Deterministic Mixed Integer Linear Programming Research Articles

Robust Design of a Closed-Loop Supply Chain Considering Multiple Recovery Options and Carbon Policies Under Uncertainty

Increasing global warming, climate change and stringent governmental legislations are driving industry practitioners and decision makers to implement various strategies to reduce carbon emissions. One of the effective approaches to mitigate carbon emissions is the implementation of closed-loop supply chain (CLSC). The key motivation for considering multiple recovery options in the CLSC is to capture the remaining economic value and to reduce carbon emissions in the collection and recovery operations. Customer's willingness to return used product depends on the acquisition price and nearness to the collection center. This research proposes a deterministic mixed-integer linear programming (MILP) model for a multi-period and multi-product CLSC network under carbon pricing and carbon trading policies consideration. The model includes different acquisition price for returned products and multiple recovery options. Further, the study takes into consideration uncertainty in procurement cost, demand, and quantity of returned products. A robust optimization approach is adopted to address uncertainty in network parameters. Numerical results show that the proposed model captures trade-offs between total cost and carbon emission. Overall, the study reveals that the carbon trading policy incurs relatively lower total cost compared to the carbon pricing policy. Repair and recycling activities in the reverse supply chain contribute significantly to the total cost and carbon emission. This study provide evidence that it is possible to achieve an optimal CLSC network with reduced carbon emission at a moderate total supply chain cost. The proposed model could be used to guide firms to choose an appropriate budget of uncertainty toward achieving a robust supply chain network.

Category Inventory Planning With Service Level Requirements and Dynamic Substitutions

We study a single‐period inventory planning problem for a category of substitutable products. This is an important practical problem facing category managers who have to maintain high service levels for constantly expanding product catalogs. We formulate the problem as a stochastic optimization model that minimizes the total stocking cost subject to service level requirements, which consist of product‐specific and category‐wide targets for inventory availability (ready rates) through the selling season. Our model accounts for stochastic customer arrivals, captures stockout‐based substitutions, and determines initial stocking quantities jointly for all products. Recognizing the challenges that these aspects pose in solving the problem, we propose an optimization‐based method that estimates the ready rates using a deterministic approximation and discretizes the selling season into a finite number of time intervals. This novel modeling approach permits us to recast the stochastic optimization model as a deterministic mixed integer linear program that can accommodate several common stockout‐based substitution schemes. We characterize the worst‐case behavior of this approach to develop performance guarantees. We also implemented and applied this model to randomly generated numerical instances featuring different types of product differentiation and varying in parameter values. We observe that the approach is robust to changes in problem parameter values and yields solutions very quickly, outperforming an enumeration‐based alternative, a practical heuristic, and an approach based on extant literature. Finally, we applied our approach to data from a re‐seller of Information Technology products. Results illustrate that our approach scales well and has the potential to generate savings in inventory costs.

A Stochastic Model for Prioritized Outpatient Scheduling in a Radiology Center

This paper discussed the scheduling problem of outpatients in a radiology center with an emphasis on priority. To more compatibility to real-world conditions, we assume that the elapsed times in different stages to be uncertain that follow from the specific distribution function. The objective is to minimize outpatients’ total spent time in a radiology center. The problem is formulated as a flexible open shop scheduling problem and a stochastic programming model. By considering the specific distribution function for uncertain variables, deterministic mixed integer linear programming (MILP) is developed such that the proposed problem can be solved by a linear programming solver in small size. Besides an effective heuristic method is proposed for the moderate size problem. To indicate the applicability of the proposed model, it has been applied to a real radiology center. The results from the proposed optimization models indicate an increase in outpatients’ satisfaction, as well as the improving of the efficiency and productivity of the radiology center.

