A study an analysis of stochastic linear programming

: So far the study of stochastic programs with recourse has been limited to the case (called by G. Dantzig programming under uncertainty) when only the right-hand sides or resources of the problem are random. In this paper the authors extend the theory to the general case when essentially all the parameters involved are random. This generalization immediately raises the problem of attributing a precise meaning to the stochastic constraints. They examine a probability formulation (satisfying the constraints almost surely) and a possibility formulation (satisfying the constraints for all values of the random parameters in the support of their joint distribution) and show them equivalent under a rather weak but curious W-condition. Finally, they prove that without restriction the equivalent deterministic form of a stochastic program with recourse is a convex program for which we obtain some additional properties when some of the parameters of the original problem are constant. The applications of the theoretical results of this paper to certain classes of stochastic programs which have arisen from practical problems will be presented in a separate paper: 'Stochastic Programs with Recourse: Special Forms.' (Author)

The study of making the best decision under risk management in a variety of areas of our lives is known as Stochastic Programming. We will go through two-stage Stochastic Linear Programming approaches for a variety of real-world choice issues, as well as how to solve them. We will achieve this by constructing stochastic linear programming models based on real-world situations like the well-known Farmer's situation and News Vendors problems. The influence of pricing, Stochastic Integer Linear Programming problem, second stage Stochastic Integer Linear Programming problem, first stage Stochastic Binary Linear Programming problem, risk aversion problem, and continuous function for random variables based on two-stage SLP with the aid of Farmer's problem will all be examined. We will address the Newsvendor’s problem with Deterministic Equivalent Stochastic Linear Programming, an extension of Deterministic Stochastic Linear Programming for risk aversion with a high number of decision variables and restrictions, utilizing the two-stage Stochastic Linear Programming approach once more. Hand calculation is a challenging way to acquire the solution to the problems. As a result, we will use the programming language AMPL to design computer solutions for tackling both farmer and newsvendor difficulties. We will also utilize MATLAB to create graphs for the farmer's problem's continuous function. Dhaka Univ. J. Sci. 72(1): 30-45, 2024 (January)

A stochastic linear goal programming model for multistage portfolio management is proposed. The model takes into account both the investment goal and risk control at each stage. A scenario generation method is proposed that acts as the basis of the portfolio management model. In particular, by matching the moments and fitting the descriptive features of the asset returns, a linear programming model is used to generate the single-stage scenarios. Scenarios for multistage portfolio management are generated by incorporating this single-stage method with the time-series model for the asset returns. Meanwhile, no arbitrage opportunity exists in the proposed method. A real case is solved via the goal programming model and the scenario generation approach which demonstrates the effectiveness of the model. We also comment on some practical issues of the approach.

Chapter 27 - Stochastic programming

Double-sided stochastic chance-constrained linear fractional programming model for managing irrigation water under uncertainty

Laboratory services in healthcare play a vital role in inpatient care. Studies have indicated laboratory data affect approximately 65% of the most critical decisions on admission, discharge, and medication. This research focuses on improving phlebotomist performance in laboratory facilities of large hospital systems. A two-stage stochastic integer linear programming (SILP) model is formulated to determine better weekly phlebotomist schedules and blood collection assignments. The objective of the two-stage SILP model is to balance the workload of the phlebotomists within and between shifts, as reducing workload imbalance will result in improved patient care. Due to the size of the two-stage SILP model, a scenario reduction model has been proposed as a solution approach. The scenario reduction heuristic is formulated as a linear programming model and the results indicate the scenarios with the largest likelihood of occurrence. These selected scenarios will be tested in the two-stage SILP model to determine weekly scheduling policies and blood draw assignments that will balance phlebotomist workload and improve overall performance.

