Mixed-integer Programming Method Research Articles

Integrating soft and hard dose-volume constraints into hierarchical constrained IMRT optimization.

Dose-volume constraints (DVCs) continue to be common features in intensity-modulated radiation therapy (IMRT) prescriptions, but they are non-convex and difficult to incorporate. We propose computationally efficient methods to incorporate dose-volume constraints (DVCs) into automated IMRT planning. We propose a two-phase approach: in phase-1, we solve a convex approximation with DVCs. Although this convex approximation does not guarantee DVC satisfaction, it provides crucial initial information about voxels likely to receive doses below DVC thresholds. Subsequently, phase-2 solves an optimization problem with maximum dose constraints imposed on those subthreshold voxels. We further categorize DVCs into hard- and soft-DVCs, where hard-DVCs are strictly enforced by the optimization and soft-DVCs are encouraged in the objective function. We tested this approach in our automated treatment planning system which is based on hierarchical constrained optimization. Performance is demonstrated on a series of paraspinal, lung, oligometastasis, and prostate cases as well as a small paraspinal case for which we can computationally afford to obtain a ground-truth by solving a non-convex optimization problem. The proposed algorithm successfully meets all the hard-DVCs while increasing the overall computational time of the baseline planning process (without DVCs) by 20%, 10%, and 11% for paraspinal, oligometastasis, and prostate cases, respectively. For a soft-DVC applied to the lung case, the dose-volume histogram curve moves toward the desired direction and the computational time is increased by 11%. For a low-resolution paraspinal case, the ground-truth solution process using mixed-integer programming methods required 15h while the proposed algorithm converges in only 2min with a proximal solution. A computationally tractable algorithm to handle hard- and soft-DVCs is developed which is capable of satisfying DVCs without any parameter tweaking. Although the algorithm is demonstrated in our in-house developed automated treatment planning system, it can potentially be used in any constrained optimization framework.

Redesign of vaccine distribution networks

AbstractIn most low‐ and middle‐income countries supported by the World Health Organization's Expanded Program on Immunization, vaccines are distributed through a legacy medical supply chain that is typically not cost‐efficient. Vaccines require storage and transport in a temperature‐controlled environment; this requires a “cold” distribution chain with capacity constraints on cold storage and cold transport. We propose an approach to redesigning the vaccine distribution chain that includes locating a set of intermediate distribution centers (DCs) and determining the flow paths from the central store (where vaccines are received into a country) through one or more of these to health clinics where vaccination actually occurs. In addition, the transport vehicles to allocate to each flow path, and the cold storage devices to use at each clinic or intermediate DC are determined. The redesigned network does not have to follow the current four‐tiered, arborescent structure commonly found in practice, but can use alternative network structures. To redesign this network optimally, we develop a mixed‐integer programming (MIP) model that can be used for small‐to‐medium‐sized problems and also present a hybrid heuristic–MIP method to obtain good solutions for larger problems. Numerical results are shown using data reflecting distribution networks in several countries in sub‐Saharan Africa.

Determining [formula omitted]- and [formula omitted]-robustness of digraphs using mixed integer linear programming

There has been an increase in the use of resilient control algorithms based on the graph theoretic properties of r- and (r,s)-robustness. These algorithms guarantee consensus of normally behaving agents in the presence of a bounded number of arbitrarily misbehaving agents if the values of the integers r and s are sufficiently large. However, determining an arbitrary graph’s robustness is a highly nontrivial problem. This paper introduces a novel method for determining the r- and (r,s)-robustness of digraphs using mixed integer linear programming; to the best of the authors’ knowledge it is the first time that mixed integer programming methods have been applied to the robustness determination problem. The approach only requires knowledge of the graph Laplacian matrix, and can be formulated with binary integer variables. Mixed integer programming algorithms such as branch-and-bound are used to iteratively tighten the lower and upper bounds on r and s. Simulations are presented which compare the performance of this approach to prior robustness determination algorithms.

