Deterministic Linear Programming Model Research Articles

On Solving Stochastic Optimization Problems

Many optimization mathematical models, associated with the technical-economic processes of real-world problems, have elements of uncertainty in their structure, which places them in stochastic optimization programming. Their diversity and complexity, due to the large uncertainty space, require special methods of solving, because there is no general solution method. Within this context, in this paper we consider the category of optimization models that can contain random variable type coefficients and/or imposed probability levels on the constraints. The purpose of the paper is to propose a methodology dedicated to these studied models. Applying the methodology leads to developing a deterministic linear programming model, associated with the initial stochastic model. In fact, the proposed methodology reduces the stochastic formulation to a deterministic formulation. The methodology is illustrated with a numerical case study based on a manufacturing problem. Solving the obtained deterministic model is carried out in the version assisted by a specialized software product (WinQSB Version 2.0). It allows for the performing of a sensitivity analysis of the optimal solution, and/or a parametric analysis relative to certain model coefficients, both also presented in the paper. The main result of the study in this paper is the proposed methodology, which is applicable on a large scale, for any mathematical model of stochastic optimization of the mentioned type, regardless of complexity, dimensions and the domain of the process to which it is associated. The numerical results obtained when applying this methodology indicate its efficiency and effectiveness in finding the solution for the studied models. The approach to this issue in the present paper is determined by the wide range of stochastic optimization problems in the various studied real-life processes and by the imperative need to adopt the best decisions in conditions of uncertainty.

Stochastic network DEA-R models for two-stage systems

PurposeThis paper aims to present two-stage network models in the presence of stochastic ratio data.Design/methodology/approachBlack-box, free-link and fix-link techniques are used to apply the internal relations of the two-stage network. A deterministic linear programming model is derived from a stochastic two-stage network data envelopment analysis (DEA) model by assuming that some basic stochastic elements are related to the inputs, outputs and intermediate products. The linkages between the overall process and the two subprocesses are proposed. The authors obtain the relation between the efficiency scores obtained from the stochastic two stage network DEA-ratio considering three different strategies involving black box, free-link and fix-link. The authors applied their proposed approach to 11 airlines in Iran.FindingsIn most of the scenarios, when alpha in particular takes any value between 0.1 and 0.4, three models from Charnes, Cooper, and Rhodes (1978), free-link and fix-link generate similar efficiency scores for the decision-making units (DMUs), While a relatively higher degree of variations in efficiency scores among the DMUs is generated when the alpha takes the value of 0.5. Comparing the results when the alpha takes the value of 0.1–0.4, the DMUs have the same ranking in terms of their efficiency scores.Originality/valueThe authors innovatively propose a deterministic linear programming model, and to the best of the authors’ knowledge, for the first time, the internal relationships of a two-stage network are analyzed by different techniques. The comparison of the results would be able to provide insights from both the policy perspective as well as the methodological perspective.

Managing Uncertain Capacities for Network Revenue Optimization

Problem definition: We study the problem of managing uncertain capacities for revenue optimization over a network of resources. The uncertainty could be due to (i) the need to reallocate initial capacities among resources or (ii) the random availability of physical capacities by the time of service execution. Academic/practical relevance: The analyzed control policy is aligned with the current industry practice, with a virtual capacity and a bid price associated with each network resource. The seller collects revenues from an arriving stream of customers. Admitted requests that cannot be accommodated within the final, effective capacities incur a penalty cost. The objective is to maximize the total cumulative net revenue (sales revenue minus penalty cost). The problem arises in practice, for instance, when airlines are subject to last-minute change of aircrafts and in cargo revenue management where the capacity left by the passengers’ load is used for freight. Methodology: We present a stochastic dynamic programming formulation for this problem and propose a stochastic gradient algorithm to approximately solve it. All limit points of our algorithm are stationary points of the approximate expected net revenue function. Results: Through an exhaustive numerical study, we show that our controls are computed efficiently and deliver revenues that are almost consistently higher than the ones obtained from benchmarks based on the widely adopted deterministic linear programming model. Managerial implications: We obtain managerial insights about the impact of the timing of the capacity uncertainty clearance, the capacity heterogeneity, the network congestion, and the penalty for not being able to accommodate the previously accepted demand. Our approach tends to offer the best performance across different parameterizations of the problem.

