Mixed-integer Nonlinear Programming Formulation Research Articles

Unified Heat Exchanger Network Synthesis via a Stageless Superstructure

The existing mathematical programming-based methods for simultaneous heat exchanger network synthesis (HENS) have used either match-centric or stage-centric superstructures. Most works over the past four decades have primarily used various modifications of essentially two superstructures whose configurational limitations are well established. We revisit a novel, promising, but unsuccessful, exchanger-centric superstructure proposed by Huang and Karimi ( Chem. Eng. Sci. 2013, 98, 231−245). We modify the superstructure and the associated mixed-integer nonlinear programming formulation substantially, and develop an efficient algorithm for its solution. The superstructure simply assumes a pool of two-stream exchangers, to which hot and cold streams are assigned. Given a sufficiently large pool, it embeds all possible heat exchanger network (HEN) configurations (in contrast to previous superstructures) including repeated matches, cross flows, bypasses, etc., and allows multiple utilities. The novel HEN configu...

A Dichotomic Algorithm for Transportation Network and Land Use Problem

Redeveloping sites to accommodate housing or new economic activities is a major urban policy challenge. This problem belongs to the class of NP-hard problems. In this paper, we present an attractive mixed integer nonlinear programming formulation for the transportation network and land use problem. We first introduce a new useful nonlinear formulation of this challenging combinatorial optimization problem. Then, an alternative to considering a linearity of the constraints is to reformulate the problem as a new exact, compact discrete linear model. The problem is solved by our new algorithm and numerical results are presented for a number of test problems in academics instances.

A flower pollination algorithm for the double-floor corridor allocation problem

This research explores the double-floor corridor allocation problem (DFCAP), which deals with the optimal arrangement of departments over two floors and then place them along both sides against a corridor. This problem is a natural extension of the corridor allocation problem (CAP) to additional floors; the layout of each floor can be regarded as an approximately independent CAP. The DFCAP is commonly observed in manufacturing and service buildings. In this study, a mixed-integer programming formulation for the DFCAP is developed, and it is able to reduce to the classical CAP model. Then a novel flower pollination algorithm is provided, which is discretised using swap pair set approach to solve the considered DFCAP. In addition, to ameliorate the algorithm, three constructive heuristic rules are developed to produce a reasonably good initial population; meanwhile, a variable neighbourhood search structure is presented to prevent prematurity in arrival at a poor local solution. Finally, several instances for the DFCAP with a size of 9 ≤ n ≤ 80 are employed in the algorithms, as well as in mixed-integer non-linear programming (MINLP) formulations, which are solved with GUROBI 7.0.1. Moreover, the above-mentioned instances are utilized to show that the proposed algorithm performs better in comparison to the state-of-the-art optimization algorithms.

Mathematical model for supply chain design with time postponement

Projetar cadeia de suprimentos é uma importante decisão estratégica e seu impacto influencia diretamente na eficiência e no nível de serviço. O projeto se torna mais complexo quando o objetivo é minimizar o custo de distribuição e utilizar a postergação de tempo na cadeia de suprimentos. Os modelos matemáticos atualmente estudados na literatura de cadeia de suprimentos consideram vários atores. Entretanto, em problemas reais existem diferentes combinações desses atores, criando fluxos próprios de transportes e aumentando a complexidade da cadeia de suprimentos. Este artigo propõe um modelo matemático para projetar a cadeia de suprimentos com postergação de tempo a partir da programação não linear inteira mista para minimizar o custo total, considerando os custos de transportes, abertura de instalações e operacionais. O modelo permite a possibilidade de uma instalação híbrida, ou seja, dois tipos de instalações abertas no mesmo local, sendo uma importante oportunidade de redução de custos. Diferentes conjuntos de instâncias foram simulados para buscar a solução ótima e analisar o comportamento da cadeia de suprimentos em diferentes tamanhos de cenários, os quais foram resolvidos usando um solver comercial e suas performances foram estudadas. O modelo proposto apresenta viabilidade em seu uso para instâncias pequenas e médias com tempo computacional suficiente para auxílio no processo de tomada de decisão.