Optimizing Disruption Recovery Operations for Wind Farms considering Power Generation Loss and Repair Time Uncertainty

The development of wind power in China shows a dramatic growth in the past decade in terms of installed capacity. However, wind power companies mainly focus on the construction of new wind farms continuously, while operations management once wind farms are built is seldom paid attention to. The problem is crucial for ensuring efficient power generation, especially when wind turbines’ performance declines over time and disruption/failure often occurs. Efficient disruption recovery operations are critical for restoring the failures of wind turbine generators as fast as possible. This paper aims to optimize the disruption recovery operations for wind farms by determining the maintenance schedule and route for multiple maintenance teams. This optimization problem is formulated as a deterministic mixed integer linear programming model with the objective of minimizing the loss of power generation due to failure. In view of the high uncertainty of repair time, a chance-constrained programming model and a cutting-plane solution algorithm are further proposed. A case study based on a real wind farm demonstrates (1) the proposed model is applicable for solving real-world-sized problems; (2) the optimal maintenance route often shows a crossing pattern, which is quite different from that of traditional vehicle routing problems; and (3) the working time limit violation for maintenance teams due to uncertain repair time can be effectively avoided. Overall, the proposed optimization model provides decision-making support for wind farm maintenance work and shows a great potential in wind farm energy management.

EVALUATION OF THE PRODUCTION PLANNING POLICY ALTERNATIVES IN A PET RESIN PRODUCTION PLANT: A CASE STUDY FROM TURKEY

Goal: The main purpose of this work is to develop several production planning models for one of bottle grade PET production plants (in Turkey) under different scenarios by considering different levels of inventory (no stock and different levels of safety stocks), different energy sourcing alternatives (natural gas/coal or both) and production throughput (different lot sizes).
 Design/Methodology/Approach: Deterministic multi-product multi-period single level mixed integer linear programming model is presented. The model is implemented for the above defined scenarios. We have also considered the total emission produced for different energy alternatives.
 Results: The models for the different scenarios have shown that significant performance improvements can be achieved by changing the parameters/policies of the production. The best model obtained has yielded 6.6% of profit improvement and 6.9% of cost reduction.
 Limitations of the investigation: Planning horizon of this work could cover the data of years 2014 to 2018. However, the actual data employed for this study is limited with one year due to unavailability of further data. In addition to this, improvement of current production planning approach of the plant by implementation of proposed model is one of the objectives of this study to see success rate of model. Though, this couldn’t be realized. We hypothetically tested the improvement provided by the proposed model, therefore we can’t analyze the actual improvement.
 Practical implications: Interactions among optimal results obtained by running model with different scenarios and their effect on selected performance variables are the main contribution of this work. Similar methodology could be used at other PET resin manufacturing plants or alternatively the production facilities having similar type of continuous manufacturing processes to find their optimal levels of inventory, production throughput and proper choice for energy sourcing alternative.
 Originality/Value: The research on the production planning for PET resin industry by considering the different scenarios (levels of inventory / energy sourcing alternatives/ production throughput) is novel as far as we know, and therefore the approach used in this study is expected to be useful for other similar plants.

Data-Driven Robust Optimization for Steam Systems in Ethylene Plants under Uncertainty

In an ethylene plant, steam system provides shaft power to compressors and pumps and heats the process streams. Modeling and optimization of a steam system is a powerful tool to bring benefits and save energy for ethylene plants. However, the uncertainty of device efficiencies and the fluctuation of the process demands cause great difficulties to traditional mathematical programming methods, which could result in suboptimal or infeasible solution. The growing data-driven optimization approaches offer new techniques to eliminate uncertainty in the process system engineering community. A data-driven robust optimization (DDRO) methodology is proposed to deal with uncertainty in the optimization of steam system in an ethylene plant. A hybrid model of extraction–exhausting steam turbine is developed, and its coefficients are considered as uncertain parameters. A deterministic mixed integer linear programming model of the steam system is formulated based on the model of the components to minimize the operating cost of the ethylene plant. The uncertain parameter set of the proposed model is derived from the historical data, and the Dirichlet process mixture model is employed to capture the features for the construction of the uncertainty set. In combination with the derived uncertainty set, a data-driven conic quadratic mixed-integer programming model is reformulated for the optimization of the steam system under uncertainty. An actual case study is utilized to validate the performance of the proposed DDRO method.

