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Practical probabilistic programming
Data accumulated about customers, products, and website users can not only help interpret the past, it can help predict the future! Probabilistic programming is a programming paradigm in which code models are used to draw probabilistic inferences from data. By applying specialized algorithms, programs assign degrees of probability to conclusions and make it possible to forecast future events like sales trends, computer system failures, experimental outcomes, and other critical concerns. This book explains how to use the PP paradigm to model application domains and express those probabilistic models in code.
Book
Mini-batch stochastic approximation methods for nonconvex stochastic composite optimization
This paper considers a class of constrained stochastic composite optimization problems whose objective function is given by the summation of a differentiable (possibly nonconvex) component, together with a certain non-differentiable (but convex) component. In order to solve these problems, we propose a randomized stochastic projected gradient (RSPG) algorithm, in which proper mini-batch of samples are taken at each iteration depending on the total budget of stochastic samples allowed. The RSPG algorithm also employs a general distance function to allow taking advantage of the geometry of the feasible region. Complexity of this algorithm is established in a unified setting, which shows nearly optimal complexity of the algorithm for convex stochastic programming. A post-optimization phase is also proposed to significantly reduce the variance of the solutions returned by the algorithm. In addition, based on the RSPG algorithm, a stochastic gradient free algorithm, which only uses the stochastic zeroth-order information, has been also discussed. Some preliminary numerical results are also provided.
Journal Article
Solving Stochastic and Bilevel Mixed-Integer Programs via a Generalized Value Function
We introduce a generalized value function of a mixed-integer program, which is simultaneously parameterized by its objective and right-hand side. We describe its fundamental properties, which we exploit through three algorithms to calculate it. We then show how this generalized value function can be used to reformulate two classes of mixed-integer optimization problems: two-stage stochastic mixed-integer programming and multifollower bilevel mixed-integer programming. For both of these problem classes, the generalized value function approach allows the solution of instances that are significantly larger than those solved in the literature in terms of the total number of variables and number of scenarios.
Journal Article
Stochastic home health care routing and scheduling problem with multiple synchronized services
The home health care (HHC) covers a wide range of health care services carried out in patients’ home in case of illness, injury or aging. Each caregiver should as far as possible adhere to the schedule set by the decision maker. However, unforeseen events would sometimes occur and delay the delivery of care services, which will qualify the service as poor or even risky. Deterministic models ignore this uncertainty, which can arise at any time and will therefore lead to non-compliance with the predefined schedule. Furthermore, patients need several care activities per day, and some of them require to be simultaneous by their nature such as dressing, getting out of bed and bathing. In this work, a stochastic programming model with recourse (SPR model) is proposed to deal with the home health care routing and scheduling problem (HHCRSP) where uncertainties in terms of traveling and caring times that may occur as well as synchronization of services are considered. The objective is to minimize the transportation cost and the expected value of recourse, which is estimated using Monte Carlo simulation. The recourse is defined as a penalty cost for patients’ delayed services and a remuneration for caregivers’ extra working time. The deterministic model is solved by CPLEX, the genetic algorithm (GA) and the general variable neighborhood search (GVNS) based heuristics. The SPR model is solved by Monte Carlo simulation embedded into the GA. Computational results highlight the efficiency of GVNS and GA based heuristics and the complexity of the SPR model in terms of CPU running times.
Journal Article
A Robust Optimization Perspective on Stochastic Programming
In this paper, we introduce an approach for constructing uncertainty sets for robust optimization using new deviation measures for random variables termed the forward and backward deviations . These deviation measures capture distributional asymmetry and lead to better approximations of chance constraints. Using a linear decision rule, we also propose a tractable approximation approach for solving a class of multistage chance-constrained stochastic linear optimization problems. An attractive feature of the framework is that we convert the original model into a second-order cone program, which is computationally tractable both in theory and in practice. We demonstrate the framework through an application of a project management problem with uncertain activity completion time.
Journal Article
Two-stage linear decision rules for multi-stage stochastic programming
Multi-stage stochastic linear programs (MSLPs) are notoriously hard to solve in general. Linear decision rules (LDRs) yield an approximation of an MSLP by restricting the decisions at each stage to be an affine function of the observed uncertain parameters. Finding an optimal LDR is a static optimization problem that provides an upper bound on the optimal value of the MSLP, and, under certain assumptions, can be formulated as an explicit linear program. Similarly, as proposed by Kuhn et al. (Math Program 130(1):177–209, 2011) a lower bound for an MSLP can be obtained by restricting decisions in the dual of the MSLP to follow an LDR. We propose a new approximation approach for MSLPs, two-stage LDRs. The idea is to require only the state variables in an MSLP to follow an LDR, which is sufficient to obtain an approximation of an MSLP that is a two-stage stochastic linear program (2SLP). We similarly propose to apply LDR only to a subset of the variables in the dual of the MSLP, which yields a 2SLP approximation of the dual that provides a lower bound on the optimal value of the MSLP. Although solving the corresponding 2SLP approximations exactly is intractable in general, we investigate how approximate solution approaches that have been developed for solving 2SLP can be applied to solve these approximation problems, and derive statistical upper and lower bounds on the optimal value of the MSLP. In addition to potentially yielding better policies and bounds, this approach requires many fewer assumptions than are required to obtain an explicit reformulation when using the standard static LDR approach. A computational study on two example problems demonstrates that using a two-stage LDR can yield significantly better primal policies and modestly better dual policies than using policies based on a static LDR.
