Further results on an abstract model for branching and its application to mixed integer programming

Mathematical Programming - A key ingredient in branch and bound (B&B) solvers for mixed-integer programming (MIP) is the selection of branching variables since poor or arbitrary selection...     

An additive utility mixed integer programming model for nonlinear discriminant analysis

Mathematical programming (MP) discriminant analysis models can be used to develop classification models for assigning observations of unknown class membership to one of a number of specified classes using values of a set of features associated with each observation. Since most MP discriminant analysis models generate linear discriminant functions, these MP models are generally used to develop linear classification models. Nonlinear classifiers may, however, have better classification performance than linear classifiers. In this paper, a mixed integer programming model is developed to generate nonlinear discriminant functions composed of monotone piecewise-linear marginal utility functions for each feature and the cut-off value for class membership. It is also shown that this model can be extended for feature selection. The performance of this new MP model for two-group discriminant analysis is compared with statistical discriminant analysis and other MP discriminant analysis models using a real problem and a number of simulated problem sets.     

A mixed integer programming model for fertiliser policy evaluation

Commercial fertiliser and trace element mixtures are widely used in crop production to supply part of the nutrient requirements of the crop. In evaluating fertiliser policy, the crop producer must consider the mixtures to be used and the blending and application policy. In this paper a mixed integer programming model is developed to determine the policy for sourcing, blending and application of commercial fertiliser and trace mixtures to supply specified crop nutrient requirements at minimum cost. Results from the model are presented and the advantages and disadvantages of the approach are discussed.     

An approximation algorithm for indefinite mixed integer quadratic programming

Mathematical Programming - In this paper, we give an algorithm that finds an $$epsilon $$ -approximate solution to a mixed integer quadratic programming (MIQP) problem. The algorithm runs in...     

Nondominated Nash points: application of biobjective mixed integer programming

We study the connection between biobjective mixed integer linear programming and normal form games with two players. We first investigate computing Nash equilibria of normal form games with two players using single-objective mixed integer linear programming. Then, we define the concept of efficient (Pareto optimal) Nash equilibria. This concept is precisely equivalent to the concept of efficient solutions in multi-objective optimization, where the solutions are Nash equilibria. We prove that the set of all points in the payoff (or objective) space of a normal form game with two players corresponding to the utilities of players in an efficient Nash equilibrium, the so-called nondominated Nash points, is finite. We demonstrate that biobjective mixed integer linear programming, where the utility of each player is an objective function, can be used to compute the set of nondominated Nash points. Finally, we illustrate how the nondominated Nash points can be used to determine the disagreement point of a bargaining problem.     

A mixed integer programming model for multiple stage adaptive testing

The last decade has seen paper-and-pencil (P&P) tests being replaced by computerized adaptive tests (CATs) within many testing programs. A CAT may yield several advantages relative to a conventional P&P test. A CAT can determine the questions or test items to administer, allowing each test form to be tailored to a test taker’s skill level. Subsequent items can be chosen to match the capability of the test taker. By adapting to a test taker’s ability, a CAT can acquire more information about a test taker while administering fewer items. A Multiple Stage Adaptive test (MST) provides a means to implement a CAT that allows review before the administration. The MST format is a hybrid between the conventional P&P and CAT formats. This paper presents mixed integer programming models for MST assembly problems. Computational results with commercial optimization software will be given and advantages of the models evaluated.     

Cross decomposition for mixed integer programming

Many methods for solving mixed integer programming problems are based either on primal or on dual decomposition, which yield, respectively, a Benders decomposition algorithm and an implicit enumeration algorithm with bounds computed via Lagrangean relaxation. These methods exploit either the primal or the dual structure of the problem. We propose a new approach, cross decomposition, which allows exploiting simultaneously both structures. The development of the cross decomposition method captures profound relationships between primal and dual decomposition. It is shown that the more constraints can be included in the Langrangean relaxation (provided the duality gap remains zero), the fewer the Benders cuts one may expect to need. If the linear programming relaxation has no duality gap, only one Benders cut is needed to verify optimality.     

An integer programming model for optimal pork marketing

Pork producers must determine when to sell pigs, which and how many pigs to sell, and to which packer(s) to sell them. We model the decision-making problem as a linear mixed-integer program that determines the marketing strategy that maximizes expected annual profit. By discretizing the barn population into appropriate weight and growth categories, we formulate an mixed-integer program that captures the effect of stocking space and shipping disruption on pig growth. We consider marketing to multiple packers via shipping policies reflecting operational sorting constraints. Utilizing data from Cargill Animal Nutrition, we implement the model to obtain solutions that characterize significant strategic departures from commonly-implemented industry rules-of-thumb and that possess the potential to increase profitability in an industry characterized by narrow profit margins.     

An integer linear programming approach for bilinear integer programming

We introduce a new Integer Linear Programming (ILP) approach for solving Integer Programming (IP) problems with bilinear objectives and linear constraints. The approach relies on a series of ILP approximations of the bilinear IP. We compare this approach with standard linearization techniques on random instances and a set of real-world product bundling problems.     

A new mixed integer programming model for curriculum balancing: Application to a Turkish university

Curriculum design is a highly important activity for the academic institutions. It is discussed in literature as a balancing academic curriculum problem (BACP). The BACP schedules courses to different semesters, while balancing the total workload per period. BACP model involves precedence relations, but the related courses are not necessarily assigned to closest periods.     