Price-and-verify: a new algorithm for recursive circle packing using Dantzig–Wolfe decomposition

Packing rings into a minimum number of rectangles is an optimization problem which appears naturally in the logistics operations of the tube industry. It encompasses two major difficulties, namely the positioning of rings in rectangles and the recursive packing of rings into other rings. This problem is known as the Recursive Circle Packing Problem (RCPP). We present the first dedicated method for solving RCPP that provides strong dual bounds based on an exact Dantzig--Wolfe reformulation of a nonconvex mixed-integer nonlinear programming formulation. The key idea of this reformulation is to break symmetry on each recursion level by enumerating one-level packings, i.e., packings of circles into other circles, and by dynamically generating packings of circles into rectangles. We use column generation techniques to design a "price-and-verify" algorithm that solves this reformulation to global optimality. Extensive computational experiments on a large test set show that our method not only computes tight dual bounds, but often produces primal solutions better than those computed by heuristics from the literature.

Reformulation, linearization, and a hybrid iterated local search algorithm for economic lot-sizing and sequencing in hybrid flow shop problems

We study an integrated economic lot-sizing and sequencing problem (ELSP) in the hybrid flow shop manufacturing setting with unlimited intermediate buffers in a finite planning horizon. The ELSP entails making two simultaneous decisions regarding (i) the manufacturing sequences of products, and (ii) their production quantity. The objective is to minimize the total cost, consisting of inventory holding and set-up costs. To solve this problem, we first develop a novel mixed-integer nonlinear programming (MINLP) model that improves an existing MINLP model in the literature. We then present a novel linearization technique that transforms these two MINLP models into effective mixed-integer linear programming (MILP) models. Additionally, we develop an effective algorithm that hybridizes the iterated local search algorithm with an approximate function. We conduct comprehensive experiments to compare the performance of MILPs+CPLEX with that of MINLPs+BARON. Additionally, our proposed algorithm is compared with four existing metaheuristic algorithms in the literature. Computational results demonstrate that our novel MINLP formulation and its linearized variant significantly improve the solvability and optimality gap of an existing MINLP formulation and its linearized variant. We also show that our new hybrid iterated local search algorithm substantially improves computational performance and optimality gap of the mathematical models and the existing algorithms in the literature, on large-size instances of the problem.

Packing circles into perimeter-minimizing convex hulls

We present and solve a new computational geometry optimization problem in which a set of circles with given radii is to be arranged in unspecified area such that the length of the boundary, i.e., the perimeter, of the convex hull enclosing the non-overlapping circles is minimized. The convex hull boundary is established by line segments and circular arcs. To tackle the problem, we derive a non-convex mixed-integer non-linear programming formulation for this circle arrangement or packing problem. Moreover, we present some theoretical insights presenting a relaxed objective function for circles with equal radius leading to the same circle arrangement as for the original objective function. If we minimize only the sum of lengths of the line segments, for selected cases of up to 10 circles we obtain gaps smaller than $$10^{-4}$$ using BARON or LINDO embedded in GAMS, while for up to 75 circles we are able to approximate the optimal solution with a gap of at most $$14\%$$ .

Algorithms for the power-p Steiner tree problem in the Euclidean plane

We study the problem of constructing minimum power-$p$ Euclidean $k$-Steiner trees in the plane. The problem is to find a tree of minimum cost spanning a set of given terminals where, as opposed to the minimum spanning tree problem, at most $k$ additional nodes (Steiner points) may be introduced anywhere in the plane. The cost of an edge is its length to the power of $p$ (where $p\geq 1$), and the cost of a network is the sum of all edge costs. We propose two heuristics: a ``beaded" minimum spanning tree heuristic; and a heuristic which alternates between minimum spanning tree construction and a local fixed topology minimisation procedure for locating the Steiner points. We show that the performance ratio $\kappa$ of the beaded-MST heuristic satisfies $\sqrt{3}^{p-1}(1+2^{1-p})\leq \kappa\leq 3(2^{p-1})$. We then provide two mixed-integer nonlinear programming formulations for the problem, and extend several important geometric properties into valid inequalities. Finally, we combine the valid inequalities with warm-starting and preprocessing to obtain computational improvements for the $p=2$ case.

