A new analytic approach to obtain the complete solution for systems of delay differential equations (DDE) based on the concept of Lambert functions is presented. The similarity with the concept of the state transition matrix in linear ordinary differential equations enables the approach to be used for general classes of linear delay differential equations using the matrix form of DDEs. The solution is in the form of an infinite series of modes written in terms of Lambert functions. Stability criteria for the individual modes, free response, and forced response for delay equations in different examples are studied, and the results are presented. The new approach is applied to obtain the stability regions for the individual modes of the linearized chatter problem in turning. The results present a necessary condition to the stability in chatter for the whole system, since it only enables the study of the individual modes, and there are an infinite number of them that contribute to the stability of the system.

An optimal solution, based on Markov Decision Theory, is presented for the capacity management problem in Reconfigurable Manufacturing Systems with stochastic market demand with a time delay between the time capacity change is ordered and the time it is delivered. The optimal policy in this paper is presented as optimal boundaries representing the optimal capacity expansion and reduction levels. The effects of change in the cost function parameters and the delay time on the optimal boundaries are presented for a capacity management scenario. The major differences between this research and the ones in inventory control lie in two folds. One is the fact that unlike inventory, capacity levels can be reduced according to the market demand. The other one is the novel approach presented in this paper to solve the delay problem which unlike the inventory control does not account for the cumulative unmet demand as a decision factor.

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This paper presents an optimal policy, based on Markov decision theory for the capacity management problem in a firm facing stochastic market demand. The firm implements a reconfigurable manufacturing system and faces a delay between the times capacity changes are ordered and the times they are delivered. Optimal policies are presented as optimal boundaries representing the optimal capacity expansion and reduction levels. To increase the robustness of the optimal policy to unexpected events, the concept of feedback control is applied to address the capacity management problem. It is shown that feedback provides sub-optimal solutions to the capacity management problem which are more robust under unexpected disturbances in market demand and unexpected events.

A new analytic approach to obtain the complete solution for systems of delay differential equations (DDE) based on the concept of Lambert functions is presented. The similarity to the concept of the state transition matrix in linear ordinary differential equations enables the approach to be used for general classes of linear DDE’s in matrix form. The solution is in the form of an infinite series of modes written in terms of Lambert functions. Results are presented for stability criteria for the individual modes, free response, and forced response in the context of specific examples. This new approach is also applied to the problem of chatter stability in a machining operation on a lathe. The results, since they are only for individual modes, and there are an infinite number of them, represent a necessary condition for system stability.