A Variable Reduction Method for Large-Scale Unit Commitment

Efficient solution methods for large-scale unit commitment (UC) problems have long been an important research topic and a challenge, especially in market clearing computation. For large-scale UC, the Lagrangian relaxation methods (LR) and the mixed integer programming methods (MIP) are most widely adopted. However, LR usually suffers from slow convergence; and the computational burden of MIP is heavy when the binary variable number is large. In this paper, a variable reduction method is proposed. First, the time-coupled constraints in the original UC problem are relaxed, and many single-period UC problems (s-UC) are obtained. Second, LR is used to solve the s-UCs. Different from traditional LR with iterative subgradient method, the optimal multipliers and the approximate UC solutions of the s-UCs are obtained by solving linear programs. Third, a criterion for choosing and fixing the UC variables in the UC problem is established; hence, the number of binary variables is reduced. Finally, the UC with reduced binary variables is solved to obtain the final UC solution. The proposed method is tested on the IEEE 118-bus system and a 6484-bus system. The results show the method is very efficient and effective.

Proposed Determination of Product-Replenishment Policy on Multi-Source Problems to Minimize Over Stock in DC PT. XYZ Area Bandung With Mixed-Integer Programming Method

PT. XYZ is one of the companies which engages in fast moving consumer goods (FMCG) sector and has soft drinks as its main product. In supporting the product distribution to its customers, PT XYZ has several administrative offices and distribution centers (DC) which spread across each region. The DC has main function as the warehouse which has storage activity before the products are distributed.
 One of the DC is located in Bandung, West Java. DC Bandung itself received product shipments from three factories, such as Cibitung, Cakung and Tambun. Related to the product shipment, DC Bandung has not determined the replenishment policy, and this condition results over stock at some periods, due to the unscheduled delivery time and undetermined product quantity. This condition gives a big impact to the total inventory cost that has to be borne by DC Bandung.
 This research was conducted to give the proposal of product replenishment policy with mixed-integer programming method, as one of the linear programming types, which aimed to minimize over stock and total holding cost in DC Bandung. Mathematical model is also involved in this research as an approximation to the real replenishment system in the DC.
 The results of application of the proposed method are in the form of analytical solution, such as order quantity, and reorder point of each product that will give an impact to the total unit inventory and the total holding cost in the DC. The results calculation of replenishment policy provide savings on total inventory costs as much as IDR 23,800,981.20

Modeling and Solution Techniques Used for Hydro Generation Scheduling

The hydro generation scheduling problem has a unit commitment sub-problem which deals with start-up/shut-down costs related hydropower units. Hydro power is the only renewable energy source for many countries, so there is a need to find better methods which give optimal hydro scheduling. In this paper, the different optimization techniques like lagrange relaxation, augmented lagrange relaxation, mixed integer programming methods, heuristic methods like genetic algorithm, fuzzy logics, nonlinear approach, stochastic programming and dynamic programming techniques are discussed. The lagrange relaxation approach deals with constraints of pumped storage hydro plants and gives efficient results. Dynamic programming handles simple constraints and it is easily adaptable but its major drawback is curse of dimensionality. However, the mixed integer nonlinear programming, mixed integer linear programming, sequential lagrange and non-linear approach deals with network constraints and head sensitive cascaded hydropower plants. The stochastic programming, fuzzy logics and simulated annealing is helpful in satisfying the ramping rate, spinning reserve and power balance constraints. Genetic algorithm has the ability to obtain the results in a short interval. Fuzzy logic never needs a mathematical formulation but it is very complex. Future work is also suggested.

A scalable stochastic programming approach for the design of flexible systems

We study the problem of designing systems in order to minimize cost while meeting a given flexibility target. Flexibility is attained by enforcing a joint chance constraint, which ensures that the system will exhibit feasible operation with a given target probability level. Unfortunately, joint chance constraints are complicated mathematical objects that often need to be reformulated using mixed-integer programming (MIP) techniques. In this work, we cast the design problem as a conflict resolution problem that seeks to minimize cost while maximizing flexibility. We propose a purely continuous relaxation of this problem that provides a significantly more scalable approach relative to MIP methods and show that the formulation delivers solutions that closely approximate the Pareto set of the original joint chance-constrained problem.

