Deterministic Mixed-integer Programming Research Articles

Data-Driven Dynamic Optimization for Real-Time Parking Reservation Considering Parking Unpunctuality

This paper examines a real-time parking reservation service that dynamically allocates a limited number of parking slots (which can be reused) to serve parking requests in a crowded area. The study takes into account two stochastic factors: (i) random arrivals of parking requests and (ii) driver parking unpunctuality, which are seldom simultaneously considered in the existing literature. This study formulates the dynamic parking resource allocation problem as a joint chance-constrained model to maximize the expected total revenue over a finite horizon. Due to the complex nature of the model considering the two stochastic factors, the study proposes a data-driven approach using joint chance constraint decomposition and sample average approximation to approximate the model to a deterministic mixed-integer programming model. Furthermore, a stratified ranked set sampling method is introduced to construct a high-quality sample set as the input to the deterministic model, and valid inequalities are designed to accelerate the solution process. Numerical experiments based on real-world data are conducted to validate the performance of the proposed approach under multiple parking scenarios and provide findings.

Multiskilled personnel assignment with k-chaining considering the learning-forgetting phenomena

Multiskilling is a workforce flexibility strategy where companies educate workers to perform a set of task types effectively. When the multiskilling plans are structured using k-chaining policies, it is possible to obtain the maximum flexibility to match the uncertain workforce demand. This work evaluates the potential benefits of multiskilled workers using a k-chaining policy with k≥2, considering the learning/forgetting phenomena to model a heterogeneous workforce. We propose a deterministic mixed-integer programming model to compute the level of required multiskilling. The mathematical formulation determines how many workers should be single-skilled and multiskilled, which task types they should be trained in, the allocation of working hours, and the expected productivity of each worker on each week of the planning horizon. We test our methodology on a case study using real, processed, and simulated data from a Chilean retail store. We perform three experiments, comparing them: zero multiskilling, k-chaining with k≥2 and homogeneous workforce, and k-chaining with k≥2 and heterogeneous workforce. We consider nine variability levels in the workforce demand for each experiment. The results show that modeling the workforce as homogeneous leads to underestimating the multiskilling level required to minimize understaffing. Incorporating heterogeneous workforce modeling through the learning-forgetting phenomena suggests more multiskilling to compensate for the lower workers’ productivity. We consider this solution is closer to the actual operation of the store. We also perform a sensitivity analysis on the learning rate parameter to evaluate the stability of the report solutions for each variability level.

Program Arrives Home Smoothly: Uncertainty-Based Routing Scheduling of Home-Based Elderly Care Programs

In China, the home-based elderly care program system plays an important role in meeting the needs of elderly individuals. The routing scheduling of home-based elderly care (RSHEC) programs is closely related to the quality of the home-based elderly care programs. The structural supply problem faced by home-based elderly care programs is a prominent problem, and the RSHEC programs are an important aspect that has rarely been studied. This paper explores RSHEC programs under uncertainty by comprehensively considering the costs of home-based elderly care, such as the fixed costs, time, and transportation costs. First, a deterministic mixed integer programming (MIP) model was constructed to solve the general routing scheduling problem. In addition, to effectively cope with the uncertainty and risk of the modern market, the robust optimization theory and algorithm model are introduced, namely, the mixed integer box set robust optimization (MIBRO) model and the mixed integer ellipsoid set robust optimization (MIERO) model. Finally, MATLAB and the Gurobi package are applied to obtain the solutions of the models. The case verification shows that the MIP model has the lowest total cost under deterministic conditions. However, the MIERO and MIBRO models can achieve more robust RSHEC programs under uncertain conditions. The results prove the effectiveness and feasibility of the optimization model and algorithm, which provides reference value for management decisions regarding home-based elderly care programs.

