E. G. Kyriakidis

Single vehicle routing with a predefined customer sequence and multiple depot returns

Abstract The optimal routing of a single vehicle with limited capacity that delivers one product to n clients according to a predefined client sequence can be determined using dynamic programming. In the present paper we propose and investigate three practical variations of this problem: (i) the case of multiple-product deliveries when each product is stored in its own compartment in the vehicle, (ii) the case of multiple-product deliveries when all products are stored together in the vehicle’s single compartment, and (iii) the case in which the vehicle picks up from and delivers a single product to each customer. Suitable dynamic programming algorithms that find the optimal routing of the vehicle are developed for each case. The efficiency of the algorithms is studied by solving large problem sets.

Markov decision models for the optimal maintenance of a production unit with an upstream buffer

We consider a manufacturing system in which a buffer has been placed between the input generator and the production unit. The input generator supplies at a constant rate the buffer with the raw material, which is pulled by the production unit. The pull-rate is greater than the input rate when the buffer is not empty. The two rates become equal as soon as the buffer is evacuated. The production unit deteriorates stochastically over time and the problem of its optimal preventive maintenance is considered. Under a suitable cost structure it is proved that the optimal average-cost policy for fixed buffer size is of control-limit type, if the repair times are geometrically distributed. Efficient Markov decision process solution algorithms that operate on the class of control-limit policies are developed, when the repair times are geometrical or follow a continuous distribution. The optimality of a control-limit policy is also proved when the production unit after the end of a maintenance remains idle until the buffer is filled up. Furthermore, numerical results are given for the optimal policy if it is permissible to leave the production unit idle whenever it is in operative condition.

Single vehicle routing problems with a predefined customer sequence, compartmentalized load and stochastic demands

We consider the problem of finding the optimal routing of a single vehicle that delivers K different products to N customers according to a particular customer order. The demands of the customers for each product are assumed to be random variables with known distributions. Each product type is stored in its dedicated compartment in the vehicle. Using a suitable dynamic programming algorithm we find the policy that satisfies the demands of the customers with the minimum total expected cost. We also prove that this policy has a specific threshold-type structure. Furthermore, we investigate a corresponding infinite-time horizon problem in which the service of the customers does not stop when the last customer has been serviced but it continues indefinitely with the same customer order. It is assumed that the demands of the customers at different tours have the same distributions. It is shown that the discounted-cost optimal policy and the average-cost optimal policy have the same threshold-type structure as the optimal policy in the original problem. The theoretical results are illustrated by numerical examples.

Optimal maintenance of two stochastically deteriorating machines with an intermediate buffer

We consider a manufacturing system in which an input generating installation transfers a raw material to a subsequent production unit. Both machines deteriorate stochastically with usage and may fail. For each machine the deteriorating process is described by some known transition probabilities between different degrees of deterioration. A buffer has been built between the two machines in order to cope with unexpected failures of the installation. A discrete-time Markov decision model is formulated for the optimal preventive maintenance of both machines. The maintenance times are geometrically distributed and the cost structure includes operating costs, storage costs, maintenance costs and costs due to the lost production. It is proved that for fixed buffer content and for fixed deterioration degree of one machine, the average-cost optimal policy initiates a preventive maintenance of the other machine if and only if its degree of deterioration exceeds some critical level. We study, by means of numerical results, the effect of the variation of some parameters on the optimal policy and on the minimum average cost. For the case in which the maintenance times follow continuous distributions, an approximate discrete-time Markov decision model is proposed.

Optimal maintenance of a production-inventory system with idle periods

In this paper we consider a production-inventory system in which an input generating installation supplies a buffer with a raw material and a production unit pulls the raw material from the buffer with constant rate. The installation deteriorates in time and the problem of its optimal preventive maintenance is considered. It is assumed that the installation after the completion of its maintenance remains idle until the buffer is evacuated. Under a suitable cost structure it is shown that the average-cost optimal policy for fixed buffer content is of control-limit type, i.e. it prescribes a preventive maintenance of the installation if and only if its degree of deterioration is greater than or equal to a critical level. Using the usual regenerative argument, the average cost of a control-limit policy is computed exactly and then, the optimal control-limit policy is determined. Furthermore, the stationary probabilities of the system under the optimal policy are computed.

A Markov decision algorithm for optimal pest control through uniform catastrophes

Abstract This paper is concerned with the problem of controlling the stochastic growth of a bounded pest population by the introduction of uniform catastrophes, whose rate is proportional to the population size. The optimality criterion is that of minimising the long-run average cost per unit time. An appealing class of policies consists of the monotone policies, which introduce catastrophes if and only if the population size is equal to or exceed some critical value x . Firstly, a necessary and sufficient condition is found under which the policy of never controlling is optimal. If this condition fails, an efficient Markov decision algorithm that generates a sequence of strictly improved monotone policies is developed. There is strong numerical evidence that the algorithm converges to the optimal policy within the wider class of all stationary policies.

Finite and infinite-horizon single vehicle routing problems with a predefined customer sequence and pickup and delivery

We consider the problem of finding the optimal routing of a single vehicle that starts its route from a depot and picks up from and delivers K different products to N customers that are served according to a predefined customer sequence. The vehicle is allowed during its route to return to the depot to unload returned products and restock with new products. The items of all products are of the same size. For each customer the demands for the products that are delivered by the vehicle and the quantity of the products that is returned to the vehicle are discrete random variables with known joint distribution. Under a suitable cost structure, it is shown that the optimal policy that serves all customers has a specific threshold-type structure. We also study a corresponding infinite-time horizon problem in which the service of the customers is not completed when the last customer has been serviced but it continues indefinitely with the same customer order. For each customer, the joint distribution of the quantities that are delivered and the quantity that is picked up is the same at each cycle. The discounted-cost optimal policy and the average-cost optimal policy have the same structure as the optimal policy in the finite-horizon problem. Numerical results are given that illustrate the structural results.

An improved algorithm for the computation of the optimal repair/replacement policy under general repairs

We consider a system which deteriorates with age and may experience a failure at any time instant. On failure, the system may be replaced or repaired. The repair can partially reset the failure intensity of the unit. Under a suitable cost structure it has been proved in the literature that the average-cost optimal policy is of control-limit type, i.e. it conducts a replacement if and only if, on the nth failure, the real age of the system is greater than or equal to a critical value. We develop an efficient special-purpose policy iteration algorithm that generates a sequence of improving control-limit policies. The value determination step of the algorithm is based on the embedding technique. There is strong numerical evidence that the algorithm converges to the optimal policy.

A single vehicle routing problem with pickups and deliveries, continuous random demands and predefined customer order

This paper extends the results of a particular capacitated vehicle routing problem with pickups and deliveries (see Pandelis et al., 2013b) to the case in which the demands for a material that is delivered to N customers and the demands for a material that is collected from the customers are continuous random variables instead of discrete ones. The customers are served according to a particular order. The optimal policy that serves all customers has a specific threshold-type structure and it is computed by a suitable efficient dynamic programming algorithm that operates over all policies having this structure. The structural result is illustrated by a numerical example.

TRANSIENT SOLUTION FOR A SIMPLE IMMIGRATION BIRTH–DEATH CATASTROPHE PROCESS

In this note, we consider a simple immigration birth–death process with total catastrophes and we obtain the transient probabilities. Our approach involves a renewal argument. It is comparatively simpler and leads to more elegant expressions than other approaches that appeared in the literature recently.