A Chance Constrained Programming Approach for No-Wait Flow Shop Scheduling Problem under the Interval-Valued Fuzzy Processing Time

This work focuses on the study of robust no-wait flow shop scheduling problem (R-NWFSP) under the interval-valued fuzzy processing time, which aims to minimize the makespan within an upper bound on total completion time. As the uncertainty of actual job processing times may cause significant differences in processing costs, a R-NWFSP model whose objective function involves interval-valued fuzzy sets (IVFSs) is proposed, and an improved SAA is designed for its efficient solution. Firstly, based on the credibility measure, chance constrained programming (CCP) is utilized to make the deterministic transformation of constraints. The uncertain NWFSP is transformed into an equivalent deterministic linear programming model. Then, in order to tackle the deterministic model efficiently, a simulated annealing algorithm (SAA) is specially designed. A powerful neighborhood search method and new acceptance criterion are applied to find better solutions. Numerical computations demonstrate the high efficiency of the SAA. In addition, a sensitivity analysis convincingly shows that the applicability of the proposed model and its solution strategy under interval-valued fuzzy sets.

A Methodology and Case Study to Assess SCOR-Make Agility Measures Under Uncertainties

An assessment of agility is not an easy task since agility has been differently defined in literature, and it is not convenient to measure by experiment in practice. The objective of this paper is to propose a methodology to assess the agility performance under uncertainties based on level 1 of SCOR-Make process metric including the upside make flexibility (AG1.1), upside make adaptability (AG1.2), and downsize make adaptability (AG1.3). The proposed methodology consists of predictive models, which are a deterministic linear programming (LP) model and LP model with uncertainties, and algorithms to assess the agility measures. A case study of a bottled-water factory is conducted to demonstrate the application of the proposed methodology. The case study shows that the proposed methodology can effectively determine the agility measures. It can also be adapted to answer other agility related practical questions that are different from the SCOR definition.

A stochastic risk-averse framework for blood donation appointment scheduling under uncertain donor arrivals

Blood is a key resource in all health care systems, usually drawn from voluntary donors. We focus on the operations management in blood collection centers, which is a key step to guarantee an adequate blood supply and a good quality of service to donors, by addressing the so-called Blood Donation Appointment Scheduling problem. Its goal is to employ appointment scheduling to balance the production of blood units between days, in order to provide a reasonably constant supply to transfusion centers and hospitals, and reduce non-alignments between physicians' working times and donor arrivals at the collection center. We consider a two-phase solution framework taken from the literature, in which a deterministic linear programming model preallocates time slots to different blood types and a prioritization policy assigns the preallocated slots to the donors when they make a reservation. However, the problem is stochastic in nature and requires consideration of the uncertain arrivals of non-booked donors. In this work, to include the uncertain arrivals, we propose three stochastic counterparts of the preallocation model based on a risk-neutral objective and two risk-averse objectives, respectively, where the Conditional Value-at-Risk is considered as the risk measure in the last two methods. The resulting stochastic frameworks have been tested considering the historical data of one of the largest Italian collection centers, the Milan Department of the "Associazione Volontari Italiani Sangue" (AVIS). Results show the effectiveness of the stochastic models, especially the mean-risk one, and the need to include the uncertainty of arrivals in order to better balance the production of blood units.

Measuring the efficiency of two-stage network processes: A satisficing DEA approach

Regular network data envelopment analysis (DEA) models deal with evaluating the performance of a set of decision-making units with a two-stage construction in the context of a deterministic data set. In the real world, however, observations may display a stochastic behavior. To the best of our knowledge, despite the existing research done with different data types, studies on two-stage processes with stochastic data are still very limited. This article proposes a two-stage network DEA model with stochastic data. The stochastic two-stage network DEA model is formulated based on the satisficing DEA models of chance-constrained programming and the leader–follower concepts. According to the probability distribution properties and under the assumption of the single random factor of the data, the probabilistic form of the model is transformed into its equivalent deterministic linear programming model. In addition, the relationship between the two stages as the leader and the follower, respectively, at different confidence levels and under different aspiration levels, is discussed. The proposed model is applied to a real case concerning 16 commercial banks in China in order to confirm the applicability of the proposed approach.