Practical approaches to chemical tanker scheduling in ports: a case study on the Port of Houston

The objective of this study was to develop practical scheduling solutions for chemical tankers visiting the Port of Houston (PoH). Chemical tanker movements represent approximately 42% of the Houston Ship Channel traffic. Historically, chemical tanker scheduling has been problematic and has resulted in long waiting times for tankers. Scheduling is difficult because chemical tankers carry several liquid cargoes and must visit multiple terminals for loading and unloading. Physical constraints (layout of the port and draft) and commercial constrains (such as terminal and personnel readiness for cargo handling operations, tank cleaning processes, and inspection requirements) create a complex scheduling problem, long waiting times, and unnecessary tanker movements in the port. These problems cause an increase in the business costs for shipowners, risk of collisions and allisions, production of additional air emissions, and decreases in the operating capacity of terminals. The recent expansion decisions for chemical and petrochemical plants in Houston, Texas, will exacerbate the problem. Significant benefits could thus be gained even for small scheduling improvements. Currently, the scheduling practice of loading/unloading activities in the PoH involves primarily the manual and de-centralized use of the “first come, first served” (FCFS) rule, which results in inefficiencies such as long waiting times and poor resource utilization. We propose two mathematical methods to address the tanker scheduling problem in the port: a mixed-integer programming (MIP) method, and a constraint programming (CP) method. The two methods are formulated as open-shop scheduling problems with sequence-dependent post-setup times. MIP yields optimum results that minimize makespan. However, computation time increases significantly as the number of tankers, or the number of terminals, increases. CP achieves better makespan results in a shorter run time, compared to MIP, for medium to large-scale problems including the problem considered in this case study. Overall, the results show that MIP is more suitable for real-time scheduling tools (hourly and daily), whereas CP is the better option for longer-horizon scheduling problems (weekly or monthly). Our models gave good alternative schedules under short optimization run times. Hence, they can afford decision makers sufficient time to complete multiple optimization scenarios and implementation setups.

Control Algorithm Using a Revised MIP Method to Deal with Livelocks in S4R of Petri Nets

Livelocks, like deadlocks, can result in serious results in running process of flexible manufacturing systems (FMSs). Current deadlock control policies (DCPs) based on mixed integer programming (MIP) cannot detect siphons that cause and cope with livelocks in Petri nets. This study proposes a revised mixed integer programming (RMIP) method to directly solve the new smart siphons (NSSs) associated with livelocks in a system of sequential systems with shared resources (S4R), a typical subclass of generalized Petri net models. Accordingly, the solved NSSs are max’-controlled by adding the corresponding control places (CPs). As a result, an original S4R system with livelocks can be converted into the live controlled Petri net system. The related theoretical analysis and an example are given to demonstrate the proposed RMIP and the corresponding control algorithm (CA).

Solutions To Storage Spaces Allocation Problem for Import Containers by Exact and Heuristic Methods

Maritime transport carries out most of the world's commercial activities. One of the elements used for maritime transportation is the container. Before moving to or from a country, containers are stacked in a container terminal, also called container yard. Container terminals perform multiple and complex operations daily. Scientific community pays attention to the container stacking operations in the yard area due to its complexity and relevance in the international commerce. In this work, we solve the problem of optimizing the storage spaces allocation to imported containers by the linear mixed integer programming method. The objective of this work is to solve storage space allocation problem for imported containers, in an acceptable computational time. We use an exact optimization method for the operations of a real container yard. We model this optimization problem mathematically as a generalized assignment problem. We present the mathematical formulation, its solution, as well as the computational experiments.

Proposal of Product-Replenishment Policy Determination on Multi-Source Problems to Minimize Over Stock in DC PT. XYZ Area Bandung Using Mixed-Integer Programming Method

XYZ is one of the companies engaging in fast moving consumer goods (FMCG) sector and produce soft drinks as its main product. In supporting the product distribution to its customers, PT XYZ has several administrative offices and distribution centers (DC) which spread across each region. The DC has main function as the warehouse which has storage activity before the products are distributed. One of the DC is located in Bandung, West Java. DC Bandung itself received product shipments from three factories, such as Cibitung, Cakung and Tambun. Related to the product shipment, DC Bandung has not determined the replenishment policy, and this condition results over stock at some periods, due to the unscheduled delivery time and undetermined product quantity. This condition gives a big impact to the total inventory cost that has to be borne by DC Bandung. This research was conducted to give the proposal of product replenishment policy using mathematical model with mixed-integer programming method as one of the linear programming types, which aimed to minimize over stock and total holding cost in DC Bandung with several constraints. Mathematical modeling is involved in this research as an approximation to the real replenishment system in the DC. The results of application of the proposed method are in the form of analytical solution, such as order quantity, and reorder point of each product that will give an impact to the total unit inventory and the total holding cost in the DC. The results calculation of replenishment policy provide savings on total inventory costs as much as IDR 23,800,981.20
 Keywords—Mixed-Integer Programming, Order Quantity, Over Stock, Replenishment Policy.

