Branch and Bound

This paper explores the use of simulated annealing (SA) for solving arbitrary combinatorialoptimisation problems. It reviews an existing code called GPSIMAN for solving0-1 problems, and evaluates it against a commercial branch-and-bound code, OSL ...

The subject of this paper is the formulation and solution of a variation of the classical binary knapsack problem. The variation that is addressed is termed the “fixed-charge knapsack problem”, in which sub-sets of variables (activities) are associated with fixed costs. These costs may represent certain set-ups and/or preparations required for the associated sub-set of activities to be scheduled. Several

Cette these est principalement consacree a l'association des methodes de points interieurs et des techniques de l'optimisation DC et DCA pour resoudre les problemes d'optimisation non convexes de grande taille. La these comporte trois parties : la premiere partie est consacree aux techniques d'optimisations locales et s'articule autour des methodes de points interieurs et de la programmation DC. Nous y developpons deux algorithmes. Apres une presentation non exhaustive de la programmation DC, des methodes de points interieurs et des proprietes essentielles de la classe des matrices quasi-definies au chapitre un, nous presentons au chapitre deux un nouvel algorithme base sur une reformulation des conditions d'optimalite de Karush-Kuhn-Tucker. Le troisieme chapitre est consacre a l'integration des techniques d'optimisation DC dans un schema de points interieurs, c'est l'algorithme IPDCA. La seconde partie de la these est consacree aux solutions ...

This paper addresses a new electricity-charging station location problem for congested intercity or regional networks, in which electric vehicles with a limited driving range require recharges for their long-distance trips. When the distribution of charging stations does not sufficiently cover all used routes, some drivers may take a detour to find charging opportunities (or switch to another transportation mode). The goal of this station location problem is to find, under a limited construction budget, an optimal set of charging station locations such that all vehicles finish their trips by choosing a self-optimal route with necessary charging opportunities while the overall network congestion caused by possible detours is minimized. The problem is written as a bi-level mixed nonlinear integer programming model, where the upper level of the model is set to regulate the selection of station locations subject to the construction budget while the lower level is used to characterize the equilibrium flow pattern of electric vehicles with the charging requirement. Selected exact and approximate algorithms, namely, the branch-and-bound algorithm and the nested partitions algorithm are adopted to solve this station location problem. While both algorithms imply a divide-and-conquer strategy, the branch-and-bound algorithm poses a deterministic, exact procedure that utilizes the bounding mechanism to prune impossible solution subspaces, whereas the nested partitions algorithm performs a stochastic search and selection process in terms of the optimality probability for near-optimal solutions. To get numerical insights on the algorithmic performance and solution behavior, we test these algorithms through a couple of benchmark network instances. A performance comparison of the algorithms indicates that, the branch-and-bound algorithm can quickly obtain the global optimum when the driving range is relatively low, while the nested partitions algorithm can find optimal solutions or extremely near-optimal solutions in all cases and typically spend on average only one fourth of the computing time of the branch-and-bound algorithm. The network flow solutions clearly show, compared to gasoline vehicle drivers, how an insufficient driving range or number of charging stations may significantly reduce the number of feasible paths and force electric vehicle drivers to choose more costly paths, which thus reshapes flow patterns and possibly increases congestion levels.

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 ...

In this paper, we propose approximate and exact algorithms for the double constrained two-dimensional guillotine cutting stock problem (DCTDC). The approximate algorithm is a two-stage procedure. The first stage attempts to produce a starting feasible solution to DCTDC by solving a single constrained two dimensional cutting problem, CDTC. If the solution to CTDC is not feasible to DCTDC, the second stage is used to eliminate non-feasibility. The exact algorithm is a branch-and-bound that uses efficient lower and upper bounding schemes. It starts with a lower bound reached by the approximate two-stage algorithm. At each internal node of the branching tree, a tailored upper bound is obtained by solving (relaxed) knapsack problems. To speed up the branch and bound, we implement, in addition to ordered data structures of lists, symmetry, duplicate, and non-feasibility detection strategies which fathom some unnecessary branches. We evaluate the performance of the algorithm on different problem instances which can become benchmark problems for the cutting and packing literature.

