Can machine learning help in solving the pallet loading optimization problem?

Solving Integer Linear Programming (ILP) models generally lies in the category of NP-hard problems. Therefore, as the size of ILP models grows, the efficiency of exact algorithms for solving the models reduced significantly and for large models it is not possible to have the result. Genetic Algorithm (GA) is a metaheuristic method capable of adjusting and redesigning parameters and operations according to the characteristics of ILP models. Still GA has huge search space for large models and parallelization is a suitable technique to tackle this problem. This paper presents a scalable parallel GA to solve large ILP models derived from behavioral synthesis of digital circuits. We show that although models have non-binary variables, only binary variables are sufficient for coding chromosomes. We also use ”unknown” values for some genes to decrease the likelihood of inconsistency in the encoded constraints. Our experiments verify the efficiency and scalability of the proposed algorithm on multicore platforms. The proposed method outperforms IBM ILOG CPLEX 12.6 and MI-LXPM algorithm where the ILP models include 550 to 2258 int / binary decision variables. Also, the results indicate that the saturation point of using parallel processing elements for solving the large ILP models is at least 60.

A bi-criteria moving-target travelling salesman problem under uncertainty

Over the course of the last decade, there have been several improvements in the performance of Integer Linear Programming (ILP) and Boolean Satisfiability (SAT) solvers. These improvements have encouraged the application of SAT and ILP techniques in modeling complex engineering problems. One such problem is the Clustering Problem in Mobile Ad-Hoc Networks (MANETs). The Clustering Problem in MANETs consists of selecting the most suitable nodes of a given MANET topology as clusterheads, and ensuring that regular nodes are connected to clusterheads such that the lifetime of the network is maximized. This paper proposes the development of an improved ILP formulation of the Clustering Problem. Additionally, various enhancements are implemented in the form of extensions to the improved formulation, including the establishment of intra-cluster communication, multihop connections and the enforcement of coverage constraints. The improved formulation and enhancements are implemented in a tool designed to visually create network topologies and cluster them using state-of-the art Generic ILP and SAT solvers. Through this tool, feasibility of using the proposed formulation and enhancements in a real-life practical environment is assessed. It is observed that the Generic ILP solvers, CPLEX, and SCIP, are able to handle large network topologies, while the 0-1 SAT-based ILP solver, BSOLO, is effective at handling the smaller scale networks. It is also observed that while these enhanced formulations enable the generation of complex network solutions, and are suitable for small scale networks, the time taken to generate the corresponding solution does not meet the strict requirements of a practical environment.

Static routing and wavelength assignment (RWA) is usually formulated as an optimization problem with the objective of minimizing wavelength channel usage or maximizing the number of connections established. In this paper, we formulate it as a priority and maximum revenue (MR) based optimization problem, which we believe will be more appealing to network operators. We describe an integer linear programming (ILP) solution which can be used to find the optimal solution for small networks. We also describe a simplified ILP (SILP) solution which can be used for both small and large networks. By means of computer simulations, the performance of the two ILP solutions are compared with the sequential R WA (SR) algorithm which we devise to mimic those algorithms commonly used for solving the maximum lightpath establishment (MLE) problem. Our results show that SILP outperforms SR significantly, and yields solutions close to the optimal solution obtained from ILP.

A method is proposed to estimate confidence intervals for the solution of integer linear programming (ILP) problems where the technological coefficients matrix and the resource vector are made up of random variables whose distribution laws are unknown and only a sample of their values is available. This method, based on the theory of order statistics, only requires knowledge of the solution of the relaxed integer linear programming (RILP) problems which correspond to the sampled random parameters. The confidence intervals obtained in this way have proved to be more accurate than those estimated by the current methods which use the integer solutions of the sampled ILP problems.

Cloud-integrated fiber-wireless (FiWi) networks inheriting advantages of optical and wireless access networks have a broad prospect in the future. As various component failures may occur in cloud-integrated FiWi networks, survivability is becoming one of the key important issues. It is necessary to provide survivability strategies for cloud-integrated FiWi networks. Hence, this paper mainly focuses on the survivability of cloud-integrated FiWi networks against multiple fibers failure. Firstly, in this paper, a novel integer linear programming (ILP) solution is proposed to tolerate the failure of multiple distribution fibers with capacity and coverage constraints in the context of urban area. Then, considering the complexity of ILP models, an efficient heuristic scheme is proposed, in order to get the approximate solutions of ILP. Simulation results and analysis give the configurations of optical network units (ONUs) and wireless routers with different constraints and show the network coverage of clients for different number of ONUs and wireless routers with ILP solution and heuristic approach, respectively.

