A stochastic production planning model with a dynamic chance constraint

This paper deals with the optimal production planning for a single product over a finite horizon. The holding and production costs are assumed quadratic as in Holt, Modigliani, Muth and Simon (HMMS) [7] model. The cumulative demand is compound Poisson and a chance constraint is included to guarantee that the inventory level is positive with a probability of at least α at each time point. The resulting stochastic optimization problem is transformed into a deterministic optimal control problem with control variable and of the optimal solution is presented. The form of state variable inequality constraints. A discussion the optimal control (production rate) is obtained as follows: if there exists a time t₁ such that t₁?[O, T]where T is the end of the planning period, then (i) produce nothing until t₁ and (ii) produce at a rate equal to the expected demand plus a ‘correction factor’ between t₁ and T. If t₁ is found to be greater than T, then the optimal decision is to produce nothing and always meet the demand from the inventory.     

Path planning for robots under stochastic uncertainty *

In order to reduce large online measurement and correction expenses, the a priori informations on the random variations of the model parameters of a robot and its working environment are taken into account already at the planning stage. Thus, instead of solving a deterministic path planning problem with a fixed nominal parameter vector, here, the optimal velocity profile along a given trajectory in work space is determined by using a stochastic optimization approach. Especially, the standard polygon of constrained motion-depending on the nominal parameter vector-is replaced by a more general set of admissible motion determined by chance constraints or more general risk constraints. Robust values (with respect to stochastic parameter variations) of the maximum, minimum velocity, acceleration, deceleration, resp., can be obtained then by solving a univariate stochastic optimization problem Considering the fields of extremal trajectories, the minimum-time path planning problem under stochastic uncertainty can be solved now by standard optimal deterministic path planning methods     

Lipschitzianity of optimal trajectories for the Bolza optimal control problem

In this paper we investigate Lipschitz continuity of optimal solutions for the Bolza optimal control problem under Tonelli’s type growth condition. Such regularity being a consequence of normal necessary conditions for optimality, we propose new sufficient conditions for normality of state-constrained nonsmooth maximum principles for absolutely continuous optimal trajectories. Furthermore we show that for unconstrained problems any minimizing sequence of controls can be slightly modified to get a new minimizing sequence with nice boundedness properties. Finally, we provide a sufficient condition for Lipschitzianity of optimal trajectories for Bolza optimal control problems with end point constraints and extend a result from (J. Math. Anal. Appl. 143, 301–316, 1989) on Lipschitzianity of minimizers for a classical problem of the calculus of variations with discontinuous Lagrangian to the nonautonomous case.     

On the Efficiency of a Global Non-differentiable Optimization Algorithm Based on the Method of Optimal Set Partitioning

The examined algorithm for global optimization of the multiextremal non-differentiable function is based on the following idea: the problem of determination of the global minimum point of the function f(x) on the set (f(x) has a finite number of local minima in this domain) is reduced to the problem of finding all local minima and their attraction spheres with a consequent choice of the global minimum point among them. This reduction is made by application of the optimal set partitioning method. The proposed algorithm is evaluated on a set of well-known one-dimensional, two-dimensional and three-dimensional test functions. Recommendations for choosing the algorithm parameters are given.     

Non-causal models in long term planning via set contractive optimal control methods

The notion of consistency for optimal plans introduced in [F.E. Kydland, E.C. Prescott, Rules rather than decisions: The inconsistency of optimal plans, J. Polit. Econ. 85 (3) (1977) 473–491] is studied in relation to dynamic programming and to multi-objective optimal control. It is demonstrated that the consistency condition for an optimal sequence of n policies for periods from 1 to n corresponding to a sequence of n economic agents’ decisions that together constitute an optimal plan for an agreed-upon social objective function is equivalent to a multi-objective problem with n+1 criteria. An optimal plan is consistent if and only if the corresponding (n+1)-criteria problem is balanced, and consistent optimal plans do exist in such economic situations. Otherwise, the consistency requirement generates the balance set and a set of Pareto solutions, and both can be computed as illustrated in the inflation–unemployment example extended for two periods with three objectives to optimize. The procedure of “consistent planning for the infinite horizon” considered in [F.E. Kydland, E.C. Prescott, Rules rather than decisions: The inconsistency of optimal plans, J. Polit. Econ. 85 (3) (1977) 473–491] is shown to concur with Bellman’s principle of optimality that may be invalid in many practical cases including the case considered in [F.E. Kydland, E.C. Prescott, Rules rather than decisions: The inconsistency of optimal plans, J. Polit. Econ. 85 (3) (1977) 473–491]. Finitely causal and non-causal models are introduced for long term planning in a multi-objective (Pareto) framework, and it is demonstrated how to deal with non-causal models using new set-contractive methods for multi-objective global optimization. The results are illustrated by examples.     

