Equivalent Deterministic Form Research Articles

Uncertain multi-product newsboy problem with chance constraint

A multi-product newsboy problem with uncertain demand and uncertain storage space is investigated under a warehousing chance constraint. It is assumed that indeterminacy may appear in demand and storage space due to rough estimation of their distributions by decision-maker. Uncertainty theory provides a new tool to deal with the human uncertainty. We develop an uncertain expected value programming model (UEVPM) based on uncertainty theory. Uncertain variables (functions from uncertainty space to the set of real numbers) are used to describe subjective estimation. The objective function is to maximize the expected profit of newsboy at a predetermined confidence level. Furthermore, we convert the chance constrained uncertain programming model to its equivalent deterministic form so as to be solved by classic integer programming method. Finally, three numerical examples are given to illustrate the potential usefulness of the study.

A Transportation Problem with Uncertain Truck Times and Unit Costs

Motivated by the emergency scheduling in a transportation network, this paper considers a transportation problem, in which, the truck times and transportation costs are assumed as uncertain variables. To meet the demand in the practical applications, two optimization objectives are considered, one is the total costs and another is the completion times. And then, a multi-objective optimization model is developed according to the situation in applications. Because there are commensurability and conflicting between the two objectives commonly, a solution does not necessarily exist that is best with respective to the two objectives. Therefore, the problem is reduced to a single objective model, which is an uncertain programming with a chance-constrain. After some analysis, its equivalent deterministic form is obtained, which is a nonlinear programming. Based on a stepwise optimization strategy, a solution method is developed to solve the problem. Finally, the computational results are provided to demonstrate the effectiveness of our model and algorithm.

Pattern-Based Modeling and Solution of Probabilistically Constrained Optimization Problems

We propose a new modeling and solution method for probabilistically constrained optimization problems. The methodology is based on the integration of the stochastic programming and combinatorial pattern recognition fields. It permits the fast solution of stochastic optimization problems in which the random variables are represented by an extremely large number of scenarios. The method involves the binarization of the probability distribution and the generation of a consistent partially defined Boolean function (pdBf) representing the combination (F,p) of the binarized probability distribution F and the enforced probability level p. We show that the pdBf representing (F,p) can be compactly extended as a disjunctive normal form (DNF). The DNF is a collection of combinatorial p-patterns, each defining sufficient conditions for a probabilistic constraint to hold. We propose two linear programming formulations for the generation of p-patterns that can be subsequently used to derive a linear programming inner approximation of the original stochastic problem. A formulation allowing for the concurrent generation of a p-pattern and the solution of the deterministic equivalent of the stochastic problem is also proposed. The number of binary variables included in the deterministic equivalent formulation is not an increasing function of the number of scenarios used to represent uncertainty. Results show that large-scale stochastic problems, in which up to 50,000 scenarios are used to describe the stochastic variables, can be consistently solved to optimality within a few seconds.

Decomposition algorithms with parametric Gomory cuts for two-stage stochastic integer programs

We consider a class of two-stage stochastic integer programs with binary variables in the rst stage and general integer variables in the second stage. We develop decomposition algorithms akin to the L-shaped or Benders’ methods by utilizing Gomory cuts to obtain iteratively tighter approximations of the second-stage integer programs. We show that the proposed methodology is exible in that it allows several modes of implementation, all of which lead to nitely convergent algorithms. We illustrate our algorithms using examples from the literature. We report computational results using the stochastic server location problem instances which suggest that our decomposition-based approach scales better with increases in the number of scenarios than a state-of-the art solver which was used to solve the deterministic equivalent formulation.

Polymorphic uncertain nonlinear programming approach for maximizing the capacity of V-belt driving

In this paper, a polymorphic uncertain nonlinear programming (PUNP) approach is developed to formulate the problem of maximizing the capacity in a system of V-belt driving with uncertainties. The constructed optimization model is found to consist of a nonlinear objective function and some nonlinear constraints with some parameters which are of uncertain nature. These uncertain parameters are interval parameters, random interval parameters, fuzzy parameters or fuzzy interval parameters. To find a robust solution of the problem, a deterministic equivalent formulation (DEF) is established for the polymorphic uncertain nonlinear programming model. For a given satisfaction level, this DEF turns out to be a nonlinear programming involving only interval parameters. A solution method, called a sampling based interactive method, is developed such that a robust solution of the original model with polymorphic uncertainties is obtained by using standard smooth optimization techniques. The proposed method is applied into a real-world design of V-belt driving, and the results indicate that both the PUNP approach and the developed algorithm are useful to the optimization problem with polymorphic uncertainty.

