George G. Polak

Risk management strategies via minimax portfolio optimization

Recent extreme economic developments nearing a worst-case scenario motivate further examination of minimax linear programming approaches for portfolio optimization. Risk measured as the worst-case return is employed and a portfolio from maximizing returns subject to a risk threshold is constructed. Minimax model properties are developed and parametric analysis of the risk threshold connects this model to expected value along a continuum, revealing an efficient frontier segmenting investors by risk preference. Divergence of minimax model results from expected value is quantified and a set of possible prior distributions expressing a degree of Knightian uncertainty corresponding to risk preference determined. The minimax model will maximize return with respect to one of these prior distributions providing valuable insight regarding an investors risk attitude and decision behavior. Linear programming models for financial firms to assist individual investors to hedge against losses by buying insurance and a model for designing variable annuities are proposed.

A hierarchical production plan for a make-to-order steel fabrication plant

A three-tiered hierarchical production plan (HPP) for a strictly make-to-order steel fabrication plant with the objective of developing a production plan and master schedule for a set of product archetypes is implemented. Data are collected from an actual steel fabrication plant located in the Midwestern section of the US. An aggregate linear programming model, a non-linear disaggregate model and a master production schedule comprise the respective tiers. Appropriate models provide the forecasts needed in the first two tiers. A production plan and master schedule based on data collected at the plant, benefits expected for its implementation and practical limitations are reported.

Optimal Clustering of Time Periods for Electricity Demand-Side Management

Several pure binary integer optimization models are developed for clustering time periods by similarity for electricity utilities seeking assistance with pricing strategies. The models include alternative objectives for characterizing various notions of within-cluster distances, admit as feasible only clusters that are contiguous, and allow for circularity, where time periods at the beginning and end of the planning cycle may be in the same cluster. Restrictions upon cluster size may conveniently be included without the need of additional constraints. The models are populated with a real-world dataset of electricity usage for 93 buildings and solutions and run-times attained by conventional optimization software are compared with those by dynamic programming, or by a greedy algorithm applicable to one of the models, that run in polynomial time. The results provide time-of-use segments that an electricity utility may employ for selective pricing for peak and off-peak time periods to influence demand for the purpose of load leveling.

Scheduling parallel assembly workstations to minimize a shared pool of labor

The problem of scheduling jobs on several identical parallel assembly workstations that constitute a production facility and share a common pool of labor within a factory environment is discussed. Labor requirements vary on each workstation depending upon the job being processed, and the workstation is paced so as to maintain a constant cycle time. The labor needed at any point in time is the sum of the labor requirements over all workstations at that instant. The workforce size is defined as the maximum level of labor required at any point during the schedule. A two-stage approach is proposed to find a schedule that minimizes workforce size: the first stage is a mixed integer linear programming model that determines starting times, and the second is a polynomial-time procedure to assign jobs to specific assembly workstations. Moreover, it is shown that the two stages yield decisions that are jointly optimal for an integrated model that determines starting times and assignments simultaneously. A heuristic procedure is then proposed for scheduling jobs such that each job is assigned a starting time according to the effect on current workforce profile, a strategy described as resource fit. A lower bound on the maximum workforce size is given by the average workforce requirement over the planning horizon. The numerical performance of both the integer programming model and the heuristics are tested on two sets of problems. Tradeoffs between the total workforce and deadline for makespan suggested by this type of scheduling problem are discussed.

On A Special Case of the Quadratic Assignment Problem with an Application to Storage-and-Retrieval Devices

In a storage-and-retrieval device, items are retrieved on demand from a storage bank by a picking mechanism. Many varieties of these robotic devices are in use in manufacturing, logistics and computer peripherals. In printed circuit board manufacturing, storage-and-retrieval is intertwined with component placement and product clustering. Under certain circumstances, the problem of assigning items by type to storage slots to minimize the expected retrieval time is a quadratic assignment problem. Although such models are very difficult to solve to optimality, an important special case considered here admits an easy solution, namely, the well known “organ pipe” arrangement of items.

Optimal block design models for course timetabling

We seek a timetable for courses offered in S sections to maximize contact among K student cohorts over T terms. For this combinatorial optimization problem we propose both integer programming and constraint programming models. The latter straightforwardly reduces the size of solution equivalence classes, thereby facilitating the search for an optimum.

Job release policy and printed circuit board assembly

A pick-and-place machine assembles printed circuit boards from electronic components. Programmed instructions control the storage and retrieval of these components from replaceable feeders. Since component requirements differ among board types, each change of feeders can improve processing time but indicates a machine setup. For the special case in which the jobs are released to the shop floor in a sequence fixed upstream, we show that an optimal setup policy can be determined efficiently by solving a shortest path problem. We then use constraint programming to model the general assembly problem and conclude by investigating a significant extension of this model.

Synchronization, cross-docking, and decoupling in supply chain networks

At any distribution centre (DC), the decision of whether to synchronise inbound and outbound flows for cross-docking, or to decouple these flows by maintaining inventory, has a significant impact on supply chain performance. Key drivers of this decision, in turn, are the sizes of the discrete lots that comprise the flows. Thus, we formulate an original optimisation model that determines order lot-sizing decisions to minimise, for given constant arc flows, the sum of ordering cost and pipeline inventory cost on arcs and buffer inventory at DCs. The model employs an average throughput as a surrogate to estimate buffer inventory at facilities at which synchronisation is not economical and therefore serves to decouple inbound and outbound flows. Perfect lot-for-lot matching of shipments would impose very restrictive constraints on supply chain operations, but equality of average throughput indicates an innovative, relaxed mode of synchronisation. This mode is practicable for cross-docking by means of bulk-breaking or consolidation of shipments. A heuristic approach based on the Lagrangian relaxation and subgradient optimisation is developed for the non-linear mixed-general integer optimisation model, which is illustrated by numerical examples and tested using a benchmark data set.

A Multi-Exchange Neighborhood Search Heuristic for an Integrated Clustering and Machine Setup Model for PCB Manufacturing

In the manufacture of printed circuit boards, electronic components are attached to a blank board by one or more pick-and-place machines. Frequent machine setups, though time-consuming, can reduce overall processing time. We consider the Integrated Clustering and Machine Setup (ICMS) model, which incorporates this tradeoff between processing time and setup time and seeks to minimize the sum of the two. Solving this model to optimality is intractable for very large-scale instances. We show that ICMS is NP-hard and consequently propose and test a heuristic based on multi-exchange neighborhood search structures. Initial numerical results arevery encouraging.

Minimizing flowtime for paired tasks

Certain service and production systems require that a single processor complete, for each job, a pair of ordered tasks separated in time by a minimal required delay. In particular, circumstances, this mode of operation can characterize a physician in a hospital Emergency Department, a painting crew at a construction site, or a work station in a job shop. Motivated by these scenarios, we formulate an applicable scheduling problem and investigate its solution. To determine an optimal schedule, we formulate an appropriate mixed integer model in which the key lever for process improvement is the batching of tasks. We further show that two special cases of this problem can be optimally solved efficiently. To expedite decision-making, we propose heuristic approaches supported by spreadsheet based software. Numerical results are then presented and insights discussed.