Onn-stage,m-reusableness rate production planning problems

In this paper, the general solutions to the optimal policy and the limit theorem forn-stage,m-reusableness rate production planning are obtained for the problem with free terminal point. In addition, general solutions to the problem with fixed terminal point are obtained. The links between the solutions to these problems are discussed.This paper has benefited from the revisions suggested by the referees. The careful and constructive reviews were appreciated by the author.     

Computation of the Optimal Policy for the Control of a Compound Immigration Process through Total Catastrophes

In this paper we consider a Markov decision model introduced by Economou (2003), in which it was proved that the optimal policy in the problem of controlling a compound immigration process through total catastrophes is of control-limit type. We show that the average cost of a control-limit policy is unimodal as a function of the critical point. This result enables us to design very efficient algorithms for the computation of the optimal policy as the bisection procedure and a special-purpose policy iteration algorithm that operates on the class of control-limit policies.AMS 2000 Subject Classification: Primary 9OC40; Secondary 6OJ25     

Hamilton-Jacobi-Bellman inequality for the average control of piecewise deterministic Markov processes

ABSTRACT

The main goal of this paper is to study the infinite-horizon long run average continuous-time optimal control problem of piecewise deterministic Markov processes (PDMPs) with the control acting continuously on the jump intensity λ and on the transition measure Q of the process. We provide conditions for the existence of a solution to an integro-differential optimality inequality, the so called Hamilton-Jacobi-Bellman (HJB) equation, and for the existence of a deterministic stationary optimal policy. These results are obtained by using the so-called vanishing discount approach, under some continuity and compactness assumptions on the parameters of the problem, as well as some non-explosive conditions for the process.     

Mathematical modeling and trajectory planning of mobile manipulators with flexible links and joints

This paper is concerned with mathematical modeling and optimal motion designing of flexible mobile manipulators. The system is composed of a multiple flexible links and flexible revolute joints manipulator mounted on a mobile platform. First, analyzing on kinematics and dynamics of the model is carried out then; open-loop optimal control approach is presented for optimal motion designing of the system. The problem is known to be complex since combined motion of the base and manipulator, non-holonomic constraint of the base and highly non-linear and complicated dynamic equations as a result of the flexible nature of both links and joints are taken into account. In the proposed method, the generalized coordinates and additional kinematic constraints are selected in such a way that the base motion coordination along the predefined path is guaranteed while the optimal motion trajectory of the end-effector is generated. This method by using Pontryagin’s minimum principle and deriving the optimality conditions converts the optimal control problem into a two point boundary value problem. A comparative assessment of the dynamic model is validated through computer simulations, and then additional simulations are done for trajectory planning of a two-link flexible mobile manipulator to demonstrate effectiveness and capability of the proposed approach.     

Short-term financial planning with uncertain receipts and disbursements

The problem of short-term financial planning is to determine an optimal credit mix to meet the short-term cash needs and an optimal investment plan for excess cash. A number of linear optimization models have been developed to solve this problem, some of which are in practical use. The purpose of this paper is to generalize the assumptions of these models concerning the available information about future receipts and disbursements. It is presupposed that the financial officer has some idea as to the amount involved which, however, cannot be specified by a probability distribution. On the contrary, we assume that these ideas only permit qualitative probability statements such as the following:“That the difference between disbursements and receipts in a certain period lies in an interval I₁ is no less probable than that it lies in an interval I₂”.For this level of information we formulate a model for short-term financial planning, and we develop a solution procedure to determine the optimum financial alternatives. Finally, the entire procedure is demonstrated by a medium sized example.     

Optimality conditions for a class of inverse optimal control problems with partial differential equations

ABSTRACT

We consider bilevel optimization problems which can be interpreted as inverse optimal control problems. The lower-level problem is an optimal control problem with a parametrized objective function. The upper-level problem is used to identify the parameters of the lower-level problem. Our main focus is the derivation of first-order necessary optimality conditions. We prove C-stationarity of local solutions of the inverse optimal control problem and give a counterexample to show that strong stationarity might be violated at a local minimizer.     

