Improvement of the branch and bound algorithm for solving the knapsack linear integer problem

The paper presents a new reformulation approach to reduce the complexity of a branch and bound algorithm for solving the knapsack linear integer problem. The branch and bound algorithm in general relies on the usual strategy of first relaxing the integer problem into a linear programing (LP) model. If the linear programming optimal solution is integer then, the optimal solution to the integer problem is available. If the linear programming optimal solution is not integer, then a variable with a fractional value is selected to create two sub-problems such that part of the feasible region is discarded without eliminating any of the feasible integer solutions. The process is repeated on all variables with fractional values until an integer solution is found. In this approach variable sum and additional constraints are generated and added to the original problem before solving. In order to do this the objective bound of knapsack problem is quickly determined. The bound is then used to generate a set of variable sum limits and four additional constraints. From the variable sum limits, initial sub-problems are constructed and solved. The optimal solution is then obtained as the best solution from all the sub-problems in terms of the objective value. The proposed procedure results in sub-problems that have reduced complexity and easier to solve than the original problem in terms of numbers of branch and bound iterations or sub-problems.The knapsack problem is a special form of the general linear integer problem. There are so many types of knapsack problems. These include the zero-one, multiple, multiple-choice, bounded, unbounded, quadratic, multi-objective, multi-dimensional, collapsing zero-one and set union knapsack problems. The zero-one knapsack problem is one in which the variables assume 0 s and 1 s only. The reason is that an item can be chosen or not chosen. In other words there is no way it is possible to have fractional amounts or items. This is the easiest class of the knapsack problems and is the only one that can be solved in polynomial by interior point algorithms and in pseudo-polynomial time by dynamic programming approaches. The multiple-choice knapsack problem is a generalization of the ordinary knapsack problem, where the set of items is partitioned into classes. The zero-one choice of taking an item is replaced by the selection of exactly one item out of each class of item

Improvement of the branch and bound algorithm for solving the knapsack linear integer problem

The paper presents a new reformulation approach to reduce the complexity of a branch and bound algorithm for solving the knapsack linear integer problem. The branch and bound algorithm in general relies on the usual strategy of first relaxing the integer problem into a linear programing (LP) model. If the linear programming optimal solution is integer then, the optimal solution to the integer problem is available. If the linear programming optimal solution is not integer, then a variable with a fractional value is selected to create two sub-problems such that part of the feasible region is discarded without eliminating any of the feasible integer solutions. The process is repeated on all variables with fractional values until an integer solution is found. In this approach variable sum and additional constraints are generated and added to the original problem before solving. In order to do this the objective bound of knapsack problem is quickly determined. The bound is then used to generate a set of variable sum limits and four additional constraints. From the variable sum limits, initial sub-problems are constructed and solved. The optimal solution is then obtained as the best solution from all the sub-problems in terms of the objective value. The proposed procedure results in sub-problems that have reduced complexity and easier to solve than the original problem in terms of numbers of branch and bound iterations or sub-problems.The knapsack problem is a special form of the general linear integer problem. There are so many types of knapsack problems. These include the zero-one, multiple, multiple-choice, bounded, unbounded, quadratic, multi-objective, multi-dimensional, collapsing zero-one and set union knapsack problems. The zero-one knapsack problem is one in which the variables assume 0 s and 1 s only. The reason is that an item can be chosen or not chosen. In other words there is no way it is possible to have fractional amounts or items. This is the easiest class of the knapsack problems and is the only one that can be solved in polynomial by interior point algorithms and in pseudo-polynomial time by dynamic programming approaches. The multiple-choice knapsack problem is a generalization of the ordinary knapsack problem, where the set of items is partitioned into classes. The zero-one choice of taking an item is replaced by the selection of exactly one item out of each class of item