Undeva, prin preajma anului '94, un elev de clasa a XI-a se gândea să-şi construiască un unit Pascal care să îl ajute la implementarea rapidă a metodei backtracking. Pentru aceasta avea la îndemână un singur atu, care era repetat la nesfârşit în toate cărţile de informatică, ce tratau această metodă: backtracking-ul se aplică problemelor a căror soluţie se poate reprezenta sub forma unui vector... Deci, la prima vedere nimic dificil. Totul consta în a specifica care sunt elementele acelui vector, care sunt condiţiile de continuare şi  care sunt condiţiile finale..., iar apoi soluţia ieşea ca pe bandă. Dar, abia mai târziu, acel elev (care de fapt eram eu) a înţeles că de fapt dificultatea unei probleme de informatică nu constă în determinarea tipului de algoritm care se aplică, ci în rezolvarea sutelor şi miilor de cazuri mărunţele care apar în jurul acesteia, cu alte cuvinte, de la soluţie care se poate reprezenta sub forma unui vector până efectiv la soluţie este o cale lungă şi spinoasă.
Puţin mai târziu, după ce am studiat bine şi celelalte metode de programare am dedus că fiecare dintre ele are un algoritm general de rezolvare, dar care nu era aşa de evident precum cel de la backtracking tocmai din cauză că la această metodă se sugera structura de date care trebuie folosită, şi anume un vector. Acum, când scriu aceste rânduri nu mi se mai pare aşa de dificilă găsirea unui algoritm general pentru o clasă de probleme. Pur şi simplu se foloseşte metoda rafinării în paşi succesivi. Adică, ca să fiu puţin mai explicit, se pleacă de la o problemă pentru care se scrie un algoritm. Apoi se analizează o altă problemă din aceeaşi clasă şi algoritmul iniţial se rescrie astfel încât să poată rezolva ambele probleme. Apoi se mai analizează încă o problemă... Şi nu trebuie să se continue această operaţie de mai mult de 10 (zece) ori, adică, cu alte cuvinte dacă se reuşeşte analizarea şi combinarea a 10 algoritmi reprezentativi, atunci rezultatul este suficient de general pentru a rezolva multe probleme din aceea clasă.

Programul nu l-am conceput ca pe un creator de elemente soft orientate pe metodele de programare. El este un mijloc şi nu un scop. De fapt este un mijloc pentru a crea un alt mijloc pe care îl voi folosi la crearea unui alt mijloc... etc, dar niciodată nu voi ajunge să creez un scop. Toată viaţa nu vom face altceva decât să creăm mijloace, adică lucruri care ne vor ajuta să creăm alte lucruri.

This paper introduces a new type of constraints, related to schedule synchronization, in the problem formulation of aircraft fleet assignment and routing problems and it proposes an optimal solution approach. This approach is based on Dantzig–Wolfe decomposition/column generation. The resulting master problem consists of flight covering constraints, as in usual applications, and of schedule synchronization constraints. The corresponding subproblem is a shortest path problem with time windows and linear costs on the time variables and it is solved by an optimal dynamic programming algorithm. This column generation procedure is embedded into a branch and bound scheme to obtain integer solutions. A dedicated branching scheme was devised in this paper where the branching decisions are imposed on the time variables. Computational experiments were conducted using weekly fleet routing and scheduling problem data coming from an European airline. The test problems are solved to optimality. A detailed result analysis highlights the advantages of this approach: an extremely short subproblem solution time and, after several improvements, a very efficient master problem solution time.

Distribusi produk (Traveling Salesman Problem-TSP) adalah masalah optimasi bagi pengecer untuk pergi ke banyak konsumen di berbagai kota dan menemukan rute terpendek  kembali ke kota asalnya. Penjelasan dari permasalahan tersebut adalah bagaimana mencari itinerary terdekat dari suatu kota dan mengunjungi semua kota lainnya dimana setiap kota hanya dapat dikunjungi satu kali dan harus kembali ke kota keberangkatan. Yang dibutuhkan di era yang mudah digunakan saat ini adalah algoritma yang dapat dengan cepat menyelesaikan TSP untuk mendapatkan solusi yang mendekati solusi optimal. Oleh karena itu, algoritma branch-and-bound digunakan untuk menentukan perjalanan terpendek. Perjalanan terpendek akan melewati kota lain hanya sekali dan kembali ke kota asal.

In this paper we propose an algorithm for the constrained two-dimensional cutting stock problem (TDC) in which a single stock sheet has to be cut into a set of small pieces, while maximizing the value of the pieces cut. The TDC problem is NP-hard in the strong sense and finds many practical applications in the cutting and packing area. The

In this chapter we discuss the Vehicle Routing Problem with Time Windows in terms of its mathematical modeling, its structure and decomposition alternatives. We then present the master problem and the subproblem for the column generation approach, respectively. Next, we illustrate a branch-and-bound framework and address acceleration strategies used to increase the efficiency of branch-and-price methods. Then, we describe generalizations of the problem and report computational results for the classic Solomon test sets. Finally, we present our conclusions and discuss some open problems.