Optimality Conditions and Synthesis for the Minimum Time Problem

We study the minimum time optimal control problem for a nonlinear system in R ⁿ with a general target. Necessary and sufficient optimality conditions are obtained. In particular, we describe a class of costates that are included in the superdifferential of the minimum time function, even in the case when this function is only lower semicontinuous. Two set-valued maps are constructed to provide time optimal synthesis.     

Computing a Stationary Base-Stock Policy for a Finite Horizon Stochastic Inventory Problem with Non-linear Shortage Costs

Abstract

We consider a periodic-review stochastic inventory problem in which demands for a single product in each of a finite number of periods are independent and identically distributed random variables. We analyze the case where shortages (stockouts) are penalized via fixed and proportional costs simultaneously. For this problem, due to the finiteness of the planning horizon and non-linearity of the shortage costs, computing the optimal inventory policy requires a substantial effort as noted in the previous literature. Hence, our paper is aimed at reducing this computational burden. As a resolution, we propose to compute “the best stationary policy.” To this end, we restrict our attention to the class of stationary base-stock policies, and show that the multi-period, stochastic, dynamic problem at hand can be reduced to a deterministic, static equivalent. Using this important result, we introduce a model for computing an optimal stationary base-stock policy for the finite horizon problem under consideration. Fundamental analytic conclusions, some numerical examples, and related research findings are also discussed.     

A model for strategic planning under uncertainty

Summary In this paper the multi-period strategic planning problem for a consumer sumer product manufacturing chain is considered. Our discussion is focused on investment decisions which, are economically optimal over the whole planning horizonT, while meeting customer demands and conforming to technological requirements. In strategic planning, time and uncertainty play important roles. The uncertainties in the model are due to different levels of forecast demands, cost estimates and equipment behaviour. The main aim of this paper is to develop and analyse a multiperiod stochastic model representing the entire manufacturing chain, from the acquisitions of raw material to the delivering of final products. The resulting optimization problem is computationally intractable because of the enormous, and sometimes unrealistic, number of scenarios that must be considered in order to identify the optimal planning strategy. We propose two different solution approaches; firstly, we apply a scenario risk analysis giving the related results of experiments on a particular real data set. We then describe and investigate an Integer Stochastic Programming formulation of the problem and propose, as a solution technique, a variation of Benders decomposition method, namely theL-shaped method.     

On the planning problem for a class of Mean Field Games

We give a result of existence and uniqueness of weak solutions to the planning problem for a class of Mean Field Games. This is a kind of optimal transportation problem consisting in the exact controllability at time T of Fokker–Planck equations obtained using drifts arising as the optimal feedbacks from a coupled backward Hamilton–Jacobi–Bellman equation.     

Valuation of variable annuities with guaranteed minimum withdrawal and death benefits via stochastic control optimization

In this paper we present a numerical valuation of variable annuities with combined Guaranteed Minimum Withdrawal Benefit (GMWB) and Guaranteed Minimum Death Benefit (GMDB) under optimal policyholder behavior solved as an optimal stochastic control problem. This product simultaneously deals with financial risk, mortality risk and human behavior. We assume that market is complete in financial risk and mortality risk is completely diversified by selling enough policies and thus the annuity price can be expressed as appropriate expectation. The computing engine employed to solve the optimal stochastic control problem is based on a robust and efficient Gauss–Hermite quadrature method with cubic spline. We present results for three different types of death benefit and show that, under the optimal policyholder behavior, adding the premium for the death benefit on top of the GMWB can be problematic for contracts with long maturities if the continuous fee structure is kept, which is ordinarily assumed for a GMWB contract. In fact for some long maturities it can be shown that the fee cannot be charged as any proportion of the account value — there is no solution to match the initial premium with the fair annuity price. On the other hand, the extra fee due to adding the death benefit can be charged upfront or in periodic installment of fixed amount, and it is cheaper than buying a separate life insurance.     