A robust optimization approach for designing a multi-echelon, multi-product, multi-period supply chain network with outsourcing

This paper addresses the problem of designing an integrated multi-echelon, multi-product, multi-period supply chain network under uncertainty in which products can be outsourced to improve responsiveness and the overall performance. The network consists of production plants, distribution centres and retailers. The demands of retailers, production capacities, and transportation costs as well as costs of opening distribution centres are subject to uncertainty. A deterministic mixed-integer linear programming (MILP) model is developed with the objective of minimising sum of the fixed, transportation, and outsourcing costs. In order to capture the uncertainties, a robust optimisation model is proposed. A set of numerical experiments using nominal and realisation data is investigated and the relative performance of the robust and deterministic models is compared based on the standard deviation and mean of the objective function values. In case of data realisation, results confirm that the proposed robust model outperforms the deterministic counterpart.

Stability-Constrained Microgrid Operation Scheduling Incorporating Frequency Control Reserve

This paper presents a two-stage microgrid scheduling strategy in which the frequency control reserve (FCR) is incorporated to ensure economic, reliable, and stable microgrid operation in a joint energy and ancillary service market environment. In addition, to using such reserve to limit frequency excursions caused by uncertainties of active loads and renewable generation, this paper is focused on determining the optimal provision of reserve for microgrid operation mode switching to assure sufficient reserve is available and readily deployable at the disposal of the microgrid operator. The scheduling problem is formulated as a two-stage chance-constrained programming (CCP) model where the power balance chance constraints are used to determine the trade-off between operational economy and stability risk exposure. The chance-constrained formulation is then approximated as a deterministic mixed-integer linear programming (MILP) model for solution. Two sets of policy studies are performed through simulation to evaluate the effectiveness of the proposed strategy in enhancing the operational performance of a microgrid through optimal scheduling.

Carbon market sensitive robust optimization model for closed loop supply chain network design under uncertainty

The adoption of closed-loop supply chain (CLSC) network is one of the effective approaches to reduce carbon emissions. In current globalization, inherent uncertainty exists in business environment so there is a need to be design robust supply chains. This paper proposes a deterministic mixed integer linear programming (MILP) model integrating economics and carbon emission considerations including selection of production technologies and transportation mode as a part of CLSC network strategic and tactical decisions. The robust counterpart of the proposed deterministic model is developed based on three alternative uncertainty sets to represent the imprecise input parameters. The robust counterpart is used to study the supply chain performance by considering the two most globally practiced carbon regulatory policies; carbon tax policy and carbon trading policy. Numerical results show that total cost of the proposed robust optimization model under each uncertainty set is greater than the total cost of deterministic model. The additional cost is due to solution space of each uncertainty set to accommodate any uncertainty level. As uncertainty level increases the overall supply chain cost worsen. Moreover, the results suggest that carbon tax rate has direct relation with overall supply chain cost whereas having carbon market trading flexibility in carbon trading policy, this policy is more efficient policy as compared to carbon tax policy. Furthermore, the proposed robust optimization model is useful for mangers to achieve not only a robust supply chain network design which can withstand any possible uncertainty level but also significant reduction in carbon emissions by choosing suitable carbon-efficient policy.

A robust bi-objective programming approach to environmental closed-loop supply chain network design under uncertainty

Imposition of strict environmental protection acts and the imperative need of the best possible allocation of resources have given birth to the concept of low carbon logistics. Environmental laws force the manufacturers to extend their existing supply chains to form a closed-loop supply chain (CLSC) through the setup of an efficient recovery system. In this paper, a multi-objective robust optimisation model is proposed for the design of CLSC network under uncertainty. First, a deterministic bi-objective mixed integer linear programming (BOMILP) model is developed for designing a CLSC network. Then, the robust counterpart of the proposed BOMILP is presented to cope with the real-world uncertainty. The first objective aims to maximise the total profit generated in the CLSC network and the second objective minimises the environmental pollution of the CLSC network. The proposed bi-objective model is solved using a multi-choice goal programming (MCGP) approach.