Glider Routing and Trajectory Optimisation in disaster assessment

In this paper, we introduce the Glider Routing and Trajectory Optimisation Problem (GRTOP), the problem of simultaneously finding optimal routes and trajectories for a fleet of gliders with the aim of surveying a set of locations. We propose a novel Mixed-Integer Nonlinear Programming (MINLP) formulation for the GRTOP, which optimises the routes as well as the trajectories along these routes, while flight dynamics is modelled as constraints. We avoid solving a non-convex problem by linearising the gliders’ flight dynamics, converting the proposed MINLP into a Mixed-Integer Second-order Cone Programming (MISOCP) problem. To allow for a more tractable formulation, the dynamical constraints are relaxed and a penalisation is added to the objective function. Several different discretisation techniques are compared. The formulation is tested on instances inspired by risk maps of flooding-prone cities across the UK and on 180 randomly generated instances.

Strategic placements of PMUs for power network observability considering redundancy measurement

This paper proposes a mixed integer nonlinear programming (MINLP) formulation with binary-valued variables for the optimal installation of the phasor measurement units (PMUs) in a power network. The MINLP framework has been solved by a two-phase branch-and-bound algorithm (BBA) for the optimal PMU placement (OPP) problem. The present work deals with the solution of the OPP problem without considering the radial buses including in the optimal solution. PMU is pre-assigned to each bus connected to a radial bus such that the radial buses are excluded from the list of potential PMU locations. The programming technique is used to determine the optimum number of PMUs and their locations to make the interconnected power network completely observable. The proposed BBA may provide multiple solutions at the lowest objective value, each time the optimizer routine restarts. Therefore, the obtained optimal solutions are being ranked regarding the measurement redundancy index. The two-phase branch-and-bound algorithm is applied to IEEE standard test systems as well as to New England 39- bus and a northern regional power grid 246- bus system. The proposed algorithm has been compared with the existing programming techniques in the recent literature. The simulation results obtained by the proposed BBA indicate the effectiveness and the efficiency to detect the desired target.

A Three-level Location-inventory Problem With Perishable Product

This article studies a three-level multi-product location-inventory problem in which location, allocation and inventory decisions are taken in a three-level supply chain including plants, distribution centers and retailers. The direct shipment from plant to retailer is allowed and two-echelon inventory and demand correlation among retailers are considered in this supply chain network. A mixed integer nonlinear programming formulation is presented to describe the problem and trade off the costs of setup, transportation, inventory and deterioration. Then it is transformed to a conic quadratic mixed integer programming formulation and solved with a standard optimization software packages-CPLEX. In addition, we demonstrate that the distribution network allowing for direct shipment from plant to retailer is more optimal than the distribution network ignoring it in the three-level supply chain network with perishable product.

A Linear Model for Dynamic Generation Expansion Planning Considering Loss of Load Probability

Computation of Loss of Load Probability (LOLP) is a challenge in Generation Expansion Planning (GEP) problem. This paper presents a dynamic GEP model considering LOLP as a reliability criterion. The objective function of the proposed dynamic GEP model includes the discounted total costs of investment cost, operation cost, and maintenance cost. The proposed LOLP constrained GEP problem is formulated as a mixed integer nonlinear programming (MINLP) problem. The MINLP formulation of GEP problem is then converted to a mixed integer linear programming (MIP) problem by applying several approximations to LOLP constraint. The utilized approximations are divided into two clusters for low and high order outages. A test case containing 12 types of installed and 5 types of candidate plants with a 14-year planning horizon is used to validate the efficiency of the proposed dynamic GEP problem. The developed MIP-based GEP model is solved using CPLEX solver and the obtained results are compared with the conventional LOLP-constrained GEP model.