Riccati equations for generalized singular estimate control systems

We consider the Bolza problem associated with boundary/point control systems governed by strongly continuous semigroups. In continuation of our work in Lasiecka and Tuffaha [I. Lasiecka and A. Tuffaha, Riccati equations for the Bolza problem arising in boundary/point control problems governed by C ₀–semigroups satisfying a singular estimate, J. Optim. Theory Appl. 136 (2008), pp. 229–246; I. Lasiecka and A. Tuffaha, A Bolza optimal synthesis problem for singular estimate control systems, Control Cybernet 38(4B) (2009), pp. 1429–1460], we yet extend the theory to a more general class of control problems that are not analytic providing sharp blow-up rates for the regularity. Solvability of the associated Riccati equations and an optimal feedback synthesis are established. The presence of unbounded control actions, such as boundary/point controls, naturally lead to a singularity at the terminal point t?=?T of the optimal control and of the corresponding feedback operator as before. The class of control systems considered in this article is a generalization to the class usually referred to in the literature as ‘Singular Estimate Control Systems’. The prototype is still that of a PDE system consisting of coupled hyperbolic parabolic dynamics interacting on an interface with point/boundary control. The distinct feature of the class considered in this article is that the degree of unboundedness in the control is stronger than that allowed in the usual singular estimate control system configuration, giving rise to less regular optimal state trajectories.     

Operational research in health services planning

Some of the characteristics which distinguish social planning from planning in the private sector, or in publicly owned industry, are identified. A survey of OR investigations (both tactical and strategic) in the area of health services planning supports the case that the mainstream OR contribution does violence to the nature of the system under study. In particular, problems are formulated in terms of (or transformed into) single objectives, to be optimized; aspects of the social world are subject to wholesale quantification, with resulting distortion; models make implausible data demands; project definition and execution reinforce the ‘scientisation’ of political debate; problems are formulated on the assumption of a single hierarchically powerful decision-maker; and problems are formulated as if they must be solved in toto at one point in time. It is suggested that we should look to the converse of these qualities for new directions in the application of operational research in social planning. Some examples are given of techniques and approaches, drawn both from within health services planning and from other areas, which exhibit these alternative characteristics.     

Efficient algorithms for combined heat and power production planning under the deregulated electricity market

Combined heat and power (CHP) production is an important energy production technology that can yield much higher total energy efficiency than separate heat and power generation. In CHP production, the heat and power production follows a joint characteristic, which means that the production planning must be done in coordination. Cost-efficient operation of a CHP system can be planned by using an optimization model. A long-term planning model decomposes into thousands of hourly models. Earlier, in the regulated electric power market, the planning problem was symmetrically driven by heat and power demand. The liberalization of the power market has created an asymmetrical planning problem, where heat production responds to the demand and power production to the volatile market price. In this paper, we utilize this asymmetry to develop novel envelope-based dual algorithms for solving the hourly CHP models efficiently. The basic idea is to transform the three-dimensional characteristic operating region for heat and power production of each CHP plant into a two-dimensional envelope by taking the power price as a parameter. Then the envelopes of each plant are used for looking up the optimal solution rapidly. We propose two versions of the algorithm: the on-line envelope construction algorithm (ECON) where the envelopes are constructed for each hour based on the power price and the off-line envelope construction algorithm (ECOFF) where envelopes are pre-computed for all different power price ranges. We derive the theoretical time complexity of the two algorithms and compare their performance empirically with realistic test models against the ILOG CPLEX solver and the Power Simplex (PS) algorithm. PS is an extremely efficient specialized primal algorithm developed for the symmetrical CHP planning problem under the regulated market. On average, when reusing previous basic solutions, ECON is 603 times faster than CPLEX and 1.3 times faster than PS. ECOFF is 1860 times faster than CPLEX and four times faster than PS.     

Forecast horizons and dynamic facility location planning

We consider a dynamic facility location model in which the objective is to find a planning horizon, *, and a first period decision,X ₁*, such thatX ₁* is a first period decision for at least one optimal policy for all problems with planning horizons equal to or longer than *. In other words, we seek a planning horizon, *, such that conditions after * do not influence the choice of the optimal initial decision,X ₁*. We call * aforecast horizon andX ₁* anoptimal initial decision. For the dynamic uncapacitated fixed charge location problem, we show that simple conditions exist such that the initial decision depends on the length of the planning horizon. Thus, a strictly optimal forecast horizon and initial policy may not exist. We therefore introduce the concepts ofe-optimal forecast horizons and -optimal initial solutions. Our computational experience inicates that such solutions can be found for practical problems. Although computing -optimal forecast horizons and initial decisions can be cumbersome, this approach offers the potential for making significantly better decisions than those generated by other approaches. To illustrate this, we show that the use of the scenario planning approach can lead to the adoption of the worst possible initial decision under conditions of future uncertainty. On the basis of our results, it appears that the forecast horizon approach offers an attractive tool for making dynamic location decisions.