Bernard F. Lamond

Bregman's balancing method

In this paper we observe that most of the independently discovered balancing methods, used in transportation planning and in other fields, are in fact special cases of a method of Bregman. Examples include the usual Kruithof or Furness method, the Evans-Kirby three dimensional balancing procedure, the Murchland multiproportional balancing procedure, the Osborne or Grad method for preconditioning matrices, the Jefferson-Scott procedure for gravity models with inequality constraints, and the method considered by Macgill for partially constrained gravity models. The convergence of all of these methods follows from Bregmans general result.

Simple bonds for finite single-server exponential tandem queues

Tandem queue configurations naturally arise in multistage stochastic systems such as assembly lines in manufacturing or multiphase transmissions in telecommunications. As finite capacity or storage constraints buffers are usually involved, the celebrated closed product form expression is generally not applicable. In this paper, a new bounding methodology for nonproduct form systems is applied to finite single-server exponential tandem queues. The methodology is based on modifying the original system into product form systems that provide bounds for some performance measure of interest. The product form modifications given for this finite tandem queue propose a computationally attractive and intuitively obvious lower and upper bound for the call congestion and throughput. Numerical results indicate that the bounds are reasonable indicators of the order of magnitude. This can be useful for quick engineering purposes as will be illustrated by an optimal design example. A formal proof of the bounds is given. This proof extends standard techniques for comparing stochastic systems and is of interest in itself.

A multiple criteria ranking procedure based on distance between partial preorders

In this paper, a multicriterion ranking procedure based on distance between partial preorders is proposed. This method consists of two phases. In the first phase, the decision maker is asked to rank alternatives with a preorder (complete or partial) for each criterion and provide complete or partial linear information about the relative importance (weights) of the criteria. In the second phase, we introduce a distance procedure to aggregate the above individual rankings into a global ranking (a partial preorder). An algorithm for the aggregation procedure is proposed, followed a numerical illustration.

Using tool life models to minimize processing time on a flexible machine

Economic tool life models are presented for machines with finite capacity tool magazines and variable processing speed capability. Single and multiple part models for minimizing the total throughput time are formulated as nonlinear, integer programs (NLIP). An algorithm is presented for the NLP relaxation and a marginal analysis approach for solving the NLIP is detailed, giving an optimal tool loading policy as well as the processing speeds for each of the part types so as to minimize the makespan. A numerical example illustrates the procedures.

Exact and Approximate Solutions of Affine Reservoir Models

We formulate an optimization model of a multiple reservoir water resource system that encompasses interbasin transfers among two or more river basins. Autocorrelated inflows are modeled with a linear autoregressive stochastic process. Benefits for each period are assumed to depend separably on storage levels and discharges with the dependence on discharges being linear. For the important special case of a single river basin, a myopic policy hence, computed easily is optimal. When the model includes the possibility of interbasin transfers, a myopic policy is optimal if the deterministic and stochastic portions of the inflow process are always nonnegative. Even if this assumption is not valid, myopic policies yield useful bounds.

A reservoir hydroelectric system: Exactly and approximately optimal policies

Abstract We analyze a discrete-time model of a single reservior whose discharges generate hydroelectric power. Inflows in successive periods are random variables and the structure of the model is invariant over time. Revenue from hydroelectric production is assumed to follow a piecewise linear function. We characterize an optimal discharge policy and describe numerical results for the computation of approximately optimal policies.

Optimizing Long-term Hydro-power Production Using Markov Decision Processes

Abstract Modelling the long-term operation of hydroelectric systems is one of the classic applications of Markov decision processes (MDP). The computation of optimal policies, for MDP models, is usually done by dynamic programming (DP) on a discretized state space. A major difficulty arises when optimizing multi-reservoir systems, because the computational complexity of DP increases exponentially with the number of sites. This so-called ‘curse of dimensionality’ has so far restricted the applicability of DP to very small systems (2 or 3 sites). Practitioners have thus had to resort to other methodologies for the long-term planning, often at the expense of rigour, and without reliable error estimates. This paper surveys recent research on MDP computation, with application to hydro-power systems. Three main approaches are discussed: (i) discrete DP, (ii) numerical approximation of the expected future reward function, and (iii) analytic solution of the DP recursion.

Minimizing the expected processing time on a flexible machine with random tool lives

We present a stochastic version of economic tool life models for machines with finite capacity tool magazines and a variable processing speed capability, where the tool life is a random variable. Using renewal theory to express the expected number of tool setups as a function of cutting speed and magazine capacity, we extend previously published deterministic mathematical programming models to the case of minimizing the expected total processing time. A numerical illustration with typical cutting tool data shows the deterministic model underestimates the optimal expected processing time by more than 8% when the coefficient of variation equals 0.3 (typical for carbide tools), and the difference exceeds 15% for single-injury tools having an exponentially distributed economic life (worst case).

Water Reservoir Applications of Markov Decision Processes

Decision problems in water resources management are usually stochastic, dynamic and multidimensional. MDP models have been used since the early fifties for the planning and operation of reservoir systems because the natural water inflows can be modeled using Markovian stochastic processes and the transition equations of mass conservation for the reservoir storages are akin to those found in inventory theory. However, the “curse of dimensionality” has been a major obstacle to the numerical solution of MDP models for systems with several reservoirs. Also, the use of optimization models for the operation of multipurpose reservoir systems is not so widespread, due to the need for negotiations between different users, with dam operators often relying on operating rules obtained by simulation models.

Piecewise affine approximations for the control of a one-reservoir hydroelectric system

Abstract We analyze the computation of optimal and approximately optimal policies for a discrete-time model of a single reservoir whose discharges generate hydroelectric power. Inflows in successive periods are random variables. Revenue from hydroelectric production is represented by a piecewise linear function. We use the special structure of optimal policies, together with piecewise affine approximations of the optimal return functions at each stage of dynamic programming, to decrease the computational effort by an order of magnitude compared with ordinary value iteration. The method is then used to obtain easily computable lower and upper bounds on the value function of an optimal policy, and a policy whose value function is between the bounds.