Optimal Maintenance for a System with Probabilistic Time Constraint

In this paper, we repair an optimum number of components in a system, repair time taken as a random variable in the constraint function. By applying probabilistic programming, constraints function is converted into an equivalent deterministic form. A numerical example is presented to illustrate the computational procedure for random time constraints, and the problem is solved by using LINGO software.

Multi-Objective Chance Constrained Capacitated Transportation Problem based on Fuzzy Goal Programming

This paper presents chance constrained multi-objective capacitated transportation problem based on fuzzy goal programming problem. Generally, in transportation problem the capacity of each origin and the demand of each destination are random in nature. The inequality constraints representing supplies and demands are probabilistically described. In many real situations, there are capacity restrictions on units of commodities which are shipped from different sources to different destinations. In the model formulation, supply and demand constraints are converted into equivalent deterministic forms. Then, we define the fuzzy goal levels of the objective functions. The fuzzy objective goals are then characterized by the associated membership functions. In the solution process, two fuzzy goal programming models are considered by minimizing negative deviational variables to obtain compromise solution. Distance function is used in order to obtain the most compromise optimal solution. In order to demonstrate the effectiveness of the proposed approach, an illustrative example of chance constrained multi- objective capacitated transportation problem is solved.

Optimization of Resource Provisioning Cost in Cloud Computing

In cloud computing, cloud providers can offer cloud consumers two provisioning plans for computing resources, namely reservation and on-demand plans. In general, cost of utilizing computing resources provisioned by reservation plan is cheaper than that provisioned by on-demand plan, since cloud consumer has to pay to provider in advance. With the reservation plan, the consumer can reduce the total resource provisioning cost. However, the best advance reservation of resources is difficult to be achieved due to uncertainty of consumer's future demand and providers' resource prices. To address this problem, an optimal cloud resource provisioning (OCRP) algorithm is proposed by formulating a stochastic programming model. The OCRP algorithm can provision computing resources for being used in multiple provisioning stages as well as a long-term plan, e.g., four stages in a quarter plan and twelve stages in a yearly plan. The demand and price uncertainty is considered in OCRP. In this paper, different approaches to obtain the solution of the OCRP algorithm are considered including deterministic equivalent formulation, sample-average approximation, and Benders decomposition. Numerical studies are extensively performed in which the results clearly show that with the OCRP algorithm, cloud consumer can successfully minimize total cost of resource provisioning in cloud computing environments.

Polymorphic uncertain nonlinear programming model and algorithm for maximizing the fatigue life of V-belt drive

In this paper, a polymorphic uncertain nonlinear programming (PUNP) model is constructed to formulate the problem of maximizing the V-belt's fatigue life according to the practical engineering design conditions. The model is converted into an equivalent interval programming only involved with interval parameters for any given degree of membership and confidence level. Then, a deterministic equivalent formulation (DEF) for the original model is obtained based on the concept of possibility degree for the order of two interval numbers. An algorithm, called sampling based algorithm, is developed to find a robust optimal design scheme for maximizing the fatigue life of the V-belt. Case study is employed to demonstrate the validity and the practicability of the constructed model and the algorithm.

Optimum allocation of redundant components: Modified E-Model Approach

In this paper, we obtain optimum allocation of redundant components in a system design when costs are considered as (with & without) random respectively in the objective function. When costs function are random, by applying Modified E-Model, objective function is converted into an equivalent deterministic form. A Numerical example is presented to illustrate the computational procedure for both (with & without) random costs respectively, and the problem is solved by using LINGO Software.