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.

Uncertain Portfolio Selection with Borrowing Constraint and Background Risk

Due to the complexity of financial markets, there exist situations where security returns and background factor returns are available mainly based on experts’ subjective beliefs, such as in the case of lack of historical data. To deal with such indeterminate quantities, uncertain variables are introduced. Based on uncertainty theory, this paper discusses the distribution function of the optimal portfolio return. Two types of new uncertain programming models, namely, the chance-mean model and the measure-mean model, are proposed to make an optimal portfolio selection decision in an uncertain environment. It is proved that there exists an equivalent relation between the chance-mean model and a deterministic linear programming model, which leads to an approach to obtain the optimal solutions of the proposed models. Finally, some numerical examples are illustrated to show the modelling ideas and the effectiveness of the models.

Application of stochastic linear programming in managerial accounting

Purpose This paper aims to focus on applications of stochastic linear programming (SLP) to managerial accounting issues by providing a theoretical foundation and practical examples. SLP models may have more implications – and broader ones – in industry practice than deterministic linear programming (DLP) models do. Design/methodology/approach This paper introduces both DLP and SLP methods. In addition, continuous and discrete SLP models are explained. Applications are demonstrated using practical examples and simulations. Findings This research work extends the current knowledge of SLP, especially concerning managerial accounting issues. Through numerical examples, SLP demonstrates its great ability of hedging against all scenarios. Originality/value This study serves as an addition to building a cumulative tradition of research on SLP in managerial accounting. Only a few SLP studies in managerial accounting have focused on the development of such an instrument. Thus, the measurement scales in this research can be used as the starting point for further refining the instrument of optimization in managerial accounting.

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.

The Sustainability of Malaysia’s Defined Contribution Pension System: Implementation of Deterministic Linear Programming

This article investigates, under deterministic linear programming model, asset allocation decision and optimal investment strategy for Malaysia’s defined contribution pension (DC) scheme -Employees Provident Fund (EPF). The model requires generation of scenarios and probabilities to represent future assets and liabilities streams. We employed Vector Autoregressive (VAR) model to generate future returns of five asset classes i.e. equity, money market instrument, Malaysia government bond with 1 and 10 years of maturity date and property. Future liabilities factors were derived from two submodels; population and salary. In population model, the future status of the EPF members was determined using a Markov Chain model. Then, the random factors of assets and liabilities were used in the asset liability model (ALM) based on linear programming (LP) and fixed mix (FM) strategy. The results of the research are grouped in two levels. First, we briefly discuss the finding of the random factor model and then we analyse the optimal investment strategy for the EPF. In terms of finding an optimal investment strategy, the FM strategy generated higher expected terminal wealth than the LP strategy. This finding suggests that FM strategy is superior to the LP strategy. In addition, we find that the higher dividend distributed to the members may result in decreasing of the expected terminal wealth of the fund for both strategies. This portrays that dividend distribution policy may affect the financial soundness of the EPF in the long run

Application of Deterministic, Stochastic and Fuzzy Linear Programming Models in Solid Waste Management Studies: Literature Review

Application of Deterministic, Stochastic and Fuzzy Linear Programming Models in Solid Waste Management Studies: Literature Review

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.

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.

Deterministic allocation models for multi-period demand fulfillment in multi-stage customer hierarchies

In manufacturing environments with heterogeneous (e.g., because of different profit margins) customers the optimal matching of available supply with dynamic demand is a challenging task as soon as supply becomes scarce. In Make-to-Stock systems, Demand Fulfillment first allocates these scarce resources (in form of Available-to-Promise — ATP) to customer segments on basis of their forecasted demand. The resulting quotas are then consumed (“promised”) when real customer orders arrive. In a multi-stage sales hierarchy, this allocation process often has to be executed level by level, on basis of decentral, aggregate information only. Decentral, multi-period, deterministic linear and non-linear programming models are proposed approximating the first-best benchmark of a central, multi-period allocation planning with full information. Roll over simulations have been performed to obtain insights about the behavior of the proposed decentral models and to demonstrate their benefits in comparison with common quantity-based methods, especially for high levels of customer heterogeneity and high shortage rates.

