Exact Algorithm Research Articles

Bounded mixed batch scheduling with job release dates and rejection

This paper investigates a bounded mixed batch scheduling problem with job release dates and rejection. The machine processes a batch containing several jobs that their number does not exceed the machine capacity. For the case with a certain release date, we present a polynomial-time exact algorithm. For the case with a constant number of release dates, a pseudo-polynomial-time exact algorithm is proposed. For the general problem, we provide a 2-approximation algorithm and a polynomial-time approximation scheme.

High-performance deep spiking neural networks with 0.3 spikes per neuron

Communication by rare, binary spikes is a key factor for the energy efficiency of biological brains. However, it is harder to train biologically-inspired spiking neural networks than artificial neural networks. This is puzzling given that theoretical results provide exact mapping algorithms from artificial to spiking neural networks with time-to-first-spike coding. In this paper we analyze in theory and simulation the learning dynamics of time-to-first-spike-networks and identify a specific instance of the vanishing-or-exploding gradient problem. While two choices of spiking neural network mappings solve this problem at initialization, only the one with a constant slope of the neuron membrane potential at threshold guarantees the equivalence of the training trajectory between spiking and artificial neural networks with rectified linear units. For specific image classification architectures comprising feed-forward dense or convolutional layers, we demonstrate that deep spiking neural network models can be effectively trained from scratch on MNIST and Fashion-MNIST datasets, or fine-tuned on large-scale datasets, such as CIFAR10, CIFAR100 and PLACES365, to achieve the exact same performance as that of artificial neural networks, surpassing previous spiking neural networks. Our approach accomplishes high-performance classification with less than 0.3 spikes per neuron, lending itself for an energy-efficient implementation. We also show that fine-tuning spiking neural networks with our robust gradient descent algorithm enables their optimization for hardware implementations with low latency and resilience to noise and quantization.

The Price of Justified Representation

In multiwinner approval voting, the goal is to select a k -member committee based on voters’ approval ballots. A well-studied concept of proportionality in this context is the justified representation (JR) axiom, which demands that no large cohesive group of voters remains unrepresented. However, the JR axiom may conflict with other desiderata, such as social welfare (the number of approvals obtained by committee members) or coverage (the number of voters who approve at least one committee member). In this article, we investigate the price of imposing the JR axiom (as well as the more demanding extended justified representation (EJR) axiom) on social welfare and coverage. Our approach is twofold: we derive worst-case bounds on the loss of welfare/coverage caused by imposing JR, and study the computational complexity of finding committees with high welfare that provide JR (obtaining hardness results, approximation algorithms, and an exact algorithm for one-dimensional preferences).

Efficient and Robust From-Point Visibility.

This article presents two from-point visibility algorithms: one aggressive and one exact. The aggressive algorithm efficiently computes a nearly complete visible set, with the guarantee of finding all triangles of a front surface, no matter how small their image footprint. The exact algorithm starts from the aggressive visible set and finds the remaining visible triangles efficiently and robustly. The algorithms are based on the idea of generalizing the set of sampling locations defined by the pixels of an image. Starting from a conventional image with one sampling location at each pixel center, the aggressive algorithm adds sampling locations to make sure that a triangle is sampled at all the pixels it touches. Thereby, the aggressive algorithm finds all triangles that are completely visible at a pixel regardless of geometric level of detail, distance from viewpoint, or view direction. The exact algorithm builds an initial visibility subdivision from the aggressive visible set, which it then uses to find most of the hidden triangles. The triangles whose visibility status is yet to be determined are processed iteratively, with the help of additional sampling locations. Since the initial visible set is almost complete, and since each additional sampling location finds a new visible triangle, the algorithm converges in a few iterations.