Journal Article
A novel multi-objective robust fuzzy stochastic programming model for sustainable agri-food supply chain: case study from an emerging economy
On the one hand, supply chain management of agri-food products under uncertain conditions has a significant impact on food security and, on the other hand, increases the profits of supply chain components. Moreover, considering the sustainability concepts leads to more social and environmental benefits. The present study investigates the canned food supply chain under uncertain conditions and sustainability concepts by considering strategic and operational decisions and different characteristics. The proposed model is a multi-echelon, multi-period, multi-product, multi-objective location-inventory-routing problem (LIRP) in which the vehicle fleet is considered heterogeneously. The objectives of this model are to (1) minimize costs, (2) minimize customer dissatisfaction, (3) maximize production throughput, and (4) maximize job opportunities. In this study, carbon cap and trade mechanism are used to minimize environmental damage. Robust fuzzy stochastic programming (RFSP) is employed to cope and control uncertainties. The multi-objective optimization problem is implemented on a real case and solved using the Torabi and Hassini (TH) method. The results of this study showed that with increasing confidence levels, the severity of the problem increased and the values of the objective functions worsened. Also, using the relative value of stochastic solution (RVSS) criterion demonstrated that the effect of utilizing the RFSP approach on the first and second objective functions was higher than that the nominal approach showed itself. Finally, sensitivity analysis is performed on two parameters: the selling price of products to foreign customers and the cost of purchasing products from farms. The results of this study showed that changing these two parameters had a significant effect on the first and second objective functions.
Journal Article
Portfolio selection problem: a review of deterministic and stochastic multiple objective programming models
The literature on portfolio selection mostly concentrates on computational analysis rather than on modelling efforts. In response, this paper provides a comprehensive literature review of multiple objective deterministic and stochastic programming models for the portfolio selection problem. First, we summarize different concepts related to portfolio selection theory, including pricing models and portfolio risk measures. Second, we report the mathematical models that are generally used to solve deterministic and stochastic multiple objective programming problems. Finally, we present how these models can be used to solve the portfolio selection problem.
Journal Article
A data-driven optimization model to response to COVID-19 pandemic: a case study
COVID-19 is a highly prevalent disease that has led to numerous predicaments for healthcare systems worldwide. Owing to the significant influx of patients and limited resources of health services, there have been several limitations associated with patients' hospitalization. These limitations can cause an increment in the COVID-19-related mortality due to the lack of appropriate medical services. They can also elevate the risk of infection in the rest of the population. The present study aims to investigate a two-phase approach to designing a supply chain network for hospitalizing patients in the existing and temporary hospitals, efficiently distributing medications and medical items needed by patients, and managing the waste created in hospitals. Since the number of future patients is uncertain, in the first phase, trained Artificial Neural Networks with historical data forecast the number of patients in future periods and generate scenarios. Through the use of the K-Means method, these scenarios are reduced. In the second phase, a multi-objective, multi-period, data-driven two-stage stochastic programming is developed using the acquired scenarios in the previous phase concerning the uncertainty and disruption in facilities. The objectives of the proposed model include maximizing the minimum allocation-to-demand ratio, minimizing the total risk of disease spread, and minimizing the total transportation time. Furthermore, a real case study is investigated in Tehran, the capital of Iran. The results showed that the areas with the highest population density and no facilities near them have been selected for the location of temporary facilities. Among temporary facilities, temporary hospitals can allocate up to 2.6% of the total demand, which puts pressure on the existing hospitals to be removed. Furthermore, the results indicated that the allocation-to-demand ratio can remain at an ideal level when disruptions occur by considering temporary facilities. Our analyses focus on: (1) Examining demand forecasting error and generated scenarios in the first phase, (2) exploring the impact of demand parameters on the allocation-to-demand ratio, total time and total risk, (3) investigating the strategy of utilizing temporary hospitals to address sudden changes in demand, (4) evaluating the effect of disruption to facilities on the supply chain network.
Journal Article