Optimal Allocation of Land and Water Resources in a Canal Command Area in the Deterministic and Stochastic Regimes

Water logging is a universal problem of irrigated agriculture and it is a serious threat to the sustainability of irrigated agriculture in many arid and semiarid regions. Limbasi branch canal command area of Mahi Right Bank Canal (MRBC) project, Gujarat, India is also found to be affected by water logging conditions. Present study deals with the formulation of the Deterministic Linear Programming (DLP) and Chance Constrained Linear Programming (CCLP) models which maximizes net return from a canal command area while simultaneously mitigating water logging conditions. The developed models are applied to the Limbasi branch canal command area. The objective function is to maximize net annual return and decision variables are the seasonal cropping pattern and seasonal water supply. Analysis shows that under optimal conditions in the DLP model, there was a 40% deviation of crop area from existing cropping pattern and Net Irrigation Requirement (NIR) of crops was satisfied by conjunctive use of 41% of canal water and 59% of groundwater (annually). There was 91.1% increase in ground water exploitation which consecutively moderated rising water table issues. Net annual return was found to increase by 46.6%. In the CCLP model, NIR of crops was considered as a stochastic variable and normal distribution was found as the best fit. The CCLP model was run from 2 to 40% risk levels and cropping pattern under 10% risk level was considered as optimal at which NIR was satisfied by conjunctive use of 53.8% of canal water and 46.2% of ground water (annually). There was 86% increase in ground water exploitation. The outcome of the study can be used to assist the water resources planners and managers in taking appropriate decisions to develop a sustainable management plan of land and water resources for an overall balance of the system.

Stochastic network DEA models for two-stage systems under the centralized control organization mechanism

This paper deals with an extension of the conventional DEA formulations by taking into account the serial relationship of two-stage processes with stochastic data. The stochastic two-stage network DEA model is proposed based on centralized control organization mechanism. The stochastic two-stage network DEA model is converted into a deterministic linear programming model under the assumption that components of inputs, outputs and intermediate products are related with some basic stochastic factors. Stochastic efficiency analysis including the relationships of the whole process and the two sub-processes are then discussed. In addition, an example of 16 commercial banks in China is used to calculate the efficiencies under different levels by applying the proposed model.

Uncertain transportation model with rough unit cost, demand and supply

In a transportation problem, the parameters like unit cost of transportation of goods or services from source to destination, supplies from the sources and demands at destinations depend on many factors which may not be deterministic in nature. To deal with the uncertainty of the parameters, random variables and fuzzy variables were used previously. Sometimes, in absence of sufficient sample observations, the uncertain parameters are estimated by the belief degree of the experts. In this paper, the aim is to investigate the transportation problem where the unit cost of transportation, supplies, demands are initially taken as rough variables based on subjective estimation of experts. Further, these rough estimates are suitably approximated as uncertain normal variables and the conceptual uncertain programming model has been developed. The model is then transformed to a deterministic linear programming model by minimizing the expected value of the uncertain objective function under the constraints at certain confidence level.

Stochastic model for hotel room pricing and upgrading

Hotel revenue management concerns allocation of limited rooms to demands from different price categories. Hotel rooms fall into groups of different amenities, each having its according prices. Sometimes, due to shortage of rooms in one category, a demand may be offered a room in a higher category but at the same lower price, called upgrading. Overbooking is also practiced because of late cancelation and no shows. Usually, upgrading is done separately from pricing optimization, which can lead to suboptimal solutions. This paper develops a deterministic linear programming (DLP) model to simultaneously consider hotel room pricing and allocation including overbooking and upgrading through a process that specifically deals with room categories. We further extend our DLP model to a stochastic one to address the uncertainties of demands. Numerical tests are carefully conducted to show the models are effective, and the system has been successfully implemented at several resort hotels.