Optimal sizing and planning of onsite generation system for manufacturing in Critical Peaking Pricing demand response program

Onsite electricity generation system in manufacturing has been traditionally considered an effective backup energy source to support the manufacturing operations when external power is not available due to natural disasters and/or power blackouts. Recently, with the increasing concerns of climate change and environmental protection, the contribution of using onsite generation system (OGS) to the manufacturing end use customers when they enroll in specific electricity demand response programs has also been gradually recognized. In this paper, we investigate the cost-effective OGS sizing problem for manufacturing practitioners when participating in Critical Peaking Pricing (CPP) demand response program. A Mixed Integer Non-Linear Programming (MINLP) formulation is proposed to identify the optimal size and utilization strategy of the OGS, as well as the corresponding production plan of the manufacturing system to minimize the overall energy related cost. Linearization strategy and metaheuristic algorithm are discussed for solving the proposed formulation with a reasonable computational cost and a good solution quality. A case study based on a real auto component manufacturing system and an existing CPP program is implemented to examine the effects of the proposed model. The results show that when utilizing the OGS appropriately sized, the total electricity related cost of the manufacturing system can be significantly reduced when participating in the CPP program.

Vehicle routing problem in a kanban controlled supply chain system considering cross-docking strategy

This paper considers an inventory-routing problem in a distribution network in which kanban is used as a means to implement just-in-time strategy. There are a set of part suppliers and cross-docks in this network, which provide parts for an assembly plant. A novel mixed integer non-linear programming formulation is developed for the problem in which the optimum number of kanbans is determined considering inventory and transportation cost. In the other word, this research aims at reducing the level of inventory and transportation cost in a kanban controlled supply chain system, in which the number of kanbans is to be determined optimally considering demand from suppliers and geographical distribution of them. A fleet of vehicles is applied to transport kanbans from suppliers to the assembly plant via two transportation strategies: direct shipment and shipment through cross-docks (indirect shipment). In the second strategy, it is possible to have routes between suppliers. The proposed problem is NP-hard based on literature, thus a memetic algorithm is introduced to solve it. Solving several examples reveals that the solving method significantly outperforms GAMS/CPLEX in reducing objective value and computational time.

Efficient planning of crude oil supplies through long-distance pipelines

This work introduces a mixed-integer nonlinear programming (MINLP) formulation for the efficient planning of crude oil supplies to a major refinery, through a long-distance pipeline. The proposed approach combines the potentials of slot-based and general precedence continuous-time representations to simultaneously address two problems that are typically decoupled: pipeline transportation and crude oil blending. General precedence sequencing variables coordinate incoming/outgoing flows to/from every tank, while crude oil batches are traced into the pipeline following a slot-based scheme. The model precisely monitors key component concentrations, keeping oil properties within admissible ranges. Since the proposed formulation is nonconvex, an efficient solution strategy is followed and a tailored relaxation is subsequently solved to assess the quality of the solutions achieved. Results show that the model is able to find very efficient solutions to real-world case studies in modest CPU times.

Optimal Conductor Size Selection in Radial Distribution Networks Using a Mixed-Integer Non-Linear Programming Formulation

In this paper a mixed-integer non-linear programming formulation for optimal conductor selection in radial distribution networks is proposed. The objective function in this problem corresponds to the minimization of power losses and costs of investment in conductors. A typical set of constraints corresponding to the operative conditions in distribution systems, as power flow balance, voltage regulation, thermal capacity and telescopic conductors distribution, among others, are employed. Three different demand scenarios are considered to evaluate their impacts in the final conductor selection. The proposed mathematical model is solved using the general algebraic modeling system (GAMS) and DICOPT solver. Two radial distribution networks with 8 and 27 nodes, respectively, are employed to verify the general performance of the mathematical model proposed.

