The household activity pattern problem of analyzing/predicting the optimal path of household members through time and space as they complete a prescribed agenda of out-of-home activities is posed as a variant of the pickup and delivery problem with time windows. The most general case of the model includes provision for vehicle transfer, selective activity participation, and ridesharing options. A series of examples are solved using generic algorithms. The model is purported to remove existing barriers to the operationalization of activity-based approaches in travel behavior analysis. 2 It is derived principally from the early work of Hagerstrand (1970) in time-space geography, in which travel and activity participation are recorded as passage through time and space, with the individual's location at any time represented by a continuous path in the spatial and temporal dimensions. As noted by Stopher et al. (1993) , however, "despite its conceptual appeal and clarity, the time-space geography structure has proved quite difficult to implement operationally." Commenting on challenges facing further development of activity-based approaches, Kitamura (1988) noted that, conceptually, Lancaster's utility formulation (1966) neatly applies to the problem, but that ". . . if a utility function can be identified at all, an array of mathematical programming methods are available. However, the problem at hand is, at the simplest, a discrete choice-continuous allocation problem with correlated multiple alternatives, combined with the traveling salesman problem, problem of collective decision-making, and household coupling constraints which is in part a logistic problem. This is an overwhelming problem. In fact no model has been constructed that determines activity patterns on the sole basis of the utility maximization principle." In his review, Kitamura further notes that existing models are largely restricted to addressing questions of activity participation and time allocation at the level of total daily time expenditure, with a wide gap existing between these models and those modeling daily activity and travel patterns. Both Kitamura (1988) and Stopher (1993) point to the STARCHILD MODEL (Recker et al., 1986a (Recker et al., ,1986b as the only known operationalized model that predicts a set of activity patterns from household decision-making information. Although based loosely on mathematical programming principles, the STARCHILD model is severely limited in that it: (1) provides no mechanism for household interaction, modeling the activity/travel patterns of each household member separately, (2) relies on a heuristic solution procedure based on exhaustive enumeration and evaluation of feasible solutions, (3) discretizes the temporal dimension and relies on pattern recognition algorithms to distinguish simple temporal displacements of similar 3 solutions, and (4) has no provision for addressing either activity or vehicle allocation decisions or for consideration of complex modal choice decisions, such as carpooling. According to Koppleman (1988) , in activity-based approaches "the research need is to develop a theoretical framework within which to relate the multiple themes of human/social behavior to the generation of the need or desire to participate in activities and the derived demand for travel." This paper attempts to provide one such framework that is believed to offer potential in the operationalization of activity-based travel demand methodologies. Specifically, the household activity pattern problem (HAPP) is posed as a variant of the pick up and delivery problem with time windows (PDPTW). In the most general case considered, the model addresses the optimization (relative to the household's utility function) of the interrelated paths through the time/space continuum of a series of household members with a prescribed activity agenda and a stable of vehicles and ridesharing options available. In the development of the model, a deliberate attempt has been made to maintain, to the extent possible, both the notation and structure of the well-known PDPTW in the hope that this would provide a conducive environment for future development and improvement. In addition, little attention has been paid to issues of model efficiency or to the efficiency of solution algorithms; rather, reliance has been placed mainly on readily available "canned" software in an effort to demonstrate the practicality of the approach. In order to take advantage of previous work involving the PDPTW, the formulation of the general HAPP involving complex elements such as ridesharing and vehicle-switching options is developed from a progression of cases in which initial restrictions that result in an equivalence to the PDPTW are gradually removed. Each member of the household has exclusive, unrestricted use of a personal vehicle and any activity can be completed by any member of the household.
An interesting extension of the PDPTW was proposed by REF , namely the household activity pattern problem.
The household activity pattern problem: General formulation and solution
Transportation Research Part B-methodological
Economics Engineering