Shocks in the warm neutral medium

Context. Atomic and molecular line emissions from shocks may provide valuable information on the injection of mechanical energy into the interstellar medium (ISM), the generation of turbulence, and the processes of phase transition between the warm neutral medium (WNM) and the cold neutral medium (CNM). Aims. In this series of papers, we investigate the properties of shocks propagating in the WNM. Our objective is to identify the tracers of these shocks, use them to interpret ancillary observations of the local diffuse matter, and provide predictions for future observations. Methods. Shocks propagating in the WNM are studied using the Paris-Durham shock code, a multi-fluid model built to follow the thermodynamical and chemical structures of shock waves at steady-state in a plane-parallel geometry. The code, designed to take into account the impact of an external radiation field, is updated to treat self-irradiated shocks at intermediate (30 < VS < 100 km s−1) and high velocity (VS ⩾ 100 km s−1), which emit ultraviolet (UV), extreme-ultraviolet (EUV), and X-ray photons. The couplings between the photons generated by the shock, the radiative precursor, and the shock structure are computed self-consistently using an exact radiative-transfer algorithm for line emission. The resulting code is explored over a wide range of parameters (0.1 ⩽ nH ⩽ 2 cm−3, 10 ⩽ VS ⩽ 500 km s−1, and 0.1 ⩽ B ⩽ 10 μG), which covers the typical conditions of the WNM in the solar neighborhood. Results. The explored physical conditions lead to the existence of a diversity of stationary magnetohydrodynamic solutions, including J-type, CJ-type, and C-type shocks. These shocks are found to naturally induce phase transition between the WNM and the CNM, provided that the postshock thermal pressure is higher than the maximum pressure of the WNM and that the maximum density allowed by magnetic compression is greater than the minimum density of the CNM. The input flux of mechanical energy is primarily reprocessed into line emissions from the X-ray to the submillimeter domain. Intermediate- and high-velocity shocks are found to generate a UV radiation field that scales as VS3 for VS < 100 km s−1 and as VS2 at higher velocities, and an X-ray radiation field that scales as VS3 for VS ⩾ 100 km s−1. Both radiation fields may extend over large distances in the preshock depending on the density of the surrounding medium and the hardness of the X-ray field, which is solely driven by the shock velocity. Conclusions. This first paper presents the thermochemical trajectories of shocks in the WNM and their associated spectra. It corresponds to a new milestone in the development of the Paris-Durham shock code and a stepping stone for the analysis of observations that will be carried out in forthcoming works.

Study on dynamic scheduling method of airport refueling vehicles based on DQN

Aiming at the low utilization rate of airport refueling vehicles and long solution time of exact algorithm caused by the uncertainty of actual flight time, a deep Q network dynamic scheduling method for refueling vehicles combining with the multi-objective deep reinforcement learning framework was proposed. Firstly, an optimization model is established to maximize the on-time rate of refueling tasks and the average proportion of idle vehicles. Then, the five state features that measure the current state of the vehicle are designed as inputs to the network. According to the two objectives, the two scheduling strategies are proposed as the action space so that the algorithm can generate the dynamic scheduling scheme based on the dynamic flight data in real time. Finally, the dynamic scheduling model for airport refueling vehicles is solved, and the effectiveness and real-time performance of the algorithm are verified by different scale examples. The results show that the average number of on-time refueling tasks per day is 9.43 more than that via manual scheduling, and the average working time of vehicles is reduced by 57.6 minutes, which shows the excellent ability of the present method in solving the dynamic scheduling problem of refueling vehicles.

A Fast Exact Algorithm for Computing the Hypervolume Contributions in 4-D Space

A Fast Exact Algorithm for Computing the Hypervolume Contributions in 4-D Space

Enhanced Dynamic Programming Approaches for Efficient Solutions to the Traveling Salesman Problem

This study aims to enhance dynamic programming techniques for efficiently solving the Traveling Salesman Problem, a fundamental combinatorial optimization challenge. Given its NP-hard classification, traditional exact algorithms become computationally infeasible as the problem size increases. The research revisits foundational dynamic programming principles, notably the Held-Karp algorithm, and identifies existing limitations. The study begins with a comprehensive literature review, followed by an analysis of the dynamic programming challenges specific to TSP. Novel algorithms are then developed, implemented, and rigorously tested against benchmark instances. Performance evaluation is conducted using metrics such as execution time, memory usage, and solution optimality across different problem sizes. Results demonstrate significant improvements in efficiency and scalability, with enhanced algorithms achieving optimal solutions in reduced time and computational resource usage. However, the exponential growth in complexity remains a challenge for larger instances. The study concludes with recommendations for future research, focusing on further algorithmic refinements and exploring hybrid approaches to address large-scale TSPs.

Managing oversaturation in BRT corridors: A new approach of timetabling for resilience enhancement using a tailored integer L-shaped algorithm

Bus rapid transit (BRT) is a high-capacity public transport system that typically operates along urban transit corridors with dense travel demand. Maintaining the efficiency and stability of the BRT is paramount for daily transport operations. Owing to the difficulty in ensuring an exclusive right-of-way along the entire route, stochastic congestion events may occur resulting from road segments without dedicated BRT lanes. This may lead to volatility in travel time and resulting in passenger stranding, determined as common-case disruptions in this study. These common-case disruptions frequently occur in the daily operation of oversaturated BRT routes. To manage and mitigate their negative impacts, a novel timetabling problem for enhancing the resilience of a BRT system was proposed to assess the ability of the system to withstand and recover from these disruptions. We formulated the problem as a two-stage stochastic mixed-integer optimization model and designed an exact algorithm based on a tailored integer L-shaped method. We then analyzed the structural properties of our model and developed several acceleration techniques to further improve the efficiency of the algorithm. The computational results show that the proposed algorithm outperforms the commercial solver in large-scale instances and can provide near-optimal solutions when the commercial solver is invalid. Besides, external comparisons with dynamic programming also demonstrate the superiority of the proposed algorithm in solution efficiency. Compared with the benchmark timetabling problem, which aims to reduce passenger waiting time, the proposed method can efficiently reduce the duration of the oversaturation period by 35.3%.