A chemical tanker scheduling problem: Port of Houston case study

The objective of this study was to develop scheduling solutions for chemical vessels in a port. Chemical tankers carry multiple liquid cargos and have to visit multiple terminals to load/unload these cargos. The practice to date in scheduling these loading/unloading activities for the Port of Houston (PoH) has primarily involved manual methods, which result in several inefficiencies such as long waiting times and inefficient resource utilisation. This paper proposes three methods for tanker-scheduling problems, formulated as an open-shop scheduling problem with sequence-dependent post-set-up times: a mixed-integer programming (MIP); a constraint programming (CP); and a dispatching rule, longest processing time (LPT) scheduling heuristic. Overall, the results showed the MIP method to be a more suitable tool for hourly and daily scheduling, whereas the CP and the LPT heuristic proved suitable for longer-horizon scheduling problems (weekly, monthly, or semi-annual) and for use as real-time scheduling tools.

Planning and Scheduling Operating Rooms with Mix Integer Programming and Meta-Heuristic Method

Planning and Scheduling Operating Rooms with Mix Integer Programming and Meta-Heuristic Method

A defender-attacker model and algorithm for maximizing weighted expected hitting time with application to conservation planning

ABSTRACTThis article studies an interdiction problem in which two agents with opposed interests, a defender and an attacker, interact in a system governed by an absorbing discrete-time Markov chain. The defender protects a subset of transient states, whereas the attacker targets a subset of the unprotected states. By changing some of the transition probabilities related to the attacked states, the attacker seeks to minimize the Weighted Expected Hitting Time (WEHT). The defender seeks to maximize the attacker’s minimum possible objective, mitigating the worst-case WEHT. Many applications can be represented by this problem; this article focuses on conservation planning. We present a defender–attacker model and algorithm for maximizing the minimum WEHT. As WEHT is not generally a convex function of the attacker’s decisions, we examine large-scale integer programming formulations and first-order approximation methods for its solution. We also develop an algorithm for solving the defender’s problem via mixed-integer programming methods augmented by supervalid inequalities. The efficacy of the proposed solution methods is then evaluated using data from a conservation case study, along with an array of randomly generated instances.

Solving discrete lot-sizing and scheduling by simulated annealing and mixed integer programming

Abstract We consider the discrete single-machine, multi-item lot-sizing and scheduling problem and we propose a Simulated Annealing (SA) approach together with a statistically-principled tuning procedure to solve it. We compare our solver with the state-of-the-art methods based on Mixed Integer Programming (MIP), both on publicly-available instances and on a set of new, more challenging ones. In addition, we propose a hybrid SA/MIP method that combines the advantages of the pure methods on the challenging instances. The outcome is that our solver is able to find near-optimal solutions in short time for all instances, including those that are not solved by MIP methods. Instances and solutions are made available on the web for inspection and future comparisons.

Prerequisites for the modelling of empty container supply chains

Abstract Containerisation of freight transport significantly facilitates the flow of traded goods between remote destinations. The most important container transport routes link Asia with North America and Europe. The seasonality and imbalance of trade are the main factors giving rise to problems related to empty container repositioning. The aim of this paper is to develop a concept of empty container supply chains and formulate prerequisites for its modelling in the search for the optimal solution with the help of linear programming and mixed integer programming methods. This paper indicates causes of the empty container relocation problem based on the literature analysis with a special focus on the Eurasian transportation route. Also, it provides a concept of empty container supply chain, prerequisites for its modelling and examples of container supply chains modelling presented in the literature. The main results of the paper include conceptualisation of an empty container supply chain and suggested modelling prerequisites. The paper contributes to research in the field of supply chain management and optimisation of transportation.