Operation planning for multiple reservoir systems is a complex and challenging problem because of inherent uncertainties in inflow forecasts. Long-term inflow predictions are required for mid-term planning. However the practice of using several inflow predictions does not fully reflect the various characteristics of the decision in uncertainty analysis, such as non-anticipating decisions or the serial correlations imbedded in the inflow. We therefore applied the stochastic linear programming (SLP) approach to tackle the uncertainties that are inherent in reservoir operation planning due to the inflow uncertainty. A SLP model is developed for coordinating the multi-reservoir operation to determine the efficient monthly target reservoir storage. The model is formulated as a multi-period, two-stage SLP based on the form of fan of individual scenarios. The inflow scenarios are generated by the multivariate periodic AR(1) model considering the serial and spatial correlations. The model becomes a large-scale, linear programming model (comprising over 120,000 columns and 80,000 rows), which was solved very quickly (in 5 minutes) using the linear programming solver CPLEX. The expected benefit of the stochastic model was analyzed quantitatively based on value of information measure. The results indicated that the solutions of the stochastic model are much more effective than those of the deterministic model with average inflows, and that this effectiveness is also maintained in real-time operation in the presence of uncertainty. The benefit of applying this stochastic model to the Nakdong River basin in Korea was presented. It implies that the use of the stochastic model in real-time operation is more effective in the presence of uncertainty.

Engineering Optimization Theory and Practice

We consider the problem of determining the cumulative distribution function and/or moments of the optimal solution value of a nonlinear program dependent upon a single random variable. This problem is difficult computationally because one must in effect determine the optimal solution to an infinite number of nonlinear programs. Bereanu [Bereanu, B., G. Peeters. 1970. A ‘Wait-and-See’ problem in stochastic linear programming. An experimental computer code. Cashiers Centre Etudes Rech. Oper. 12 (3) 133–148.] has provided an algorithm to solve the distribution problem in the linear case based on extensions of the methods of parametric linear programming. (See also [Bereanu, B. 1967. On stochastic linear programming, distribution problems: stochastic technology matrix. Z. f. Wahrscheinlichkeitstheorie u. oerw. Gerbieter 8 148–152; Bereanu, B. 1971. The distribution problem in stochastic linear programming: the Cartesian integration method. Center of Mathematical Statistics of the Academy of RSR, Bucharest, 71–103 (mimeographed); Bereanu, B. 1970. Renewal processes and some stochastic programming problems in economics. SIAM J. Appl. Math. 19 308–322; Bereanu, B. 1973. The Cartesian integration method in stochastic linear programming. L. Collatz, W. Wetterlink, eds. Numerische Methoden bei Optimierungsaufgaben. Springer-Verlag Publishing Co., Inc., Basel; Prekopa, A. 1966. On the probability distribution of the optimum of a random linear program. SIAM J. Control 4 211–222.] for the analysis of more general linear programs.) This paper presents an extremely simple algorithm to solve the problem in the special case when all functions in the nonlinear program are homogeneous. In this instance the infinite class of optimal solutions are known linear homogeneous transformations of the optimal solution to a single nonlinear program. The distribution function may then be determined by substitution of an easily calculated variable into the distribution function of the random variable. The results are useful in the solution and analysis of a number of financial optimization problems. Problems from the analysis of optimal capital accumulation and portfolio separation are treated in some detail.

This paper gives an algorithm for L-shaped linear programs which arise naturally in optimal control problems with state constraints and stochastic linear programs (which can be represented in this form with an infinite number of linear constraints). The first section describes a cutting hyperplane algorithm which is shown to be equivalent to a partial decomposition algorithm of the dual program. The two last sections are devoted to applications of the cutting hyperplane algorithm to a linear optimal control problem and stochastic programming problems.