Exploiting Symmetry in Linear and Integer Linear Programming

This thesis explores two algorithmic approaches for exploiting symmetries in linear and integer linear programs. The first is orbital crossover, a novel method of crossover designed to exploit symmetry in linear programs. Symmetry has long been considered a curse in combinatorial optimization problems, but significant progress has been made. Up until recently, symmetry exploitation in linear programs was not worth the upfront cost of symmetry detection. However, recent results involving a generalization of symmetries, equitable partitions, has made the upfront cost much more manageable. The motivation for orbital crossover is that many highly symmetric integer linear programs exist, and thus, solving symmetric linear programs is of major interest in order to efficiently solve symmetric integer linear programs. The results of this work indicate that a specialized linear programming algorithm that exploits symmetry is likely to be useful in the toolbox of linear programming solvers. The second algorithm is orbital cut generation. The main issue brought forward by symmetric integer linear programs is multiple symmetric solutions having an equivalent objective value. This massively increases the search space for algorithms such as branch and bound or branch and cut. Orbital cut generation aims to tackle the issues of multiple equivalent symmetric solutions using symmetrically valid inequalities. Chapter 2 shows how to effectively exploit symmetry in integer linear programs by generating symmetric cutting planes that remove multiple symmetric solutions in one go. Further, the method is strengthened using symmetry to aggregate integer linear programs and generate cutting planes in aggregate spaces before lifting them to the original problem

Capacity Planning in Stable Matching

We introduce the problem of jointly increasing school capacities and finding a student-optimal assignment in the expanded market. Due to the impossibility of efficiently solving the problem with classical methods, we generalize existent mathematical programming formulations of stability constraints to our setting, most of which result in integer quadratically-constrained programs. In addition, we propose a novel mixed-integer linear programming formulation that is exponentially large on the problem size. We show that its stability constraints can be separated by exploiting the objective function, leading to an effective cutting-plane algorithm. We conclude the theoretical analysis of the problem by discussing some mechanism properties. On the computational side, we evaluate the performance of our approaches in a detailed study, and we find that our cutting-plane method outperforms our generalization of existing mixed-integer approaches. We also propose two heuristics that are effective for large instances of the problem. Finally, we use the Chilean school choice system data to demonstrate the impact of capacity planning under stability conditions. Our results show that each additional seat can benefit multiple students and that we can effectively target the assignment of previously unassigned students or improve the assignment of several students through improvement chains. These insights empower the decision-maker in tuning the matching algorithm to provide a fair application-oriented solution

A New Class of Compact Formulations for Vehicle Routing Problems

This paper introduces a novel compact mixed integer linear programming (MILP) formulation and a discretization discovery-based solution approach for the Vehicle Routing Problem with Time Windows (VRPTW). We aim to solve the optimization problem efficiently by constraining the linear programming (LP) solutions to use only flows corresponding to time and capacity-feasible routes that are locally elementary (prohibiting cycles of customers localized in space). We employ a discretization discovery algorithm to refine the LP relaxation iteratively. This iterative process alternates between two steps: (1) increasing time/capacity/elementarity enforcement to increase the LP objective, albeit at the expense of increased complexity (more variables and constraints), and (2) decreasing enforcement without decreasing the LP objective to reduce complexity. This iterative approach ensures we produce an LP relaxation that closely approximates the optimal MILP objective with minimal complexity, facilitating an efficient solution via an off-the-shelf MILP solver. The effectiveness of our method is demonstrated through empirical evaluations on classical VRPTW instances. We showcase the efficiency of solving the final MILP and multiple iterations of LP relaxations, highlighting the decreased integrality gap of the final LP relaxation. We believe that our approach holds promise for addressing a wide range of routing problems within and beyond the VRPTW domain