Safety constraints and optimal operation of large‐scale nuclear power plant participating in peak load regulation of power system

Comprehensively considering the operation cost and safety constraints of nuclear power, an optimal operation scheme of large-scale nuclear power plant participating in peak load regulation of power system is proposed. After quantitatively analysing the peak load regulation cost of nuclear power, the optimal objective is set to minimise the total operation cost including the fuel cost, the start-stop cost, and the peak load regulation cost. On the basis of the normal constraints of power scheduling, safety constraints of the nuclear power unit's daily output performance and power characteristic constraints of the pumped storage unit are considered. The mixed integer programming method is used to achieve the solution of the optimal operation scheme. Simulation results from a regional power system show that the proposed optimal operation scheme can effectively reflect the real characteristic of nuclear power and apply to power dispatching analysis under the background of the large-scale access of nuclear power. In comparison to other two typical schemes of peak load regulation, the proposed optimal operation scheme of peak load regulation is more economic and efficient.

Operating Rooms Planning and Scheduling with Mix Integer Programming and Meta-Heuristic Method

The surgery department is required to improve its performance by efficient usage of its available resources. Surgery department interdependencies with other hospital functions and departments are numerous and complex, which causes some restrictions on surgery ward performance and consequently hospital performance. Operating room (OR) schedule determines day to day operations that take a place at the surgery department. Creating such a schedule is a complex task, due to a multitude of constraints, preferences and objectives that planners need to take into account. Meanwhile, the schedule has major effects on the performance of surgery department and the hospital. This paper provides a review of recent operational research on operating room planning and scheduling with focus on multi-objective simulation optimization with use meta-heuristic method: in operating room, as a practical suggestion for hospitals. Recent researches were collected based on some keywords including simulation optimization, and operating room by google scholar, Scopus, and Pubmed. The findings showed that, there are not much practical suggestion in this case and still there is a gap and needs to be discovered.

碳限额与交易机制下供应链网络设计

本文结合碳足迹及经典的供应链网络模型等相关理论，在传统的三级供应链网络模型中加入了碳限额与交易机制，全面分析了碳限额与交易机制对运营成本和供应链网络的影响。在经典供应链网络模型基础上运用0-1混合整数规划的方法构建了考虑碳限额与交易机制的供应链网络优化模型。从消费者、供应商和生产商三个角度对模型进行数值模拟对比分析，综合考虑了在碳限额与交易机制下各影响因子的变动对企业运营成本和供应链网络的影响。进一步验证了企业可以运用构建的模型来制定出合理的供应链网络，实现减排的同时降低企业的运输成本和运营成本。 This paper combines the theory of carbon footprint and basic supply chain model, add the carbon cap and trade mechanism into the basic supply chain model. And a comprehensive analysis of the impact of carbon cap and trade mechanism on operational costs and network flow has been made. Based upon basic transportation model, and taking into consideration of carbon cap and trade mechanism, we construct an optimized transportation network model applying 0-1 mixed integer programming method. Through the numerical simulation analysis of the model, we consider the influence on the enterprise’s operational cost and strategy under the cap and trade policy from three aspects, such as consumers, suppliers, manufactures. The industry can use the model to develop a reasonable transportation route to reduce carbon emissions and operational costs.

Scenario analysis of carbon emissions' anti-driving effect on Qingdao's energy structure adjustment with an optimization model, Part II: Energy system planning and management

In this study, an inexact multistage stochastic mixed-integer programming (IMSMP) method was developed for supporting regional-scale energy system planning (EPS) associated with multiple uncertainties presented as discrete intervals, probability distributions and their combinations. An IMSMP-based energy system planning (IMSMP-ESP) model was formulated for Qingdao to demonstrate its applicability. Solutions which can provide optimal patterns of energy resources generation, conversion, transmission, allocation and facility capacity expansion schemes have been obtained. The results can help local decision makers generate cost-effective energy system management schemes and gain a comprehensive tradeoff between economic objectives and environmental requirements. Moreover, taking the CO2 emissions scenarios mentioned in Part I into consideration, the anti-driving effect of carbon emissions on energy structure adjustment was studied based on the developed model and scenario analysis. Several suggestions can be concluded from the results: (a) to ensure the smooth realization of low-carbon and sustainable development, appropriate price control and fiscal subsidy on high-cost energy resources should be considered by the decision-makers; (b) compared with coal, natural gas utilization should be strongly encouraged in order to insure that Qingdao could reach the carbon discharges peak value in 2020; (c) to guarantee Qingdao's power supply security in the future, the construction of new power plants should be emphasised instead of enhancing the transmission capacity of grid infrastructure.