Deterministic Mixed-integer Linear Programming Model Research Articles

En-route charge scheduling for an electric bus network: Stochasticity and real-world practice

Bus electrification, with its high energy conversion efficiency and zero tailpipe emissions, has a significant impact on addressing the petroleum crisis and lowering carbon emissions in urban transportation. However, long charging times and uncertain operational risks hinder their wider adoption. This paper explores the robust en-route charge scheduling problem for electric buses to overcome uncertain energy consumption while ensuring charging accessibility. The problem is first described as a deterministic mixed-integer linear programming model that employs a time-of-use electricity price to balance the charging demand. Additionally, a robust counterpart model is formulated to account for stochastic energy. Numerical studies are designed to evaluate the proposed model in a real network, followed by a sensitivity analysis to investigate the impact of battery type, fleet composition, and depth of discharge on the charging schedule. The results show that the robust optimization model provides system feasibility against uncertainty with a comparable price and emissions.

A novel stochastic programming model under endogenous uncertainty for the CCS-EOR planning problem

Carbon-capture-and-storage (CCS) is one of the leading technologies to reduce CO2 emissions. A commercial way to deploy CCS on a large scale is to sequestrate CO2 in depleted oil reservoirs and to combine it with enhanced oil recovery (EOR) operations. In this manner, not only the CO2 emission is reduced, but also the oil production increases. The collaborative CCS-EOR planning problem determines the proper allocation of available CO2 to depleted reservoirs and the scheduling of the EOR operations. This problem is of great importance, especially when there are multiple oil reservoirs. This paper presents a deterministic mixed-integer linear programming model as an improvement of an existing model in the literature. Then, it is extended to a multistage stochastic model with endogenous uncertainty in which the parameters expressing the initial oil yields and the periodic depletion factor of oil yields associated with reservoirs are uncertain, and the time of uncertainty realization is decision-dependent. Our deterministic model is computationally more efficient than the existing model in the literature, due to the reduction of binary variables to about one-third. Also, providing the possibility of selecting pipeline types among different options as well as incorporating uncertainty may lead to a significant cost-saving. The proposed models are examined over two case-studies taken from the literature. The results indicate that in comparison to the deterministic model, the cost-saving achieved by incorporating uncertainty is about 8.8%, on average.

A joint location-distribution-inventory model for a multi-echelon supply chain network considering multi-sourcing, drop-shipping, and lateral transhipment

Competitive pressure in the global marketplace requires enterprises to establish efficient distribution planning and inventory management in a supply chain. This study proposes a framework to optimise distribution and inventory decisions, integrating decisions of opening or closing facilities in a multi-echelon supply chain network under non-stationary demand. The model adopts distribution requirements planning (DRP) and considers flexible distribution strategies, including multiple sourcing, drop-shipping, and lateral transhipment to minimise the total expected cost. The original stochastic model has been transformed into an equivalent deterministic mixed-integer linear programming model by using the sample average approximation (SAA) technique. Numerical experiments are conducted to validate the model and its practical applications. The model that adopted multiple flexible distribution strategies outperforms the models that adopted a single distribution strategy. This study is useful for enterprises who sell products through their physical stores and online stores on e-commerce platforms or company private websites.

Robust strategic planning of dynamic wireless charging infrastructure for electric buses

Electromobility in public bus systems is growing rapidly and experiencing a fundamental transformation in their infrastructure and operations. The dilemma of limited driving range and charging time of battery electric buses (BEBs) hinders their adoption. A novel approach to address BEB limitations is to utilize dynamic wireless charging (DWC) technology that allows buses to charge while in motion. This paper aims to analyze robust strategic planning of DWC and BEB fleet scheduling based on a real bus network at Binghamton University. The problem is first formulated as a new deterministic mixed-integer linear programming model to simultaneously optimize both the charging planning problem and fleet scheduling problem in an integrated fashion. To address the uncertainty of energy demand and charging time, a robust counterpart model (RCM) has been derived. To increase RCM flexibility, the battery status variable is formulated in a cumulative form. Dependent and independent budget uncertainty sets have been developed to control the robustness. A sensitivity analysis has been conducted to study the system behavior in response to different charging types, auxiliary energy demand, depth of discharge, charging options at terminals, battery degradation, and electricity cost. The deterministic model shows that eight homogeneous BEBs are required to operate on the selected routes with a battery capacity of 54.01 kWh and a total cost of $3,636,347. The results show that joint planning of charging infrastructure and fleet scheduling can save 19.2% of total cost compared to disjoint planning. The RCM results in 10 BEBs to ensure feasiblility against uncertainty.