Metabolic and Bariatric Surgery Outcomes in African American Patients: a Single Institution Experience.

The incidence of obesity in African Americans (AAs) is higher than in non-AA in the USA. Previous studies using large national databases report that AA patients have worse outcomes than non-AA patients. To assess perioperative outcomes among AA patients after MBS at a center of excellence (COE) that serves a large, diverse patient population. University Hospital METHODS: A retrospective analysis was performed on patients undergoing MBS between 2010 and 2020 at our two accredited MBSAQIP (Metabolic and Bariatric Surgery Accreditation and Quality Improvement Program) COEs where the AA population makes up over 35% of the population. Preoperative variables were compared using unpaired t-test or chi-squared test where appropriate. Thirty-day outcomes were compared following propensity score matching (exact algorithm) of demographics and comorbidities. Overall, 5742 patients (AA = 2058, 36%) had Roux-en-Y gastric bypass (AA = 1028, 26%) or sleeve gastrectomy (AA = 1030, 27%). AA patients were more often female (90.2% vs. 80.2%, p < 0.001) and had higher rates of hypertension (56.3% vs. 47.8%, p < 0.001), while non-AA patients had higher rates of hyperlipidemia (27.3% vs. 20.7%, p < 0.001) and obstructive sleep apnea (41.2% vs. 37.1%, p = 0.0024). Matched data showed that AA patients had higher rates of pulmonary embolism (PE) (0.3% vs. 0.1%, p = 0.020) and more emergency department visits (7.0% vs. 5.1%, p = 0.012) but no differences in mortality, readmission, reintervention, or reoperation rates. In a diverse area, AA patients who underwent MBS had similar perioperative outcomes compared to non-AA patients except that they experienced higher rates of PE. They also experienced higher rates of emergency department visits but had similar readmission rates.

Optimization of machine schedules and vehicle routes in the mobile additive manufacturing mode

Online shopping giant Amazon proposes to use 3D printers mounted on vehicles to print customer purchases on the fly and deliver them immediately. Because of the overlap between production and transportation processes, such a mobile manufacturing mode offers potential savings in lead times and inventory costs. However, the decisions associated with the two processes are intertwined, making it challenging to achieve satisfactory machine schedules and vehicle routes. To address the situation and fully understand the benefits of the mode, we study a related optimisation problem of finding routes for vehicles and production schedules for 3D printers mounted on them with minimum travel costs. First, we construct a mixed-integer programming model to formulate the problem. Then, we develop an exact algorithm to solve it. Specifically, we propose two propositions about the optimal production schedule of machines in the route and develop a labelling algorithm for the pricing problem. And we devise a problem-specific heuristic labelling algorithm to speed up column generation. We use two sizes of instances with five problem parameter settings (including 290 instances) to validate the effectiveness of the algorithm. In addition, we provide some managerial insights for decision-makers by comparing the novel mode with the traditional manufacturing mode.

A matheuristic for the resource-constrained project scheduling problem

This paper presents a matheuristic solution algorithm to solve the well-known resource-constrained project scheduling problem (RCPSP). The problem makes use of a restricted neighbourhood method using an activity selection and a search space restriction module and implements them as two alternative search algorithms. The first algorithm makes use of the best-performing components of the branch-and-bound procedures from the literature, and embeds them into a greedy neighbourhood search. The second matheuristic implements the exact branch-and-bound procedures into a known and well-performing meta-heuristic search algorithm.Computational experiments have been carried out on seven different datasets consisting of 10,000+ project instances. Experiments reveal that the choice of exact algorithm is key in finding high-quality solutions, and illustrate that the trade-off between selecting an activity set size and search space restriction depends on the specific implementation. The computational tests demonstrate that the matheuristic discovered 24 new best known solutions that could not be found by either a meta-heuristic or an exact method individually. Moreover, a new benchmark dataset has been proposed that can be used to develop new matheuristic search procedures to solve the problem consisting of 461 instances from the literature.

