Leyuan Shi

Nested Partitions Method for Global Optimization

We propose a new randomized method for solving global optimization problems. This method, the Nested Partitions (NP) method, systematically partitions the feasible region and concentrates the search in regions that are the most promising. The most promising region is selected in each iteration based on information obtained from random sampling of the entire feasible region and local search. The method hence combines global and local search. We first develop the method for discrete problems and then show that the method can be extended to continuous global optimization. The method is shown to converge with probability one to a global optimum in finite time. In addition, we provide bounds on the expected number of iterations required for convergence, and we suggest two stopping criteria. Numerical examples are also presented to demonstrate the effectiveness of the method.

Nested Partitions Method for Stochastic Optimization

The nested partitions (NP) method is a recently proposed new alternative for global optimization. Primarily aimed at problems with large but finite feasible regions, the method employs a global sampling strategy that is continuously adapted via a partitioning of the feasible region. In this paper we adapt the original NP method to stochastic optimization where the performance is estimated using simulation. We prove asymptotic convergence of the new method and present a numerical example to illustrate its potential.

An Optimization Framework for Product Design

An important problem in the product design and development process is to use the part-worths preferences of potential customers to design a new product such that market share is maximized. The authors present a new optimization framework for this problem, the nested partitions (NP) method. This method is globally convergent and may utilize existing heuristic methods to speed its convergence. We incorporate several known heuristics into this framework and demonstrate through numerical experiments that using the NP method results in superior product designs. Our numerical results suggest that the new framework is particularly useful for designing complex products with many attributes.

Optimal buffer allocation in production lines

The optimal allocation of buffers in production lines is an important research issue in the design of a manufacturing system. We present a new hybrid algorithm for this complex design problem: the hybrid Nested Partitions (NP) and Tabu Search (TS) method. The Nested Partitions method is globally convergent and can utilize many of the existing heuristic methods to speed up its convergence. In this paper, we incorporate the Tabu Search heuristic into the NP framework and demonstrate through numerical examples that using the hybrid method results in superior solutions. Our numerical results illustrate that the new algorithm is very efficient for buffer allocation problems in large production lines.

New parallel randomized algorithms for the traveling salesman problem

Abstract We recently developed a new randomized optimization framework, the Nested Partitions (NP) method. This approach uses partitioning, global random sampling, and local search heuristics to create a Markov chain that has global optima as its absorbing states. This new method combines global and local search in a natural way and it is highly matched to emerging massively parallel processing capabilities. In this paper, we apply the NP method to the Travelling Salesman Problem . Preliminary numerical results show that the NP method generates high-quality solutions compared to well-known heuristic methods, and that it can be a very promising alternative for finding a solution to the TSP. Scope and purpose The traveling salesman problem involves finding the shortest route between a number of cities. This route must visit each of the cities exactly once and begin and finish in the same city. As easy as it is to describe, this problem is notoriously difficult to solve. It is widely believed that there is no efficient algorithm that can solve it accurately. On the other hand, this problem is very important since it has many applications in such areas as routing robots through automatic warehouses and drilling holes in printed circuit boards. We present a new method, the Nested Partitions method, for solving the traveling salesman problem. The method is very flexible in that it is capable of finding good solutions rapidly and given enough time will identify the optimal solution. This new method is also highly matched with parallel processing capabilities.

Some topics for simulation optimization

We give a tutorial introduction to simulation optimization. We begin by classifying the problem setting according to the decision variables and constraints, putting the setting in the simulation context, and then summarize the main approaches to simulation optimization. We then discuss three topics in more depth: optimal computing budget allocation, stochastic gradient estimation, and the nested partitions method. We conclude by briefly discussing some related research and currently available simulation optimization software.

An optimization framework for solving capacitated multi-level lot-sizing problems with backlogging

This paper proposes two new mixed integer programming models for capacitated multi-level lot-sizing problems with backlogging, whose linear programming relaxations provide good lower bounds on the optimal solution value. We show that both of these strong formulations yield the same lower bounds. In addition to these theoretical results, we propose a new, effective optimization framework that achieves high quality solutions in reasonable computational time. Computational results show that the proposed optimization framework is superior to other well-known approaches on several important performance dimensions.

Selection of beam orientations in intensity-modulated radiation therapy using single-beam indices and integer programming

While the process of IMRT planning involves optimization of the dose distribution, the procedure for selecting the beam inputs for this process continues to be largely trial-and-error. We have developed an integer programming (IP) optimization method to optimize beam orientation using mean organ-at-risk (MOD) data from single-beam plans. Two test cases were selected in which one organ-at-risk (OAR) and four OARs were simulated, respectively, along with a PTV. Beam orientation space was discretized in 10 degrees increments. For each beam orientation, a single-beam plan without intensity modulation and without constraints on OAR dose was generated and normalized to yield a mean PTV dose of 2 Gy and the corresponding MOD was calculated. The degree of OAR sparing was related to the average OAR MODs resulting from the beam orientations utilized with improvements of up to 10% at some dose levels. On the other hand, OAR DVHs in the IMRT plans were insensitive to beam numbers (in the 6-9 range) for similar average single-beam MODs. These MOD data were input to an IP optimization process, which then selected specified numbers of beam angles as inputs to a treatment planning system. Our results show that sets of beam angles with lower average single-beam MODs produce IMRT plans with better OAR sparing than manually selected beam angles. To optimize beam orientations, weights were assigned to each OAR following MOD input to the IP which was subsequently solved using the branch-and-cut algorithm. Seven-beam orientations obtained from solving the IP were applied to the test case with four OARs and the resulting plan with a dose prescription of 63 Gy was compared with an equi-spaced beam plan. The IP selected beams produced dose-volume improvements of up to 40% for OARs proximal to the PTV. Further improvement in the DVH can be obtained by increasing the weights assigned to these OARs but at the expense of the remaining OARs.

The allocation of berths and quay cranes by using a sub-gradient optimization technique

This paper examines the allocation of berths and quay cranes for vessels arriving at container terminals. Previous work in this area has either ignored or deemphasized the coverage range limitations of quay cranes or imposed too loose or too rigid assumptions about whether quay cranes can be adjusted during loading and discharging, thus making the extracted models less applicable to the actual situation. This paper takes into consideration the coverage ranges of quay cranes and allows for limited adjustments of quay cranes during loading and discharging. A mixed integer programming model is constructed, and a sub-gradient optimization algorithm is applied to solving the problem. Using actual data, the performance of the algorithm is demonstrated.

A real-options-based analysis for supply chain decisions

Flexibility allows firms to compete more effectively in a world of short product life cycles, rapid product development, and substantial demand and/or price uncertainty. We develop a supply chain model in which a manufacturing firm can have the flexibility to select different suppliers, plant locations, and market regions and there can be an implementation time lag for the supply chain operations. We use a real options approach to estimate the value of flexibility and to determine the optimum strategy to manage the flexibility under uncertainty in the currency exchange rate. To price the operational flexibility, we develop a Monte Carlo simulation technique that is able to incorporate a large number of variables into the valuation. We show that without considering time lag impact, the value of the operational flexibility can be significantly overestimated.