Optimal crop planning using a chance constrained linear programming model

A monthly time-stepped chance constrained linear programming (CCLP) model was developed to derive optimal cropping patterns and optimal operational strategies for the Sri Ram Sagar Project, with stochastic inflows. The stochastic nature of the inflows was incorporated into the model in its equivalent deterministic form. These equivalent deterministic inflow values were estimated from the annual and monthly probability distribution of observed inflows, and named the chance constrained linear programming model-annual and chance constrained linear programming model-monthly, respectively. The models were solved for nine different dependable inflow levels, namely for 50, 55, 60, 65, 70, 75, 80, 85 and 90% in each CCLP. The results of the models were compared with respect to net benefit, irrigation intensity, total cropped area, optimal cropping pattern, optimal releases, evaporation loss and initial storages. Based on the results obtained, it can be concluded that, in CCLP optimization, the probability distribution of ‘model time period’ (t-month in this case) should be considered rather than the probability of ‘planning time period’ (year in this case).

Build-to-order supply chain network design under supply and demand uncertainties

Supply chain disruptions are unintended, unwanted situations resulting in a negative supply chain performance. We study the supply chain network design under supply and demand uncertainty with embedded supply chain disruption mitigation strategies, postponement with downward substitution, centralized stocking and supplier sourcing base. We designed an integrated supply-side, manufacturing and demand-side operations network in such that the total expected operating cost is minimized. We modeled it in a deterministic equivalent formulation. An L-shaped decomposition with an additional decomposition step in the master problem is proposed. The computational results showed that parallel sourcing has a cost advantage against single sourcing under supply disruptions. In addition, the build-to-order (BTO) manufacturing mitigation process has its greatest impact with high variations on demands and is integrated with the component downward substitution. Lastly, the manufacturer needs to order differentiated components to cover its requirement for maximal product demand to prevent the loss of sale, even with fewer modules in stock.

The facility location problem with Bernoulli demands

This paper presents the facility location problem with Bernoulli demands. In this capacitated discrete location stochastic problem the goal is to define an a priori solution for the locations of the facilities and for the allocation of customers to the operating facilities that minimizes the sum of the fixed costs of the open facilities plus the expected value of the recourse function. The problem is formulated as a two-stage stochastic program and two different recourse actions are considered. For each of them, a closed form is presented for the recourse function and a deterministic equivalent formulation is obtained for the case in which the probability of demand is the same for all customers. Numerical results from computational experiments are presented and analyzed.

A chance constrained approach for a gas processing plant with uncertain feed conditions

In this work, we propose a new optimization model for gas processing plant under uncertain feed flow rate and composition. Deterministic model has initially being formulated; subsequently, linear steady state chance constrained models with time-dependent uncertainties under single and joint chance constraints have been developed. The chance constrained models have then been converted to their equivalent deterministic form so as to be solved. Case study has been taken for real plant to effectively implement the proposed model. The solution from the optimization provides prior-decision for the in-operation plant to produce the desired product at certain confidence level by satisfying the process constraints. Sensitivity analysis has been made to see the effect of each uncertain feed component flows. Final computational results ratify the need of taking optimal decisions using the chance constrained approach which comprises the profitability and reliability of the process compared to deterministic, “worst case”, and two-stage programming approaches.

An approximation-based approach for fuzzy multi-period production planning problem with credibility objective

This paper develops a fuzzy multi-period production planning and sourcing problem with credibility objective, in which a manufacturer has a number of plants or subcontractors. According to the credibility service levels set by customers in advance, the manufacturer has to satisfy different product demands. In the proposed production problem, production cost, inventory cost and product demands are uncertain and characterized by fuzzy variables. The problem is to determine when and how many products are manufactured so as to maximize the credibility of the fuzzy costs not exceeding a given allowable invested capital, and this credibility can be regarded as the investment risk criteria in fuzzy decision systems. In the case when the fuzzy parameters are mutually independent gamma distributions, we can turn the service level constraints into their equivalent deterministic forms. However, in this situation the exact analytical expression for the credibility objective is unavailable, thus conventional optimization algorithms cannot be used to solve our production planning problems. To overcome this obstacle, we adopt an approximation scheme to compute the credibility objective, and deal with the convergence about the computational method. Furthermore, we develop two heuristic solution methods. The first is a combination of the approximation method and a particle swarm optimization (PSO) algorithm, and the second is a hybrid algorithm by integrating the approximation method, a neural network (NN), and the PSO algorithm. Finally, we consider one 6-product source, 6-period production planning problem, and compare the effectiveness of two algorithms via numerical experiments.