On a fair and risk‐averse urban air mobility resource allocation problem under demand and capacity uncertainties

AbstractUrban air mobility (UAM) is an emerging air transportation mode to alleviate the ground traffic burden and achieve zero direct aviation emissions. Due to the potential economic scaling effects, the UAM traffic flow is expected to increase dramatically once implemented, and its market can be substantially large. To be prepared for the era of UAM, we study the fair and risk‐averse urban air mobility resource allocation model (FairUAM) under passenger demand and airspace capacity uncertainties for fair, safe, and efficient aircraft operations. FairUAM is a two‐stage model, where the first stage is the aircraft resource allocation, and the second stage is to fairly and efficiently assign the ground and airspace delays to each aircraft provided the realization of random airspace capacities and passenger demand. We show that FairUAM is NP‐hard even when there is no delay assignment decision or no aircraft allocation decision. Thus, we recast FairUAM as a mixed‐integer linear program (MILP) and explore model properties and strengthen the model formulation by developing multiple families of valid inequalities. The stronger formulation allows us to develop a customized exact decomposition algorithm with both benders and L‐shaped cuts, which significantly outperforms the off‐the‐shelf solvers. Finally, we numerically demonstrate the effectiveness of the proposed method and draw managerial insights when applying FairUAM to a real‐world network.

A two‐stage adaptive robust model for designing a reliable blood supply chain network with disruption considerations in disaster situations

AbstractWe consider multi‐period blood supply chain network design in disaster situations that involve blood donor groups, permanent and temporary blood collection facilities, blood banks, and hospitals. We use a discrete scenario set to model the uncertain blood supply and demand, and the unforeseeable disruptions in permanent blood collection facilities, blood banks, and road links arising from a disaster, where instead of complete failure, disrupted permanent blood collection facilities and blood blanks may only lose part of their capacities. To design a reliable blood supply network to mitigate the possible disruptions, we present a two‐stage adaptive robust model that integrates the location, inventory, and allocation decisions incorporating a blood sharing strategy, where blood can be delivered from a disrupted/non‐disrupted blood bank to disrupted blood banks to enhance the flexibility of the relief network. For this novel problem, we devise an exact algorithm that integrates column‐and‐constraint generation and Benders decomposition and introduce several non‐trivial acceleration techniques to speed up the solution generation process. We conduct extensive numerical studies on random data sets to evaluate the algorithmic performance. We also conduct a case study in Tehran to demonstrate its real‐life applicability and examine the impacts of key model parameters on the solutions. The numerical results verify the benefits of our model over typical benchmarks, that is, deterministic and stochastic models, and the superiority of our solution algorithm over the CPLEX solver and two well‐known solution approaches, that is, column‐and‐constraint generation and Benders decomposition. Finally, based on the numerical results, we derive managerial insights from the analytical findings.

Energy-efficient scheduling in an identical parallel machine environment with peak power consumption and deadline constraints

Energy-efficient scheduling is an essential means of achieving sustainability in manufacturing systems. This paper defines and addresses the problem of scheduling n jobs on m identical parallel machines in which peak power consumption and deadline constraints exist simultaneously. The objective is to maximize the total value of the selected jobs. We show that this problem is equivalent to a special case of the rectangular knapsack problem, based on which four properties are observed. To solve the problem, an effective mixed integer linear programming model is proposed based on the properties, and it is much more efficient compared to the performance of modeling methods inspired by other works. Furthermore, three effective decoding methods are proposed and embedded into genetic algorithms (GAs). Comparisons with the exact algorithm (i.e., the proposed model) show that our GAs can lead to good-quality solutions within one second for small instances. Meanwhile, experimental results for large instances indicate that the proposed GAs can obtain near-optimal or satisfactory solutions. Finally, results also show that the proposed GA-DD significantly outperforms the existing matheuristic algorithm.

Algorithms for Densest Subgraphs of Vertex-Weighted Graphs

Finding the densest subgraph has tremendous potential in computer vision and social network research, among other domains. In computer vision, it can demonstrate essential structures, and in social network research, it aids in identifying closely associated communities. The densest subgraph problem is finding a subgraph with maximum mean density. However, most densest subgraph-finding algorithms are based on edge-weighted graphs, where edge weights can only represent a single value dimension, whereas practical applications involve multiple dimensions. To resolve the challenge, we propose two algorithms for resolving the densest subgraph problem in a vertex-weighted graph. First, we present an exact algorithm that builds upon Goldberg’s original algorithm. Through theoretical exploration and analysis, we rigorously verify our proposed algorithm’s correctness and confirm that it can efficiently run in polynomial time O(n(n + m)log2n) is its temporal complexity. Our approach can be applied to identify closely related subgroups demonstrating the maximum average density in real-life situations. Additionally, we consistently offer an approximation algorithm that guarantees an accurate approximation ratio of 2. In conclusion, our contributions enrich theoretical foundations for addressing the densest subgraph problem.

