Tran.rppn Rer -E “0, 20B. No 1. pp 19-39. 1986 axI-ooo’86 $3 cm+ 00 Pruned m Ihe U S A 0 1986 Pegamon Press Ltd GRAVITY MODELS WITH MULTIPLE OBJECTIVES- THEORY AND APPLICATIONS ASA HALLEFJORD and KURT J~RNSTEN LinkGping Institute of Technology, Department of Mathematics. S-581 83 LinkGping. Sweden (Received 1 May 1984; in revised form 13 September 1984) Abstract-The gravity model has been widely used to predict trip matrices in transportation planning, both as a descriptive and an optimizing model. In this paper we present multiobjective programming formulations of the gravity model. The models use the concept of entropy, both as a measure of interactivity in the system, and as a distance measure, for measuring the distance from a reference or target point suggested by the planner. We also present gravity model formulations which do not require consistent estimates for the demand and supply in the different sites. These relaxed multiobjective gravity models include objectives for the demands and supplies, where the distance from some n priori given values is measured in entropy terms. The use of the models is illustrated on 1975 census data from the county of estergbtland, Sweden. I. TRIP DISTRIBUTION MODELS In traffic planning it is of great interest to study the commuting pattern in a region. Often, predictions about an uncertain future have to be made. At best there are historical data at hand about trip patterns in the past. Also census data should give reliable information about where people live and where they work. Traffic planners often concentrate on travel between work and place of residence, since these travels are believed to constitute a large part of private travels. Moreover, the work journeys are easier to predict than for instance shopping tours or leisure trips. This paper deals with the problem of predicting a future trip matrix, where we want to allocate trips originating in a specific zone among a number of destinations, to create a trip matrix with certain desired properties. We can here think of the origins as being zones where people live and destinations as zones where they work. Of course, these may in some cases coincide, but in the model formulations they can be thought of as being distinct. One way to create a trip matrix is to use a descriptive model, where the number of trips from origin i to destination j, ti,, is given by t,, = r,s,fk-,). Here f(c,,) is a so-called deterrence function, used to describe how tij depends on some gen- eralized cost cil of travelling from i to j. r, and s, are balancing factors used to make the predictions feasible, that is to satisfy the marginal total constraints of the prediction year. It is assumed that the number of people living in zone i is known and therefore Z, ti, must equal this number. The same thing applies to destinations and C, t,,. The deterrence function may contain information about some observed trip pattern in some base year. In the following we shall denote by tt the number of trips from i zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLK to j in some base year. I and J denote the number of origin and destination zones. The marginal total constraints can mathematically be expressed as c t, = A,, i = 1,. . ,I zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR zt,,=B,, j=l,..., J where the Ai’s and the Bj’s are predictions on the total number of trips from origin i and the total number of trips to destination j, respectively. In this model we require that the estimates 19