Public Hospital Inpatient Room Allocation and Patient Scheduling Considering Equity

This article studies the optimal allocation of inpatient rooms for multiple types of patients in public hospitals and the patient scheduling problem with planned acceptance ratios (ARs). For public hospitals, it is important to allocate limited resources to multiple types of patients and manage patient access for maximizing hospital revenue and upholding service equity. The problem is formulated as two-stage models. Considering uncertainties in patients’ arrival and length of stay, we first propose a nonlinear stochastic programming (NSP) model for inpatient room allocation with the objective of maximizing revenue under the constraints of maintaining equity. To solve this problem, we transform the complex NSP model into a deterministic mixed-integer linear programming model, which is solved by CPLEX, by reformulating the chance constraints as knapsack constraints based on a linearization technique and a simulation model. Given the allocated capacity and planned AR, we further propose a two-stage stochastic mixed-integer program combined with a goal program model to optimize patient scheduling. To solve the model, a Benders decomposition based on the sample average approximation approach is proposed. The real data-based experimental results demonstrate the applicability and effectiveness of our models and approaches. The impacts of some parameters on the objective and decisions are also explored. A simulation procedure is developed to compare the performances of different patient scheduling methods, from which the results show that our proposed approach outperforms a benchmark policy. Note to Practitioners —Inpatient rooms are critical resources for hospitals. Against the background of aging populations and environmental problems, the twofold predicament—involving escalating healthcare demands and insufficient room-based resources—has led to the necessity for hospitals to operate more effectively and efficiently. Capacity management and patient scheduling are thus the two most important operations for hospital management. Because of the self-financing feature of hospitals and the quasi-public nature of medical services, it is important for public hospitals to judiciously allocate limited room capacities to multiple types of patients for balancing revenue and equity and schedule the arrival demands dynamically according to the planned capacity and acceptance ratio. In this article, we propose mathematical models and solution approaches for these two-stage problems, and their applicability and effectiveness are demonstrated by experiments based on real data. The managerial insights suggest that hospitals should improve the service equity gradually according to their financial situation because of the increasing marginal cost and should apply the scientific approaches and techniques, rather than by their experiences, to aid their management for better performance. By using approaches proposed in this article, hospital managers can be equipped with a decision support tool for effective capacity allocation and patient scheduling decisions. The parameters of our models could be tuned based on the preferences of different hospitals. Furthermore, these approaches can be applied to other settings with similar problems, such as government budget allocation considering both utility and equity, and service system management with waiting time requirement.

Optimal operation of building microgrids – comparison with mixed-integer linear and continuous non-linear programming approaches

PurposeThis paper aims to present two mathematical models to solve the Energy Management problem of a building microgrid (MG). In particular, it proposes a deterministic mixed integer linear programming (MILP) and non-linear programming (NLP) formulations. This paper focuses on the modelling process and the optimization performances for both approaches regarding optimal operation of near-zero energy buildings connected to an electric MG with a 24-h time horizon.Design/methodology/approachA general architecture of a MG is detailed, involving energy storage systems, distributed generation and a thermal reduced model of the grid-connected building. A continuous non-linear model is detailed along with linearizations for the mixed-integer liner formulation. Multi-physic, non-linear and non-convex phenomena are detailed, such as ventilation and air quality models.FindingsResults show that both approaches are relevant for solving the energy management problem of the building MG.Originality/valueIntroduction and modelling of the thermal loads within the MG. The resulting linear program handles the mutli-objective trade-off between discomfort and the cost of use taking into account air quality criterion. Linearization and modelling of the ventilation system behaviour, which is generally non-linear and non-convex equality constraints, involving air quality model, heat transfer and ventilation power. Comparison of both MILP and NLP methods on a general use case provides a solution that can be interpreted for implementation.