An inexact stochastic mixed integer linear semi-infinite programming (ISMISIP) model is developed for municipal solid waste (MSW) management under uncertainty. By incorporating stochastic programming (SP), integer programming and interval semi-infinite programming (ISIP) within a general waste management problem, the model can simultaneously handle programming problems with coefficients expressed as probability distribution functions, intervals and functional intervals. Compared with those inexact programming models without introducing functional interval coefficients, the ISMISIP model has the following advantages that: (1) since parameters are represented as functional intervals, the parameter’s dynamic feature (i.e., the constraint should be satisfied under all possible levels within its range) can be reflected, and (2) it is applicable to practical problems as the solution method does not generate more complicated intermediate models (He and Huang, Technical Report, 2004; He et al. J Air Waste Manage Assoc, 2007). Moreover, the ISMISIP model is proposed upon the previous inexact mixed integer linear semi-infinite programming (IMISIP) model by assuming capacities of the landfill, WTE and composting facilities to be stochastic. Thus it has the improved capabilities in (1) identifying schemes regarding to the waste allocation and facility expansions with a minimized system cost and (2) addressing tradeoffs among environmental, economic and system reliability level.

Rolling schedule approaches for supply chain operations planning

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.

Marine fisheries play an important role in the economic development of Indonesia. Besides being the most affordable source of animal protein in the diet of most people in the country, this industrial sector could provide employment to thousands who live at coastal area. We consider the management of small scale traditional business at North Sumatera Province which processes fish into several local seafood products. The inherent uncertainty of data (for example, demand and fish availability), together with the sequential evolution of data over time leads the production planning problem to be a linear mixed-integer stochastic program. We use a scenario generation based approach for solving the model. The result shows the amount of each fish processed product and the number of workforce needed in each horizon planning. References Birge J. R., Louveaux F. V. Introduction to stochastic programming. New York: Springer; 1997. Mawengkang H., Saib Suwilo, Opim S. Sitompul. Revision Modeling of Two-Stage Stochastic Programming Problem. Journal of Industrial System 2006; 7 (4):6--10. Mawengkang H., Suherman. A Heuristic Method of Scenario Generation in Multi-Stage Decision Problem under Uncertainty. Journal of Industrial System 2007; 8(2) : 98--105. Mulvey J. M., Van derbei R., Zenios S. Robust optimization of large scale systems. Operations research 1995;43 (2): 264--281. Rico-Ramirez V. Two-stage stochastic linear programming: a tutorial. SIAG/OPT Views-and-News 2002; 13 (1): 8--14. Sen S., Higle J. L. Introductory tutorial on stochastic linear programming models. Interfaces 1999; 29 (2): 33--61. Van der Vlerk M. H., Haneveld W. K. Stochastic integer programming: General models and algorithms. Annals of Operations Research,85:39--57,1999.

Many planning problems involve choosing a set of optimal decisions for a system in the face of uncertainty of elements that may play a central role in the way the system is analyzed and operated. During the past decade, there has been a renewed interest in the modelling, analysis, and solution of such problems due to a remarkable development of both new theoretical results and novel computational techniques in stochastic optimization. A prominent approach is to develop upper and lower bounding approximations to the problem along with procedures to sharpen bounds until an acceptable tolerance is satisfied. The contributions of this dissertation are concerned with the latter approach. The thesis first studies the stochastic linear programming problem with randomness in both the objective coefficients and the constraints. A convex concave saddle property of the value function is utilized to derive new bounding techniques which generalize previously known results. These approximations require discretizing bounded domains of the random variables in such a way that tight upper and lower bounds result. Such techniques will prove attractive with the recent advances in large scale linear programming. The above results are also extended to obtain new upper and lower bounds when the domains of random variables are unbounded. While these bounds are tight, the approximating models are large-scale deterministic linear programs. In particular, with a proposed order-cone decomposition for the domains, these linear programs are well-structured, thus enabling one to use efficient techniques for solution, such as parallel computation. The thesis next considers convex stochastic programs. Using aggregation concepts from the deterministic literature, new bounds are developed for the problem which are computable using standard convex programming algorithms. Finally, the discussion is focused on a stochastic convex program arising in a certain resource allocation problem. Exploiting the problem structure, bounds are developed via the Karush-Kuhn-Tucker conditions. Rather than discretizing domains, these approximations advocate replacing difficult multidimensional integrals by a series of simple univariate integrals. Such practice allows one to preserve differentiability properties so that smooth convex programming methods can be applied for solution.