Inexact Fixed-Point Proximity Algorithm for the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} Sparse Regularization Problem

We study inexact fixed-point proximity algorithms for solving a class of sparse regularization problems involving the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} norm. Specifically, the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} model has an objective function that is the sum of a convex fidelity term and a Moreau envelope of the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} norm regularization term. Such an ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} model is non-convex. Existing exact algorithms for solving the problems require the availability of closed-form formulas for the proximity operator of convex functions involved in the objective function. When such formulas are not available, numerical computation of the proximity operator becomes inevitable. This leads to inexact iteration algorithms. We investigate in this paper how the numerical error for every step of the iteration should be controlled to ensure global convergence of the inexact algorithms. We establish a theoretical result that guarantees the sequence generated by the proposed inexact algorithm converges to a local minimizer of the optimization problem. We implement the proposed algorithms for three applications of practical importance in machine learning and image science, which include regression, classification, and image deblurring. The numerical results demonstrate the convergence of the proposed algorithm and confirm that local minimizers of the ℓ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _0$$\\end{document} models found by the proposed inexact algorithm outperform global minimizers of the corresponding ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document} models, in terms of approximation accuracy and sparsity of the solutions.

An exact algorithm for simultaneous pickup and delivery problem with split demand and time windows

This study introduces a new variant of the vehicle routing problem (VRP) called the simultaneous pickup and delivery problem with split demand and time windows (SPDP-SDTW). The motivation behind this study stems from real-life urban and rural delivery scenarios, encompassing features such as split demand, simultaneous pickup and delivery, many-to-many pickup and delivery, and time windows. The study thoroughly investigates the properties of the optimal solution for the SPDP-SDTW. Based on these properties, an arc flow model is developed for the SPDP-SDTW. Dantzig Wolfe (DW) decomposition techniques are employed to obtain the master problem and the pricing subproblem. In order to effectively address the SPDP-SDTW, an improved branch and price (I-BP) algorithm is proposed, incorporating a tailored column generation (CG) algorithm, branching strategies, and dual stabilization strategies. The proposed CG algorithm provides a framework that combines the improved adaptive degree heuristic (I-AGH) algorithm and the solver Gurobi. This integration substantially mitigates the computational burden involved in solving the subproblem. Extensive computational experiments conducted on datasets of varying sizes, including small, medium, and large instances, consistently demonstrate that the I-BP algorithm performs the best in both solution quality and computational efficiency when compared to existing exact and heuristic algorithms.

A non-anticipative learning-optimization framework for solving multi-stage stochastic programs

AbstractWe present a non-anticipative learning- and scenario-based prediction-optimization (ScenPredOpt) framework that combines deep learning, heuristics, and mathematical solvers for solving combinatorial problems under uncertainty. Specifically, we transform neural machine translation frameworks to predict the optimal solutions of scenario-based multi-stage stochastic programs. The learning models are trained efficiently using the input and solution data of the multi-stage single-scenario deterministic problems. Then our ScenPredOpt framework creates a mapping from the inputs used in training into an output of predictions that are close to optimal solutions. We present a Non-anticipative Encoder-Decoder with Attention (NEDA) approach, which ensures the non-anticipativity property of multi-stage stochastic programs and, thus, time consistency by calibrating the learned information based on the problem’s scenario tree and adjusting the hidden states of the neural network. In our ScenPredOpt framework, the percent predicted variables used for the solution are iteratively reduced through a relaxation of the problem to eliminate infeasibility. Then, a linear relaxation-based heuristic is performed to further reduce the solution time. Finally, a mathematical solver is used to generate the complete solution. We present the results on two NP-Hard sequential optimization problems under uncertainty: stochastic multi-item capacitated lot-sizing and stochastic multistage multidimensional knapsack. The results show that the solution time can be reduced by a factor of 599 with an optimality gap of only 0.08%. We compare the results of the ScenPredOpt framework with cutting-edge exact and heuristic solution algorithms for the problems studied and find that our framework is more effective. Additionally, the computational results demonstrate that ScenPredOpt can solve instances with a larger number of items and scenarios than the trained ones. Our non-anticipative learning-optimization approach can be beneficial for stochastic programming problems involving binary variables that are solved repeatedly with various types of dimensions and similar decisions at each period.

Mathematical models and an effective exact algorithm for unrelated parallel machine scheduling with family setup times and machine cost

Mathematical models and an effective exact algorithm for unrelated parallel machine scheduling with family setup times and machine cost