A Modified Priority-Based Encoding for Design of a Closed-Loop Supply Chain Network Using a Discrete League Championship Algorithm

In a closed-loop supply chain network, the aim is to ensure a smooth flow of materials and attaining the maximum value from returning and end-of-life goods. This paper presents a single-objective deterministic mixed integer linear programming (MILP) model for the closed-loop supply chain (CLSC) network design problem consisting of plants, collection centers, disposal centers, and customer zones. Our model minimizes the total costs comprising fixed opening cost of plants, collection, disposal centers, and transportation costs of products among the nodes. As supply chain network design problems belong to the class of NP-hard problems, a novel league championship algorithm (LCA) with a modified priority-based encoding is applied to find a near-optimal solution. New operators are defined for the LCA to search the discrete space. Numerical comparison of our proposed encoding with the existing approaches in the literature is indicative of the high quality performance of the proposed encoding.

Detailed Scheduling of Tree-like Pipeline Networks with Multiple Refineries

In the oil supply chain, the refined petroleum products are transported by various transportation modes, such as rail, road, vessel and pipeline. The latter provides one of the safest and cheapest ways to connect production areas to local markets. This paper addresses the operational scheduling of a multi-product tree-like pipeline connecting several refineries to multiple distribution centers under demand uncertainty. A new deterministic mixed-integer linear programming (MILP) model is first presented, and then a two-stage stochastic model is proposed. The aim of this model is to meet depot requirements at the minimum total cost including pumping and stoppages costs. The efficiency and utility of the proposed model is shown by two numerical examples, which one of them uses the industrial and real data.

Weight reduction technology and supply chain network design under carbon emission restriction

As policies and regulations related to environmental protection and resource constraints are becoming increasingly tougher, corporations may face the difficulty of determining the optimal trade-offs between economic performance and environmental concerns when selecting product technology and designing supply chain networks. This paper considers weight reduction technology selection and network design problem in a real-world corporation in China which produces, sells and recycles polyethylene terephthalate (PET) bottles used for soft drinks. The problem is addressed while taking consideration of future regulations of carbon emissions restrictions. First, a deterministic mixed-integer linear programming model is developed to analyze the influence of economic cost and carbon emissions for different selections in terms of the weight of PET bottle, raw material purchasing, vehicle routing, facility location, manufacturing and recycling plans, etc. Then, the robust counterpart of the proposed mixed-integer linear programming model is used to deal with the uncertainty in supply chain network resulting from the weight reduction. Finally, results show that though weight reduction is both cost-effective and environmentally beneficial, the increased cost due to the switching of the filling procedure from hot-filling to aseptic cold-filling and the incumbent uncertainties have impacts on the location of the Pareto frontier. Besides, we observe that the feasible range between economic cost and carbon emission shrinks with weightreduction; and the threshold of restricted volume of carbon emission decreases with the increase of uncertainty in the supply chain network.

On the Reliability of Optimization Results for Trigeneration Systems in Buildings, in the Presence of Price Uncertainties and Erroneous Load Estimation

Cogeneration and trigeneration plants are widely recognized as promising technologies for increasing energy efficiency in buildings. However, their overall potential is scarcely exploited, due to the difficulties in achieving economic viability and the risk of investment related to uncertainties in future energy loads and prices. Several stochastic optimization models have been proposed in the literature to account for uncertainties, but these instruments share in a common reliance on user-defined probability functions for each stochastic parameter. Being such functions hard to predict, in this paper an analysis of the influence of erroneous estimation of the uncertain energy loads and prices on the optimal plant design and operation is proposed. With reference to a hotel building, a number of realistic scenarios is developed, exploring all the most frequent errors occurring in the estimation of energy loads and prices. Then, profit-oriented optimizations are performed for the examined scenarios, by means of a deterministic mixed integer linear programming algorithm. From a comparison between the achieved results, it emerges that: (i) the plant profitability is prevalently influenced by the average “spark-spread” (i.e., ratio between electricity and fuel price) and, secondarily, from the shape of the daily price profiles; (ii) the “optimal sizes” of the main components are scarcely influenced by the daily load profiles, while they are more strictly related with the average “power to heat” and “power to cooling” ratios of the building.