Multi-product multi-region supply chain optimisation for seasonal crops

There is a large demand for seasonal crops year-round within Australia, even when they are considered out of season. The demand is satisfied by continually moving production throughout the year to climates where the crops are in season. Management of the supply chain for a major national grower is challenging for several reasons. Due to the large number of planting decisions, it is standard practice for a team of production planners to create the annual production plan. For fresh food production, the supply chain does not contain intermediate storage requiring that the production plan is carefully timed not to waste resources with overproduction. In this paper, we develop a supply chain model that simultaneously manages the production of multiple crops across many growing regions. Production is set to satisfy the demand of multiple end-products while considering the packing plants' throughput capacity, each growing region's harvest capacity, and farm capacity. A time delay may be applied when moving between stages of the supply chain due to the geographic scale being modelled. A deterministic Mixed Integer Program is used to find the optimal planting plan, which minimises the deviation from demand for all products year-round at a minimal cost. Due to the excessive runtime for solving the model, a heuristic solution method is introduced. Numerical experiments demonstrate the advantage of the proposed model over the current manual planning process, which can solve the problem faster and with less deviation across the planning horizon.

A branch-and-price algorithm for robust parallel batch scheduling problem with uncertain size

PurposeIn intelligent scheduling, parallel batch processing can reasonably allocate production resources and reduce the production cost per unit product. Hence, the research on a parallel batch scheduling problem (PBSP) with uncertain job size is of great significance to realize the flexibility of product production and mass customization of personalized products.Design/methodology/approachThe authors propose a robust formulation in which the job size is defined by budget constrained support. For obtaining the robust solution of the robust PBSP, the authors propose an exact algorithm based on branch-and-price framework, where the pricing subproblem can be reduced to a robust shortest path problem with resource constraints. The robust subproblem is transformed into a deterministic mixed integer programming by duality. A series of deterministic shortest path problems with resource constraints is derived from the programming for which the authors design an efficient label-setting algorithm with a strong dominance rule.FindingsThe authors test the performance of the proposed algorithm on the extension of benchmark instances in literature and compare the infeasible rate of robust and deterministic solutions in simulated scenarios. The authors' results show the efficiency of the authors' algorithm and importance of incorporating uncertainties in the problem.Originality/valueThis work is the first to study the PBSP with uncertain size. To solve this problem, the authors design an efficient exact algorithm based on Dantzig–Wolfe decomposition. This can not only enrich the intelligent manufacturing theory related to parallel batch scheduling but also provide ideas for relevant enterprises to solve problems.

Optimisation of long-term quarry production scheduling under geological uncertainty to supply raw materials to a cement plant

ABSTRACT The success of a cement production project depends on the supply of raw materials. Long-term quarry production scheduling (LTQPS) based on resource models is essential to maintain a consistent supply to cement plants. Geological uncertainty is inherent due to sparse exploration data in resource models and significant risk factors for not achieving production targets. This research proposes a stochastic framework for LTQPS that considers the impact of geological uncertainty on raw material supply. A clustering algorithm uses multiple simulated deposit models to aggregate blocks into mining cuts. A new stochastic mixed-integer programming model is formulated with two objectives: to minimise the cost for developing the raw mix and the risk of not meeting production targets. The proposed framework is implemented successfully in a limestone deposit in Southern Vietnam, resulting in an increase of 5 million tons (Mt) and a 30% reduction in unit cost over the deterministic mixed-integer programming model.

Distributionally robust chance-constrained programs with right-hand side uncertainty under Wasserstein ambiguity

We consider exact deterministic mixed-integer programming (MIP) reformulations of distributionally robust chance-constrained programs (DR-CCP) with random right-hand sides over Wasserstein ambiguity sets. The existing MIP formulations are known to have weak continuous relaxation bounds, and, consequently, for hard instances with small radius, or with large problem sizes, the branch-and-bound based solution processes suffer from large optimality gaps even after hours of computation time. This significantly hinders the practical application of the DR-CCP paradigm. Motivated by these challenges, we conduct a polyhedral study to strengthen these formulations. We reveal several hidden connections between DR-CCP and its nominal counterpart (the sample average approximation), mixing sets, and robust 0–1 programming. By exploiting these connections in combination, we provide an improved formulation and two classes of valid inequalities for DR-CCP. We test the impact of our results on a stochastic transportation problem numerically. Our experiments demonstrate the effectiveness of our approach; in particular our improved formulation and proposed valid inequalities reduce the overall solution times remarkably. Moreover, this allows us to significantly scale up the problem sizes that can be handled in such DR-CCP formulations by reducing the solution times from hours to seconds.

