Interactive Bi-Level Multi-Objective Integer Non-linear Programming Problem

Abstract This paper presents a bi-level multi-objective integer non-linear programming (BLMINP) problem with linear or non-linear constraints and an interactive algorithm for solving such model. At the first phase of the solution algorithm to avoid the complexity of non convexity of this problem, we begin by finding the convex hull of its original set of constraints using the cutting-plane algorithm to convert the BLMINP problem to an equivalent bi-level multi-objective non-linear programming (BLMNP) problem. At the second phase the algorithm simplifies an equivalent (BLMNP) problem by transforming it into separate multi-objective decision-making problems with hierarchical structure, and solving it by using ε -constraint method to avoid the difficulty associated with non-convex mathematical programming. In addition, the author put forward the satisfactoriness concept as the first-level decision-maker preference. Finally, an illustrative numerical example is given to demonstrate the obtained results. Mathematics Subject Classification: 90C29; 90C30; 41A58; 90C1

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

Optimising a nonlinear utility function in multi-objective integer programming

In this paper we develop an algorithm to optimise a nonlinear utility function of multiple objectives over the integer efficient set. Our approach is based on identifying and updating bounds on the individual objectives as well as the optimal utility value. This is done using already known solutions, linear programming relaxations, utility function inversion, and integer programming. We develop a general optimisation algorithm for use with k objectives, and we illustrate our approach using a tri-objective integer programming problem.Comment: 11 pages, 2 tables; v3: minor revisions, to appear in Journal of Global Optimizatio

Decomposition, Reformulation, and Diving in University Course Timetabling

In many real-life optimisation problems, there are multiple interacting components in a solution. For example, different components might specify assignments to different kinds of resource. Often, each component is associated with different sets of soft constraints, and so with different measures of soft constraint violation. The goal is then to minimise a linear combination of such measures. This paper studies an approach to such problems, which can be thought of as multiphase exploitation of multiple objective-/value-restricted submodels. In this approach, only one computationally difficult component of a problem and the associated subset of objectives is considered at first. This produces partial solutions, which define interesting neighbourhoods in the search space of the complete problem. Often, it is possible to pick the initial component so that variable aggregation can be performed at the first stage, and the neighbourhoods to be explored next are guaranteed to contain feasible solutions. Using integer programming, it is then easy to implement heuristics producing solutions with bounds on their quality. Our study is performed on a university course timetabling problem used in the 2007 International Timetabling Competition, also known as the Udine Course Timetabling Problem. In the proposed heuristic, an objective-restricted neighbourhood generator produces assignments of periods to events, with decreasing numbers of violations of two period-related soft constraints. Those are relaxed into assignments of events to days, which define neighbourhoods that are easier to search with respect to all four soft constraints. Integer programming formulations for all subproblems are given and evaluated using ILOG CPLEX 11. The wider applicability of this approach is analysed and discussed.Comment: 45 pages, 7 figures. Improved typesetting of figures and table

Bin Packing and Related Problems: General Arc-flow Formulation with Graph Compression

We present an exact method, based on an arc-flow formulation with side constraints, for solving bin packing and cutting stock problems --- including multi-constraint variants --- by simply representing all the patterns in a very compact graph. Our method includes a graph compression algorithm that usually reduces the size of the underlying graph substantially without weakening the model. As opposed to our method, which provides strong models, conventional models are usually highly symmetric and provide very weak lower bounds. Our formulation is equivalent to Gilmore and Gomory's, thus providing a very strong linear relaxation. However, instead of using column-generation in an iterative process, the method constructs a graph, where paths from the source to the target node represent every valid packing pattern. The same method, without any problem-specific parameterization, was used to solve a large variety of instances from several different cutting and packing problems. In this paper, we deal with vector packing, graph coloring, bin packing, cutting stock, cardinality constrained bin packing, cutting stock with cutting knife limitation, cutting stock with binary patterns, bin packing with conflicts, and cutting stock with binary patterns and forbidden pairs. We report computational results obtained with many benchmark test data sets, all of them showing a large advantage of this formulation with respect to the traditional ones

Stochastic multi-period multi-product multi-objective Aggregate Production Planning model in multi-echelon supply chain

In this paper a multi-period multi-product multi-objective aggregate production planning (APP) model is proposed for an uncertain multi-echelon supply chain considering financial risk, customer satisfaction, and human resource training. Three conflictive objective functions and several sets of real constraints are considered concurrently in the proposed APP model. Some parameters of the proposed model are assumed to be uncertain and handled through a two-stage stochastic programming (TSSP) approach. The proposed TSSP is solved using three multi-objective solution procedures, i.e., the goal attainment technique, the modified ε-constraint method, and STEM method. The whole procedure is applied in an automotive resin and oil supply chain as a real case study wherein the efficacy and applicability of the proposed approaches are illustrated in comparison with existing experimental production planning method

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