A robust multi-objective global supplier selection model under currency fluctuation and price discount

Robust supplier selection problem, in a scenario-based approach has been proposed, when the demand and exchange rates are subject to uncertainties. First, a deterministic multi-objective mixed integer linear programming is developed; then, the robust counterpart of the proposed mixed integer linear programming is presented using the recent extension in robust optimization theory. We discuss decision variables, respectively, by a two-stage stochastic planning model, a robust stochastic optimization planning model which integrates worst case scenario in modeling approach and finally by equivalent deterministic planning model. The experimental study is carried out to compare the performances of the three models. Robust model resulted in remarkable cost saving and it illustrated that to cope with such uncertainties, we should consider them in advance in our planning. In our case study different supplier were selected due to this uncertainties and since supplier selection is a strategic decision, it is crucial to consider these uncertainties in planning approach.

The Influence of Environmental Constraints on the Water Value

The establishment of more severe hydrological environmental constraints, usually as seasonal minimum flows (ϕ) and maximum ramping rates (ρ), on hydropower operation is a growing trend. This paper presents a study on the influence of ϕ and ρ on the water values (WV) of a real hydropower plant that participates in the Spanish day-ahead electricity market. For this purpose, a master-slave algorithm, based on stochastic dynamic programming (SDP) and deterministic mixed integer linear programming (DMILP), is used on a real hydropower plant. The master module, based on SDP, has a yearly planning period with weekly time steps and considers three state variables: stored water volume in the reservoir at the beginning of each week; weekly water inflow; and average weekly energy price. The slave module, based on DMILP, has a weekly planning period with hourly time steps and considers many features of the hydropower plant operation, such as: start-up costs, evaporation, wear and tear costs, etc. The results indicate that WV of a hydropower plant are very sensitive to the presence of these constraints; ϕ especially during the wettest season and ρ during the driest one. As the severity of ϕ and ρ increase, WV increase and decrease, respectively.

Robust and Fuzzy Optimisation Models for a Flow shop Scheduling Problem with Sequence Dependent Setup Times: A real case study on a PCB assembly company

In this paper, authors considered a flow shop scheduling problem with sequence dependent set-up times in an uncertain environment. Its objective function is to minimise weighted mean completion time. As for uncertainty, set-up and processing times are considered not to be deterministic. Authors propose two different approaches to deal with uncertainty of input data: robust optimisation (RO) and fuzzy optimisation. First, a deterministic mixed-integer linear programming model is presented for the general problem. Then, its robust counterpart of the proposed model is dealt with. Afterwards, the fuzzy flow shop model is developed. Moreover, a real case study on Tehran-Madar Company which is a producer of printed circuit board and OEMs is studied. Finally, a considerable discussion is held on comparison of all three approaches of namely deterministic, fuzzy and ROs based on some generated numerical examples.

Enabling Urban Logistics Services at La Poste through Multi-Echelon Location-Routing

We present a large-scale static and deterministic mixed-integer linear programming (MILP) model solving a two-echelon capacitated location-routing problem (2E-CLRP) with modal choice in the context of urban logistics services (ULS). This model aims to support the development of profitable ULS by guiding the strategic decision making of postal operators as they design an optimal facility network and vehicle fleet for the centralized consolidation and transportation of inbound and outbound urban freight flows. After comprehensively analyzing operating data from La Poste, we identify the key determinants of an optimal infrastructure and fleet design for the centralized coordination and consolidation of urban freight flows under a global service time constraint. Further, we discuss the optimal design’s sensitivity to changes in the input data and parameters of the 2E-CLRP model. The main theoretical contributions of this work are an optimal routing cost estimation formula and an optimization heuristic. Together these allow us to solve the large-scale MILP problem presented here within a reasonable time and with little loss of precision.