Robust multi-class multi-period patient scheduling with wait time targets

Long wait times for health care services is a known challenge in most health care systems. This is partially due to limited capacity and increased demand, but also due to sub-optimal scheduling policies. In this paper, we consider a health system in which patients are prioritized based on their acuity level. We assume that there is a wait time target for each acuity level to ensure that patients of lower acuity do not wait for an unreasonable amount of time while higher acuity patients are being served. We apply a robust optimization (RO) approach to schedule patients over a multi-period finite horizon considering the wait targets. First, we present a deterministic mixed-integer programming model which considers patient priorities, available capacity, and wait time targets for each priority level. We then investigate the robust counterpart of the model by considering uncertainty in demand and employing the notion of budget of uncertainty. Finally, we numerically compare the proposed robust model with the deterministic method. Our results demonstrate that the proposed robust approach provides solutions with higher service levels and lower wait times. Our results also provide insights on how expanding capacity and choosing the level of uncertainty affect the performance of the system.

Two-Stage Stochastic Programming for the Refined Oil Secondary Distribution With Uncertain Demand and Limited Inventory Capacity

In recent years, more and more oil companies adopt initiative delivery mode to make the refined oil secondary distribution scheme. In this work, we focus on the optimization problem of refined oil secondary distribution based on the initiative distribution mode considering stochastic demand and the limited inventory capacity of each petrol station. We present a two-stage stochastic programming model that determines the replenishment quantity of each petrol station based on its existing stock and the available supply quantity of each oil depot, as well as transportation schedule. When the uncertainty in demand can be captured via a finite set of scenarios, the two-stage stochastic programming model is transformed into an equivalent deterministic mixed integer programming model that can be efficiently solved by CPLEX solver. The effectiveness of the two-stage stochastic programming model is verified by simulation on extensive computer-generated instances. To solve practical problems with a large number of scenarios, we propose a method to reduce the problem scale by merging similar scenarios. We demonstrate that compared to the optimal solution obtained from the model with all scenarios, the gap corresponding to the model with merged scenarios is always less than 1%. The results of the sensitivity analysis show that an increase in the inventory capacity leads to a decrease in the total cost within a certain range. The results of this study can help companies making refined oil secondary distribution plan.

When to go electric? A parallel bus fleet replacement study

United States in 2017 emitted about 14.36% of the total global Greenhouse Gas (GHG), 27% of which comes from the transportation sector. In order to address some of these emission sources, alternative fuel technology vehicles are becoming more progressive and market ready. Transit agencies are making an effort to reduce their carbon footprint by adopting these technologies. The overarching objective of this paper is to aid transit agencies make more informed decisions regarding the process of replacing a diesel fleet with alternative-technology buses to minimize GHG emissions. This study investigates the complete course of fleet replacement using a deterministic mixed integer programming. Bus fleet replacement is optimized by minimizing the Life Cycle Cost (LCC) of owning and operating a fleet of buses and required infrastructures while reducing GHG emission simultaneously. Buses operated by Connecticut Department of Transportation (CTDOT) were used as a case-study. Results also show a significant reduction in both cost and emission for optimized replacement schedule vs the unoptimized one. Results also show that a fleet consisting of 79% Battery Electric Bus and 21% Diesel Hybrid Bus yields the least cost solution which conforms to the other operational and environmental constraints. This study also includes various sensitivity tests, that illustrates that although the magnitude of the results may vary depending on the input data, the direction remains the same. The problem formulated in this study can help any transit agencies determine the most optimized solution to their fleet replacement problem under customizable constraints or desired set of outcomes.