Computing non-stationary (s, S) policies using mixed integer linear programming

This paper addresses the single-item single-stocking location non-stationary stochastic lot sizing problem under the (s, S) control policy. We first present a mixed integer non-linear programming (MINLP) formulation for determining near-optimal (s, S) policy parameters. To tackle larger instances, we then combine the previously introduced MINLP model and a binary search approach. These models can be reformulated as mixed integer linear programming (MILP) models which can be easily implemented and solved by using off-the-shelf optimization software. Computational experiments demonstrate that optimality gaps of these models are less than 0.3% of the optimal policy cost and computational times are reasonable.

A global MINLP approach to symbolic regression

Symbolic regression methods generate expression trees that simultaneously define the functional form of a regression model and the regression parameter values. As a result, the regression problem can search many nonlinear functional forms using only the specification of simple mathematical operators such as addition, subtraction, multiplication, and division, among others. Currently, state-of-the-art symbolic regression methods leverage genetic algorithms and adaptive programming techniques. Genetic algorithms lack optimality certifications and are typically stochastic in nature. In contrast, we propose an optimization formulation for the rigorous deterministic optimization of the symbolic regression problem. We present a mixed-integer nonlinear programming (MINLP) formulation to solve the symbolic regression problem as well as several alternative models to eliminate redundancies and symmetries. We demonstrate this symbolic regression technique using an array of experiments based upon literature instances. We then use a set of 24 MINLPs from symbolic regression to compare the performance of five local and five global MINLP solvers. Finally, we use larger instances to demonstrate that a portfolio of models provides an effective solution mechanism for problems of the size typically addressed in the symbolic regression literature.

Optimal Allocation of LDOs and Decoupling Capacitors within a Distributed On-Chip Power Grid

Parallel on-chip voltage regulation, where multiple regulators are connected to the same power grid, has recently attracted significant attention with the proliferation of small on-chip voltage regulators. In this article, the number, size, and location of parallel low-dropout (LDO) regulators and intentional decoupling capacitors are optimized using mixed integer non-linear programming formulation. The proposed optimization function concurrently considers multiple objectives such as area, power noise, and overall power consumption. Certain objectives are optimized by putting constraints on the other objectives with the proposed technique. Additional constraints have been added to avoid the overlap of LDOs and decoupling capacitors in the optimization process. The results of an optimized LDO allocation in the POWER8 chip is compared with the recent LDO allocation in the same IBM chip in a case study where a 20% reduction in the noise is achieved. The results of the proposed multi-criteria objective function under a different area, power, and noise constraints are also evaluated with a sample ISPD’11 benchmark circuits in another case study.

Multiobjective Dynamic VAR Planning Strategy With Different Shunt Compensation Technologies

High concentrations of induction motor loads can impose stress on transmission and distribution systems, leading to voltage instability in some situations. Properly sized and coordinated reactive power sources will provide for improved operation. We present a strategy for finding an optimal mix (type, size, and location) of dynamic shunt reactive compensation devices. The planning strategy is subject to satisfying steady-state, dynamic and transient performance criteria such as fault-induced delayed voltage recovery limits, as well as criteria related to single ( N –1) contingency and load disturbance events. Shunt reactive power compensation devices considered include mechanically switched capacitor banks, static reactive power compensators and static synchronous compensators. The proposed strategy employs a large number of multitimescale time-domain simulations suitable for use with high performance computing clusters and a genetic algorithm to solve the mixed-integer nonlinear programming formulation using parallel computation capabilities. The method is applied to a New England IEEE 39-bus system with assumed high penetration of induction motors. A comprehensive study shows that performance enhancement and significant cost reduction can be achieved using an optimum combination of various shunt compensator technologies.