a mixed integer linear programming model to minimize the total travelling time is proposed. Test problems that are generated based on real data for computational analysis and results are discussed. Main contr...
In classical scheduling problem, it is assumed that all orders must be processed. In the order ac... more In classical scheduling problem, it is assumed that all orders must be processed. In the order acceptance and scheduling (OAS) problem, some orders are rejected due to limited capacity. In make-to-order production environment, in which the OAS problem occurs, accepting all orders may cause overloads, delay in deliveries and unsatisfied customers. Oguz et al. (2010) introduced the OAS problem with sequence-dependent setup times and release dates. In this paper, we propose a new mixed integer programming formulation with O(n2) decision variables and O(n2) constraints for the same problem. We conduct a computational analysis comparing the performance of our formulation with Oguz et al. (2010) formulation. We use the benchmark instances, which are available in the literature. We observe that our formulation can solve all the instances up to 50 orders in a reasonable time, while Oguz et al. (2010) formulation can solve only the instances with 10 orders in the same time limit.
Abstract The Traveling Repairman Problem (TRP) is one of the most important variants of the Trave... more Abstract The Traveling Repairman Problem (TRP) is one of the most important variants of the Traveling Salesman Problem (TSP). The objective function of TRP is to find a Hamiltonian path or tour starting from the origin while minimizing the total latency (waiting or delay time) for all customers. The latency of a customer is defined as the time passed from the beginning of a tour (or path) until a customer’s service is completed. TRP with time windows (TRPTW) is the case where the earliest and latest times for visiting each customer are restricted by prescribed time windows. The literature on TRPTW is scarce. We only found one formulation for TRPTW and one formulation for its variant. In this paper, we propose four new mathematical models for TRPTW with O(n2) binary variables and O(n2) constraints. We computationally analyze the performance of existing and new formulations by solving symmetric and asymmetric benchmark instances with CPLEX 12.5.0.1 and compare the results in terms of CPU times and optimality gap. We observed that our two formulations were extremely faster than existing formulations, and they could optimally solve symmetric instances up to 150 nodes and asymmetric instances up to 131 nodes within seconds.
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