The Network Function Virtualization (NFV) is very promising for efficient provisioning of network services and is attracting a lot of attention. NFV can be implemented in commercial off-the-shelf servers or Physical Machines (PMs), and many network services can be offered as a sequence of Virtual Network Functions (VNFs), known as VNF chains. Furthermore, many existing network devices (e.g., switches) and collocated PMs are underutilized or over-provisioned, resulting in low power-efficiency. In order to achieve more energy efficient systems, this work aims at designing the placement of VNFs such that the total power consumption in network nodes and PMs is minimized, while meeting the delay and capacity requirements of the foreseen demands. Based on existing switch and PM power models, we formulate an Integer Linear Programming (ILP) model to find the optimal solution. We also propose a heuristic based on the concept of Blocking Islands (BI), and a baseline heuristic based on the Betweenness Centrality (BC) property of the graph. Both heuristics and the ILP solutions have been compared in terms of total power consumption, delay, demands acceptance rate, and computation time. Our simulation results suggest that BI-based heuristic is superior compared with the BC-based heuristic, and very close to the optimal solution obtained from the ILP in terms of total power consumption and demands acceptance rate. Compared to the ILP, the proposed BI-based heuristic is significantly faster and results in 22% lower end-to-end delay, with a penalty of consuming 6% more power in average.

In order for transport networks to cost-effectively provide higher capacity, it is expected that channel bit-rates beyond 100 Gb/s will be accomplished by resorting to a flexible WDM grid with variable channel spacing. Among the implications of this concept is the need for planning tools that fully exploit the additional degrees of freedom enabled by a flexible grid to further optimize network cost and spectral efficiency. This paper proposes an optimization framework to minimize the transponder and regenerator deployment cost in a translucent WDM network featuring channel bit-rates of 40, 100 and 400 Gb/s and multiple transmission formats per bit-rate, each characterized by its own spectral width, optical reach and cost properties. Firstly, we formulate the problem via a novel Integer Linear Programming (ILP) model, whose resolution finds the optimal (cheapest) feasible network configuration. Secondly, we propose an efficient heuristic called Narrowest First-Iterative Cost Reduction (NF-ICR) to handle network scenarios for which solving the ILP entails an unreasonable computational burden. The NF-ICR heuristic is shown to provide tight optimality bounds where the benchmark given by the ILP solution is attainable. For larger networks, we show that the use of a flexible grid and multiple format options for each bit-rate results in around 10% less cost in transponders and regenerators for metro networks, and a substantial increase in the total traffic load supported by the network. We also conclude that a distinction emerges between metro/regional scenarios and long-haul networks with long paths, wherein the shorter transparent reach of 400 Gb/s channels drives up the cost due to extra regeneration, favoring the use of parallelized solutions of lower bit-rate channels.

Grid-based location problems (GBLPs) can be used to solve location problems in business, engineering, resource exploitation, and even in the field of medical sciences. To solve these decision problems, an integer linear programming (ILP) model is designed and developed to provide the optimal solution for GBLPs considering fixed cost criteria. Preliminary results show that the ILP model is efficient in solving small to moderate-sized problems. However, this ILP model becomes intractable in solving large-scale instances. Therefore, a decomposition heuristic is proposed to solve these large-scale GBLPs, which demonstrates significant reduction of solution runtimes. To benchmark the proposed heuristic, results are compared with the exact solution via ILP. The experimental results show that the proposed method significantly outperforms the exact method in runtime with minimal (and in most cases, no) loss of optimality.

This study presents a novel integer linear programming (ILP) model and a modified branch and bound (MBB) algorithm to minimize the total project inventory cost for the project material requirements planning (PMRP) problem. Twelve two-step lot-sizing rules are compared with the ILP model based on an experiment involving the single-material and two-material requirement problems. According to the results of the experiment, the ILP model outperforms the twelve two-step lot-sizing rules with an average reduction in the total project inventory cost of about 28.54%. This study further assesses the performance of the twelve two-step lot-sizing rules with respect to several important parameters of the PMRP problem. The results indicate that the performance of the twelve two-step lot-sizing rules is significantly affected by the rule itself, the network density, and the value of the percentage of activities with material requirements (PERC). On the other hand, the effectiveness of the MBB algorithm is also evaluated by comparing it with the ILP model. The results of the comparison reveal that the solution obtained by the MBB algorithm is very close to the optimal solution, and that the proposed algorithm does not consume much computational time. The approaches presented in this study, in which project scheduling and lot sizing are considered simultaneously, could provide project managers with useful tools for making better PMRP decisions.