Benders decomposition approach with heuristic improvements for the robust foodgrain supply network design problem

We investigate the problem of designing a supply chain for foodgrains distribution under uncertainty, which is motivated from the Indian public distribution system (PDS). Initially, a deterministic mixed-integer programming (MIP) model is formulated to determine the strategic decisions of storage and transportation capacity hiring and the tactical decisions of procurement, inventory, and transportation planning to minimise the total cost over a finite planning horizon. The storage capacity hiring decision is characterised by the options of hiring under long-term contract for the entire planning horizon and from spot market for individual period. Next, we extend the model to consider the uncertainties associated with supply, demand and procurement parameters, and propose a scenario-based robust optimisation model to minimum the total relative regret. An MIP-based fix-and-optimise heuristic is proposed to efficiently solve the deterministic equivalent problem of each scenario. To solve the robust model, we implement Benders decomposition algorithm (BDA) with several heuristic techniques including warm-start strategy to build an initial feasible solution, trust region for master problem, and logical inequalities to accelerate the performance of the BDA. Finally, the effectiveness of the solution approach is demonstrated through extensive computational experiments using the test instances simulating the Indian PDS.

Mixed-integer programming models for optimal constellation scheduling given cloud cover uncertainty

We consider the problem of scheduling observations on a constellation of remote sensors, to maximize the aggregate quality of the collections obtained. While automated tools exist to schedule remote sensors, they are often based on heuristic scheduling techniques, which typically fail to provide bounds on the quality of the resultant schedules. To address this issue, we first introduce a novel deterministic mixed-integer programming (MIP) model for scheduling a constellation of one to n satellites, which relies on extensive pre-computations associated with orbital propagators and sensor collection simulators to mitigate model size and complexity. Our MIP model captures realistic and complex constellation-target geometries, with solutions providing optimality guarantees. We then extend our base deterministic MIP model to obtain two-stage and three-stage stochastic MIP models that proactively schedule to maximize expected collection quality across a set of scenarios representing cloud cover uncertainty. Our experimental results on instances of one and two satellites demonstrate that our stochastic MIP models yield significantly improved collection quality relative to our base deterministic MIP model. We further demonstrate that commercial off-the-shelf MIP solvers can produce provably optimal or near-optimal schedules from these models in time frames suitable for sensor operations.

Optimizing an equilibrium supply chain network design problem by an improved hybrid biogeography based optimization algorithm

This paper presents a new equilibrium optimization method for supply chain network design (SCND) problem under uncertainty, where the uncertain transportation costs and customer demands are characterized by both probability and possibility distributions. We introduce cost risk level constraint and joint service level constraint in the proposed optimization model. When the random parameters follow normal distributions, we reduce the risk level constraint and the joint service level constraint into their equivalent credibility constraints. Furthermore, we employ a sequence of discrete possibility distributions to approximate continuous possibility distributions. To enhance solution efficiency, we introduce the dominance set and efficient valid inequalities into deterministic mixed-integer programming (MIP) model, and preprocess the valid inequalities to obtain a simplified nonlinear programming model. After that, a hybrid biogeography based optimization (BBO) algorithm incorporating new solution presentation and local search operations is designed to solve the simplified optimization model. Finally, we conduct some numerical experiments via an application example to demonstrate the effectiveness of the designed hybrid BBO.

Integrated Semiconductor Supply Chain Production Planning

In this paper, we propose a deterministic mixed-integer programming model to schedule weekly production quantities for semiconductor manufacturing in order to meet forecasted demand over a six-month planning horizon. The model considers facility capacities, facility qualifying, limits on the number of facilities to use, and unit-level minimum inventory requirements. Model validity is demonstrated with a simple test case based on an industry dataset. A case study is then presented using actual industry data. Minimum inventory requirement levels are varied and compared to results obtained from traditional safety stock policies.

A tractable two-stage robust winner determination model for truckload service procurement via combinatorial auctions