Reliability stochastic optimization for a series system with interval component reliability via genetic algorithm

This paper deals with chance constraints based reliability stochastic optimization problem in the series system. This problem can be formulated as a nonlinear integer programming problem of maximizing the overall system reliability under chance constraints due to resources. The assumption of traditional reliability optimization problem is that the reliability of a component is known as a fixed quantity which lies in the open interval (0, 1). However, in real life situations, the reliability of an individual component may vary due to some realistic factors and it is sensible to treat this as a positive imprecise number and this imprecise number is represented by an interval valued number. In this work, we have formulated the reliability optimization problem as a chance constraints based reliability stochastic optimization problem with interval valued reliabilities of components. Then, the chance constraints of the problem are converted into the equivalent deterministic form. The transformed problem has been formulated as an unconstrained integer programming problem with interval coefficients by Big-M penalty technique. Then to solve this problem, we have developed a real coded genetic algorithm (GA) for integer variables with tournament selection, uniform crossover and one-neighborhood mutation. To illustrate the model two numerical examples have been solved by our developed GA. Finally to study the stability of our developed GA with respect to the different GA parameters, sensitivity analyses have been done graphically.

DUALITY FOR LINEAR CHANCE-CONSTRAINED OPTIMIZATION PROBLEMS

In this paper we deal with linear chance-constrained optimization problems, a class of problems which naturally arise in practical applications in finance, engineering, transportation and scheduling, where decisions are made in presence of uncertainty. After giving the deterministic equivalent formulation of a linear chance-constrained optimization problem we construct a conjugate dual problem to it. Then we provide for this primal-dual pair weak sufficient conditions which ensure strong duality. In this way we generalize some results recently given in the literature. We also apply the general duality scheme to a portfolio optimization problem, a fact that allows us to derive necessary and sufficient optimality conditions for it.

A CLASS OF MIX DESIGN PROBLEMS: FORMULATION, SOLUTION METHODS AND APPLICATIONS

AbstractIn this paper, we consider a linear program with only equality constraints but containing interval and random coefficients. We first address the linear program with interval coefficients, and establish some structural properties of this linear program. On this basis, a solution method is proposed. We then move on to consider the linear program with random coefficients. Using the chance constraint approach and a new approach, the satisfaction degree approach, we obtain the two respective deterministic equivalent formulations. Then the results and the numerical solution methods obtained for these two linear models are applied to the original linear problem which contains both interval and random coefficients. By way of illustration, we consider a practical problem, where the optimal mixing proportions need to be determined for the mix slurry in the production process of aluminium with sintering. This gives rise to a linear program with interval and random coefficients. Its deterministic equivalent formulations are presented. Preliminary numerical examples show that the proposed models and the solution methods are promising.

Solving non-linear portfolio optimization problems with the primal-dual interior point method

Stochastic programming is recognized as a powerful tool to help decision making under uncertainty in financial planning. The deterministic equivalent formulations of these stochastic programs have huge dimensions even for moderate numbers of assets, time stages and scenarios per time stage. So far models treated by mathematical programming approaches have been limited to simple linear or quadratic models due to the inability of currently available solvers to solve NLP problems of typical sizes. However stochastic programming problems are highly structured. The key to the efficient solution of such problems is therefore the ability to exploit their structure. Interior point methods are well-suited to the solution of very large non-linear optimization problems. In this paper we exploit this feature and show how portfolio optimization problems with sizes measured in millions of constraints and decision variables, featuring constraints on semi-variance, skewness or non-linear utility functions in the objective, can be solved with the state-of-the-art solver.

Decoupled approach to multidisciplinary design optimization under uncertainty

We propose solution methods for multidisciplinary design optimization (MDO) under uncertainty. This is a class of stochastic optimization problems that engineers are often faced with in a realistic design process of complex systems. Our approach integrates solution methods for reliability-based design optimization (RBDO) with solution methods for deterministic MDO problems. The integration is enabled by the use of a deterministic equivalent formulation and the first order Taylor’s approximation in these RBDO methods. We discuss three specific combinations: the RBDO methods with the multidisciplinary feasibility method, the all-at-once method, and the individual disciplinary feasibility method. Numerical examples are provided to demonstrate the procedure.