Recently there has been a lot of focus on developing deep learning models for symbolic reasoning tasks. One such task involves solving combinatorial problems, which can be viewed as instances of a constraint satisfaction problem, albeit with unknown constraints. The task of the neural model is then to discover the unknown constraints using the training data consisting of many solved instances. There are broadly two approaches for learning the unknown constraints: the first approach creates a purely neural model that maps an input puzzle directly to its solution, thereby representing the constraints implicitly in the model’s weights [1, 2, 3], and the second approach invokes a symbolic reasoner, such as an Integer Linear Program (ILP) solver, learning the constraints explicitly in the solver’s language, e.g., linear inequalities for an ILP solver [4, 5]. In this chapter, we discuss about both the implicit and explicit approaches. In the implicit approach, we review three neural architectures for solving combinatorial problems, viz., Neural Logic Machines (NLM) [2], Recurrent Relational Networks (RRN) [1], and its extensions proposed by Nandwani et al. [3] to tackle output space invariance (solving 16×16 sudoku after training on 9×9 sudokus). Under the second approach having explicit representation of constraints, we present two methods (CombOptNet [5] and ILP–Loss[4]) for end-to-end training of a Neural-ILP architecture: a neuro–symbolic model where a neural perception layer is followed by an ILP layer to represent reasoning. CombOptNet proposed by Paulus et al. [5] trains slowly owing to a call to an ILP solver in each learning iteration. On the other hand, ILP–Loss proposed by Nandwani et al. [4] is solver–free during training and thus much more scalable. In the end, we present specific experiments from [3] and [4] that compare the different methods discussed in this chapter.

The proper mapping of an application on a multi-core platform and the scheduling of its tasks are key elements to achieve the maximum performance. In this article, a novel hybrid approach based on integrating the Logic-Based Benders Decomposition (LBBD) principle with a pure Integer Linear Programming (ILP) model is introduced for mapping applications described by Directed Acyclic Graphs (DAGs) on platforms consisting of heterogeneous cores. The LBBD approach combines two optimization techniques with complementary strengths, namely ILP and Constraint Programming (CP), and is employed as a cut generation scheme. The generated constraints are utilized by the ILP model to cut possible assignment combinations aiming at improving the solution or proving the optimality of the best-found one. The introduced approach was applied both on synthetic DAGs and on DAGs derived from real applications. Through the proposed approach, many problems were optimally solved that could not be solved by any of the above methods (ILP, LBBD) alone within a time limit of 2 hours, while the overall solution time was also significantly decreased. Specifically, the hybrid method exhibited speedups equal to 4.2× for the synthetic instances and 10× for the real-application DAGs over the LBBD approach and two orders of magnitude over the ILP model.

This paper proposes a test scheduling method for stuck-at faults in a CHAIN interconnect, which is an asynchronous on-chip interconnect architecture, with scan ability. Special data transfer which is permitted only during test, is exploited to realize a more flexible test schedule than that of a conventional approach. Integer linear programming (ILP) models considering such special data transfer are developed according to the types of modules under test in a CHAIN interconnect. The obtained models are processed by using an ILP solver. This framework can not only obtain optimal test schedules but also easily introduce additional constraints such as a test power budget. Experimental results using benchmark circuits show that the proposed method can reduce test application time compared to that achieved by the conventional method.

ABSTRACTIn this article we present a game that can be used as a tool to educate students and managers on the issues in supply chain (SC), inventory management. The game has a bilevel demand with one level during regular times and another during sale times. The game could be played in two modes (independence and cooperation) and has been field tested in engineering and business classes. Players developed an appreciation for fluctuating demand and its impact on the costs and performance of a SC. They also learned the benefits and a monetary evaluation approach for cooperation. Our statistical analysis revealed that, as the game progressed, the performance of the teams improved. We present an integer linear programming (ILP) model to evaluate the performance of the teams. Because it is a post facto analysis, while the game is played without knowing the materialized retailer demand for the period, the ILP solution is not a tight lower bound on the total cost of the SC. However, it could be used to compare performance across teams. As an alternative, we also present a possible distribution of total SC costs that could be used as another reference without actually solving an ILP.

This paper presents a technique for the optimal design of signal processing system. This technique optimizes area of the design subject to throughput constraints. An Integer Linear Programming (IP) model is formed and mathematical modeling approach is used for solving the constrained optimization problem. The approach adds more sample points in the discrete design space by allowing hybrid architectures. The solution from the IP solver is passed to a code generator engine, which generates RTL Verilog code of the final design.