A combinatorial auction is one of the adopted mechanisms for truckload (TL) service procurement. In such an auction, the shipper faces a well-known winner determination problem (WDP): the shipper, as the auctioneer, is given bids submitted by a group of carriers. In most literature, WDP is modeled as a deterministic mixed-integer program (MIP) and is solved by standard MIP algorithms. However, in practice, the exact shipping demand is unavailable until after the auction. This shipment volume uncertainty has a significant impact on the solution to WDP. Therefore, a deterministic winner determination model with an estimate of shipment volume may not provide solutions that attain low procurement costs. This paper proposes a new tractable two-stage robust optimization (RO) approach to solve WDP for TL service procurement under shipment volume uncertainty. Assuming that only historical data is available, we propose a data-driven approach based on the central limit theorem (CLT) to construct polyhedral uncertainty sets. In particular, we consider two random cases: independent shipment volume and correlated shipment volume. A two-stage RO model with integer first-stage decision variables and continuous recourse variables is then formulated. We develop a reformulation solution method and use numerical tests to demonstrate that it is much more computationally efficient than the widely adopted Benders’ type constraint generation algorithm. We demonstrate by numerical tests that real-world sized instances of TL service procurement problems can be solved by our proposed robust method. Moreover, we compare our robust approach with benchmark and show that it is more tractable and robust to uncertainty.

Toward Optimal Day-Ahead Scheduling and Operation Control of Microgrids Under Uncertainty

In this paper, we present optimal day-ahead scheduling and control models for microgrids under uncertainty. The traditional deterministic mixed integer programming used in wholesale market fails to produce adequate results, especially when variations and risks are high. We develop stochastic programming optimization models with risk neutral and risk-averse options. We also characterize microgrids with their mean design capacity configuration and operational volatility. A dissimilarity measure determines the relative volatility between two microgrids. We demonstrate that risk-based decision-making in microgrids can significantly change the day-ahead scheduling and control outcomes, and capacity configuration and volatility measures are determining factors for microgrid risks and savings.

Probabilistic Set Covering with Correlations

We consider two variants of a probabilistic set covering (PSC) problem. The first variant assumes that there is uncertainty regarding whether a selected set can cover an item, and the objective is to determine a minimum-cost combination of sets so that each item is covered with a prespecified probability. The second variant seeks to maximize the minimum probability that a selected set can cover all items. To date, literature on this problem has focused on the special case in which uncertainties are independent. In this paper, we formulate deterministic mixed-integer programming models for distributionally robust PSC problems with correlated uncertainties. By exploiting the supermodularity of certain substructures and analyzing their polyhedral properties, we develop strong valid inequalities to strengthen the formulations. Computational results illustrate that our modeling approach can outperform formulations in which correlations are ignored and that our algorithms can significantly reduce overall computation time.

Stochastic optimization of the daily operation of wind farm and pumped-hydro-storage plant

Abstract This paper proposes a new coordination operation mode of wind farm (WF) and pumped-hydro-storage plant (PHSP) based on day-ahead wind power output forecasts. Firstly, a deterministic mixed integer programming (MIP) formulation is built considering the constraints of unit total startup and shutdown frequencies, as well as unit state exclusion between pumping and generating. Secondly, the paper presents the chance-constrained and scenario-based optimization formulations to deal with errors of wind power forecast. These two formulations target on maximizing the expected profit of joint operation of WF and PHSP. Case studies and sensitivity analyses are conducted, which demonstrate that the coordination of WF and PHSP can greatly alleviate the negative effect of wind power fluctuations on power grid and also increase remarkable profit.

Inventory policies for humanitarian aid during hurricanes

In this work we present a stochastic programming model minimizing costs, to support the decision process of inventory policy which best satisfies the demand for food in shelters when hurricane winds are about to impact a town. In this model we consider perishable products as well as the first in first out (FIFO) system for their consumption. In order to make the model closer to reality ordering cost is time-varying and we add a penalty cost in case the shortage exceeds a known limit for two days in a row. Finally the cost to dispose of expired food is greater than the purchase cost of the product since throwing away food has ethical implications. Starting from a stochastic programming model, we present a procedure to transform it to a deterministic mixed integer programming model (MIP) with non-convex objective function over its entire domain, which closely states the situation in reality. Preliminary computational results and discussion are presented.

Optimal Policies for Crude Oil Selection

In this paper, we investigate the crude oil selection problem derived from the production of a large typical oil refinery. A deterministic mixed integer programming model for making purchasing decisions in the refinery is proposed in which practical transportation requirements are considered. The objective is to select the varieties of crude oil and determine the purchasing, production and inventory quantities under complicated material-balance constraints and capacity constraints for minimizing the sum of purchasing, order, inventory and transportation costs. A numerical example indicates the extensive application of the presented model.