Eugene Khmelnitsky

On an optimal control problem of train operation

This paper examines the operation of a train on a variable grade profile subject to arbitrary speed restrictions. The purpose of the study is to determine a detailed program for traction and brake applications, which minimizes energy consumption in moving the train along a given route for a given time. Stated in the form of optimal control, this problem is solved by constructing a numerical algorithm which essentially exploits analytical properties of the optimal solution obtained from the maximum principle analysis. Due to its analytical origin, the algorithm has inherent accuracy and adequate quick-operation that are demonstrated in numerical examples.

One-machine n-part-type optimal setup scheduling: analytical characterization of switching surfaces

The authors consider optimal setup scheduling of a single reliable machine. Production flow of n different part types and the setup process are described by differential equations. Setup change rates are control variables. Necessary conditions on optimal setup changes are characterized analytically, and optimal setup change times are derived for a given setup change sequence. The linearization of optimal setup switching surfaces is derived, indicating the existence of attractors observed in numerical optimal solutions. The approach developed in this paper establishes a strong basis for studying multimachine production systems and for constructing tractable near-optimal numerical solution techniques.

Optimizing a dynamic order-picking process

This research studies the problem of batching orders in a dynamic, finite-horizon environment to minimize order tardiness and overtime costs of the pickers. The problem introduces the following trade-off: at every period, the picker has to decide whether to go on a tour and pick the accumulated orders, or to wait for more orders to arrive. By waiting, the picker risks higher tardiness of existing orders on the account of lower tardiness of future orders. We use a Markov decision process (MDP) based approach to set an optimal decision making policy. In order to evaluate the potential improvement of the proposed approach in practice, we compare the optimal policy with two naive heuristics: (1) “Go on tour immediately after an order arrives”, and, (2) “Wait as long as the current orders can be picked and supplied on time”. The optimal policy shows a considerable improvement over the naive heuristics, in the range of 7–99%, where the specific values depend on the picking process parameters. We have found that one measure, the slack percentage of the picking process, associated with the difference between the promised lead time and the single item picking time, predicts quite accurately the cost reduction generated by the optimal policy. Since relatively small-scale problems could be solved by the optimal algorithm, a heuristic was developed, based on the structure and properties of the optimal solutions. Numerical results show that the proposed heuristic, MDP-H, outperforms the naive heuristics in all experiments. As compared to the optimal solution, MDP-H provides close to optimal results for a slack of up to 40%.

A consignment system where suppliers cannot verify retailer's sales reports

Abstract Newspapers are often sold through stores via consignment arrangements, which involve vendor (publisher) managed inventory and revenue sharing. Since retailers are not required to actually return unsold copies, it is said that some of them occasionally under-report sales. That hurts the publisher on the short run and could also interfere with his rational stocking decisions as these are based, to some extent, on previous sales reports. We construct a discounted dynamic framework for the retailers optimal reporting as a function of the publishers delivery–response function to these reports, and a similar average-cost model. It turns out the optimal report does not depend on actual sales. We then show that the publishers resulting delivery response function is the same as it would be in an integrated system. Thus the retailers untruthful behavior actually causes the system to behave optimally. Had the retailer been verifiably truthful, the system would not be coordinated.

Maximum principle-based methods for production scheduling with partially sequence-dependent setups

This paper introduces a continuous-time optimal control approach to cope with scheduling problems in complex manufacturing systems. These systems are controlled under arbitrary changing demand profiles with respect to multi-level bills of materials, flexible flow shops with setup effects and finite production capacities. The suggested methods extend a general optimal control approach to handle with partially sequence-dependent setups on one hierarchical level along with production scheduling. The optimal scheduling method and its enhanced modifications are derived based on the maximum principle analysis and compared to the traditional two-level hierarchical approach.

A maximum principle based combined method for scheduling in a flexible manufacturing system

A continuous time dynamic model of discrete scheduling problems for a large class of manufacturing systems is considered in the present paper. The realistic manufacturing based on multi-level bills of materials, flexible machines, controllable buffers and deterministic demand profiles is modeled in the canonical form of optimal control. Carrying buffer costs are minimized by controlling production rates of all machines that can be set up instantly. The maximum principle for the model is studied and properties of the optimal production regimes are revealed. The solution method developed rests on the iterative approach generalizing the method of projected gradient, but takes advantage of the analytical properties of the optimal solution to reduce significantly computational efforts. Computational experiments presented demonstrate effectiveness of the approach in comparison with pure iterative method.

An optimal control method for aggregate production planning in large-scale manufacturing systems with capacity expansion and deterioration

An optimal control approach to continuous-time aggregate production planning problems is presented. The proposed approach describes the production and capacity evolution (expansion, sell and deterioration) processes in the form of differential equations with regular production, subcontracting and capacity change rates controllable on one hierarchical level. In this way, the traditional disadvantages of the two-level problem consideration (one level for strategic capacity planning and the other for production smoothing) are avoided. Analytical properties for optimal production and capacity control regimes and conditions for their changeover are derived by the maximum principle. Based on these results, an insight into the optimal behaviour of the production system is gained and a fast numerical method is developed to identify and sequence the optimal regimes for arbitrary demand profiles. A computational example illustrates the effectiveness of the approach.

A time-decomposition method for sequence-dependent setup scheduling under pressing demand conditions

This paper develops a method for continuous-time scheduling problems in flexible manufacturing systems. The objective is to find the optimal schedule subject to different production constraints: precedence constraints (bills of materials), sequence-dependent setup times, finite machine capacities, and pressing demands. Differential equations along with mixed constraints are used to model production and setup processes in a canonical form of optimal control. The proposed approach to the search for the optimal solution is based on the maximum principle analysis and time-decomposition methodology. To develop fast near-optimal solution algorithms for sizable problems, we replace the general problem with a number of sub-problems so that solving them iteratively provides tight lower and upper estimates of the optimal solution.

Balancing facilities in aggregate production planning: Make-to-order and make-to-stock environments

The paper concerns an optimal control approach to continuous-time aggregate production planning in make-to-stock and make-to-order environments. The dynamics of such a production system are modelled by purchased, in-process and finished inventory flows through multiple facilities of finite capacity. The objective of the make-to-order production is to track a given customer demand as closely as possible, while in the make-to-stock production it is to keep the purchased inventory minimal when filling the stocks with volumes of products scheduled to be completed by the end of the planning horizon. The optimal behaviour of the system is studied with the aid of the maximum principle which allows new analytical results to be derived. Consequently, fast numerical and analytical algorithms for balancing facilities in the two corresponding environments are suggested.

An optimal control model for continuous time production and setup scheduling

SUMMARY This paper concerns a new approach to continuous time optimal scheduling problems for a large class of manufacturing systems. The proposed approach states the problem in terms of optimal control with setup and production rates controllable on one hierarchical level. This allows for the traditional disadvantages of the two-level problem consideration (one level for defining the target production rates, and the other for scheduling the setup changes) to be avoided and stable control strategies to be obtained. Analysis of the maximum principle results in setup conditions of the optimal schedule and special regimes to which the optimal tends between subsequent setups. Based on these results, a numerical method is developed to define the sequence of the special regimes and the timing for getting into and out of them. An example illustrates the effectiveness of the approach.