Joint pricing and production planning of multiple products

Many industries are beginning to use innovative pricing techniques to improve inventory control, capacity utilisation, and ultimately the profit of the firm. In manufacturing, the coordination of pricing and production decisions offers significant opportunities to improve supply chain performance by better matching supply and demand. This integration of pricing, production and distribution decisions in retail or manufacturing environments is still in its early stages in many companies. Importantly it has the potential to radically improve supply chain efficiencies in much the same way as revenue management has changed the management of the airline, hotel and car rental industries. These developments raise the need and interest of having models that integrate production decisions, inventory control and pricing strategies.In this thesis, we focus on joint pricing and production planning, where prices and production values are determined in coordination over a multiperiod horizon with non-perishable inventory. We specifically look at multiproduct systems with either constant or dynamic pricing. The fundamental problem is: when the capacity limitations and other parameters like production, holding, and backordering costs are given, what the optimal values are for production quantities, and inventory and backorder levels for each item as well as a price at which the firm commits to sell the products over the total planning horizon. Our aim is to develop models and solution strategies that are practical to implement for real sized problems.We initially formulate the problem of time-varying pricing and production planning of multiple products over a multiperiod horizon as a nonlinear programming problem. When backorders are not allowed, we show that if the demand/price function is linear, as a special case of the without backorders model, the problem becomes a Quadratic Programming problem which has only linear constraints. Existing solution methods for Quadratic Programming problem are discussed. We then present the case of allowed backorders. This assumption makes the problem more difficult to handle, because the constraint set changes to a non-convex set. We modify the nonlinear constraints to obtain an alternative formulation with a convex set of constraints. By this modification the problem becomes a Mixed Integer Nonlinear Programming problem over a linear set of constraints. The integer variables are all binary variables. The limitation of obtaining the optimal solution of the developed models is discussed. We describe our strategy to overcome the computational difficulties to solve the models.We tackle the main nonlinear problem with backorders through solving an easier case when prices are constant. This resulting model involves a nonlinear objective function and some nonlinear constraints. Our strategy to reduce the level of difficulty is to utilise a method that solves the relaxed problem which considers only linear constraints. However, our method keeps track of the feasibility with respect to the nonlinear constraints in the original problem. The developed model which is a combination of Linear Programming (LP) and Nonlinear Programming (NLP) is solved iteratively. The solution strategy for the constant pricing case constructs a tree search in breadth-first manner. The detailed algorithm is presented. This algorithm is practical to implement, as we demonstrate through a small but practical size numerical example.The algorithm for the constant pricing case is extended to the more general problem. More specifically, we reformulate the timevariant problem in which there are multi blocks of constant pricing problems. The developed model is a combination of Linear Programming (LP) and linearly constrained Nonlinear Programming (NLP) which is solved iteratively. Iterations consist of two main stages: finding the value of LP’s objective function for a known basis, solving a very smaller size NLP problem. The detailed algorithm is presented and a practical size numerical example is used to implement the algorithm. The significance of this algorithm is that it can be applied to large scale problems which are not easily solved with the existing commercial packages

Comparing Optimization Methods for Radiation Therapy Patient Scheduling using Different Objectives

Radiation therapy (RT) is one of the most common technologies used to treat cancer. To better use resources in RT, optimization models can be used to automatically create patient schedules, a task that today is done manually in almost all clinics. This paper presents a comprehensive study of different optimization methods for modeling and solving the RT patient scheduling problem. The results can be used as decision support when implementing an automatic scheduling algorithm in practice. We introduce an Integer Linear Programming (IP) model, a column generation IP model (CG-IP), and a Constraint Programming model. Patients are scheduled on multiple machine types considering their priority for treatment, session duration and allowed machines, while taking expected future patient arrivals into account. Different cancer centers may have different scheduling objectives, and therefore each model is solved using multiple different objective functions, including minimizing waiting times, and maximizing the fulfillment of patients' preferences for treatment times. The test data is generated from historical data from Iridium Netwerk, a large cancer center in Belgium with 10 linear accelerators. The results demonstrate that the CG-IP model can solve all the different problem instances to a mean optimality gap of less than 1% within one hour. The proposed methodology provides a tool for automated scheduling of RT treatments and can be generally applied to RT centers.Comment: 20 pages, 4 figures, Submitted to Operations Research Foru

An exact method for a discrete multiobjective linear fractional optimization

Integer linear fractional programming problem with multiple objective MOILFP is an important field of research and has not received as much attention as did multiple objective linear fractional programming. In this work, we develop a branch and cut algorithm based on continuous fractional optimization, for generating the whole integer efficient solutions of the MOILFP problem. The basic idea of the computation phase of the algorithm is to optimize one of the fractional objective functions, then generate an integer feasible solution. Using the reduced gradients of the objective functions, an efficient cut is built and a part of the feasible domain not containing efficient solutions is truncated by adding this cut. A sample problem is solved using this algorithm, and the main practical advantages of the algorithm are indicated

Multi-objective integer programming: An improved recursive algorithm

This paper introduces an improved recursive algorithm to generate the set of all nondominated objective vectors for the Multi-Objective Integer Programming (MOIP) problem. We significantly improve the earlier recursive algorithm of \"Ozlen and Azizo\u{g}lu by using the set of already solved subproblems and their solutions to avoid solving a large number of IPs. A numerical example is presented to explain the workings of the algorithm, and we conduct a series of computational experiments to show the savings that can be obtained. As our experiments show, the improvement becomes more significant as the problems grow larger in terms of the number of objectives.Comment: 11 pages, 6 tables; v2: added more details and a computational stud