A multi-objective mixed integer linear programming model for thesis defence scheduling

In this paper, we address the thesis defence scheduling problem, a critical academic scheduling management process, which has been overshadowed in the literature by its counterparts, course timetabling and exam scheduling. Specifically, we address the single defence assignment type of thesis defence scheduling problems, where each committee is assigned to a single defence, scheduled for a specific day, hour and room. We formulate a multi-objective mixed-integer linear programming model, which aims to be applicable to a broader set of cases than other single defence assignment models present in the literature, which have a focus on the characteristics of their universities. For such a purpose, we introduce a different decision variable, propose constraint formulations that are not regulation and policy specific, and cover and offer new takes on the more common objectives seen in the literature. We also include new objective functions based on our experience with the problem at our university and by applying knowledge from other academic scheduling problems. We also propose a two-stage solution approach. The first stage is employed to find the number of schedulable defences, enabling the optimisation of instances with unschedulable defences. The second stage is an implementation of the augmented $ϵ$-constraint method, which allows for the search of a set of different and non-dominated solutions while skipping redundant iterations. The methodology is tested for case-studies from our university, significantly outperforming the solutions found by human schedulers. A novel instance generator for thesis scheduling problems is presented. Its main benefit is the generation of the availability of committee members and rooms in availability and unavailability blocks, resembling their real-world counterparts. A set of 96 randomly generated instances of varying sizes is solved and analysed regarding their relative computational performance, the number of schedulable defences and the distribution of the considered types of iterations. The proposed method can find the optimal number of schedulable defences and present non-dominated solutions within the set time limits for every tested instance.     

A mixed integer linear programming model for optimal sovereign debt issuance

Governments borrow funds to finance the excess of cash payments or interest payments over receipts, usually by issuing fixed income debt and index-linked debt. The goal of this work is to propose a stochastic optimization-based approach to determine the composition of the portfolio issued over a series of government auctions for the fixed income debt, to minimize the cost of servicing debt while controlling risk and maintaining market liquidity. We show that this debt issuance problem can be modeled as a mixed integer linear programming problem with a receding horizon. The stochastic model for the interest rates is calibrated using a Kalman filter and the future interest rates are represented using a recombining trinomial lattice for the purpose of scenario-based optimization. The use of a latent factor interest rate model and a recombining lattice provides us with a realistic, yet very tractable scenario generator and allows us to do a multi-stage stochastic optimization involving integer variables on an ordinary desktop in a matter of seconds. This, in turn, facilitates frequent re-calibration of the interest rate model and re-optimization of the issuance throughout the budgetary year allows us to respond to the changes in the interest rate environment. We successfully demonstrate the utility of our approach by out-of-sample back-testing on the UK debt issuance data.     

Strong formulations for mixed integer programming: A survey

We attempt to motivate and survey recent research on the use of strong valid inequalities and reformulation to solve mixed integer programming problems.     

Large-scale mixed integer programming: Benders-type heuristics

The limited success of exact methods for solving integer programming problems has prompted the development of heuristic procedures which have performed surprisingly well in their search for near-optimal solutions. The present paper constitutes an attempt to generalize these procedures to the mixed integer case. The approach is based on the utilization of the Benders partitioning method (BPM) which separates the integer from the continuous variables. A number of alternatives are presented for integrating IP heuristics in the BPM thus alleviating its main limitation: the necessity of solving a sequence of integer master problems. The rationale and its usefulness are illustrated on large-scale applications from the fields of power systems and network flows.     

An integer programming model for hierarchical workforce scheduling problem

In this paper, an integer programming model for the hierarchical workforce problem under the compressed workweeks is developed. The model is based on the integer programming formulation developed by Billionnet [A. Billionnet, Integer programming to schedule a hierarchical workforce with variable demands, European Journal of Operational Research 114 (1999) 105–114] for the hierarchical workforce problem. In our model, workers can be assigned to alternative shifts in a day during the course of a week, whereas all workers are assigned to one shift type in Billionnet’s model. The main idea of this paper is to use compressed workweeks in order to save worker costs. This case is also suitable for the practice. The proposed model is illustrated on the Billionnet’s example problem and the obtained results are compared with the Billionnet’s model results.     

Using mixed integer programming to design employee rosters

This paper describes the problem of rostering a workforce so as to optimise a weighted sum of three criteria while satisfying several constraints. The rostering entailed deciding on a pattern of working days and breaks over a period of (typically) one year. Demand had to meet 24?hours each day and 365?days each year.It was possible to formulate this problem as a mixed integer program and, with some experimentation, solve it using an ‘off the shelf’ linear programming package. The results obtained are compared with rosters the client now uses.     

A mixed integer programming model for advanced planning and scheduling (APS)

This paper presents a mixed integer programming (MIP) model which succeeds in a system integration of the production planning and shop floor scheduling problems. The proposed advanced planning and scheduling (APS) model explicitly considers capacity constraints, operation sequences, lead times and due dates in a multi-order environment. The objective of the model is to seek the minimum cost of both production idle time and tardiness or earliness penalty of an order. The output of the model is operation schedules with order starting time and finish time. Numerical result shows that the suggested APS model can favorably produce optimal schedules.     

A mixed integer programming model of a multicohort single-species fishery

Many marine fisheries are under pressure from overfishing. Fisheriesmanagement is a complex process because of the need to considerthe interaction of the biological components of the fishery,the technical characteristics of the fishing fleet, and theeconomic aspects of the fishing industry. In this paper, a mixedinteger programming (MIP) model for determining the policy tomaximize the long-run economic benefit from a single-speciesmulticohort fishery is developed. The model takes account ofthe biological, technical, and economic characteristics of thefishery, using integer variables to model the fishing activities.An iterative procedure for solving the model using commercialMIP software is described, and the viability of this procedureis illustrated using data for the western mackerel fishery.