A two-stage stochastic optimization model for the retail multiskilled personnel scheduling problem: a k-chaining policy with k≥2.

Considering an uncertain demand, this study evaluates the potential benefits of using a multiskilled workforce through a k-chaining policy with k≥2. For the service sector and, particularly for the retail industry, we initially propose a deterministic mixed-integer linear programming model that determines how many employees should be multiskilled, in which and how many departments they should be trained, and how their weekly working hours will be assigned. Then, the deterministic model is reformulated using a two-stage stochastic optimization (TSSO) model to explicitly incorporate the uncertain personnel demand. The methodology is tested for a case study using real and simulated data derived from a Chilean retail store. We also compare the TSSO approach solutions with the myopic approaches' solutions (i.e., zero and total multiskilling). The case study is oriented to answer two key questions: how much multiskilling to add and how to add it. Results show that TSSO approach solutions always report maximum reliability for all levels of demand variability considered. It was also observed that, for high levels of demand variability, a k-chaining policy with k≥2 is more cost-effective than a 2-chaining policy. Finally, to evaluate the conservatism level in the solutions reported by the TSSO approach, two truncation types in the probability density function (pdf) associated with the personnel demand were considered. Results show that, if the pdf is only truncated at zero (more conservative truncation) the levels of required multiskilling are higher than when the pdf is truncated at 5th and 95th percentiles (less conservative truncation).

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.

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.

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.

A robust optimisation model for manufacturing cell design problem under uncertainty

The concern about significant changes in the manufacturing environment (such as operating/part costs and part demands) has spurred an interest in designing robust dynamic cellular manufacturing systems (RDCMS). Here, first a deterministic mixed-integer linear programming (MILP) model for cell formation is proposed. Cost parameters of model and demand fluctuations are subject to uncertainty. The objective function calculates machine costs, operating cost, internal part production cost, inter-cellular material handling cost, backorder cost, inventory holding cost and subcontracting. Then the robust counterpart of the proposed mixed integer linear programming model is presented by using the recent extensions in robust optimisation theory. Finally, to assess the robustness of the solutions obtained by the novel robust optimisation model, they are compared to those generated by the deterministic mixed-integer linear programming model in a number of realisations under different test problems. Numerical tests show the power of the proposed robust model in handling uncertainty in parameters and generating robust optimal solutions.

A robust optimization approach to closed-loop supply chain network design under uncertainty

The concern about significant changes in the business environment (such as customer demands and transportation costs) has spurred an interest in designing scalable and robust supply chains. This paper proposes a robust optimization model for handling the inherent uncertainty of input data in a closed-loop supply chain network design problem. First, a deterministic mixed-integer linear programming model is developed for designing a closed-loop supply chain network. Then, the robust counterpart of the proposed mixed-integer linear programming model is presented by using the recent extensions in robust optimization theory. Finally, to assess the robustness of the solutions obtained by the novel robust optimization model, they are compared to those generated by the deterministic mixed-integer linear programming model in a number of realizations under different test problems.

Chance constrained programming models for refinery short-term crude oil scheduling problem

The main objective of this work is to put forward chance constrained mixed-integer nonlinear stochastic and fuzzy programming models for refinery short-term crude oil scheduling problem under demands uncertainty of distillation units. The scheduling problem studied has characteristics of discrete events and continuous events coexistence, multistage, multiproduct, nonlinear, uncertainty and large scale. At first, the two models are transformed into their equivalent stochastic and fuzzy mixed-integer linear programming (MILP) models by using the method of Quesada and Grossmann [I. Quesada, I E. Grossmann, Global optimization of bilinear process networks with multicomponent flows, Comput. Chem. Eng. 19 (12) (1995) 1219–1242], respectively. After that, the stochastic equivalent model is converted into its deterministic MILP model through probabilistic theory. The fuzzy equivalent model is transformed into its crisp MILP model relies on the fuzzy theory presented by Liu and Iwamura [B.D. Liu, K. Iwamura, Chance constrained programming with fuzzy parameters, Fuzzy Sets Syst. 94 (2) (1998) 227–237] for the first time in this area. Finally, the two crisp MILP models are solved in LINGO 8.0 based on scheduling time discretization. A case study which has 267 continuous variables, 68 binary variables and 320 constraints is effectively solved with the solution approaches proposed.