Theory and Methodology A model for allocated versus actual costs in assignment and transportation problems Chris Pilot a, * , Steve Pilot b a Department of Arts and Science, Maine Maritime Academy, Castine, ME 04420, USA b University of Massachusetts at Lowell, Lowell, MA 01854, USA Received 10 March 1997; accepted 18 August 1997 Abstract We present a simple mathematical model which will relate the actual cost spent in accomplishing a task to the dollars budgeted for that task. In the speci®c instances of assignment and transportation problems we show how to minimize total dollars spent given total dollars allocated. We show furthermore how to quantitatively measure the work done along each arc in such problems. The total work, which will measure how ®xed costs are realized across various arcs for a given prescribed eort, can then be minimized. It is shown that this, in general, leads to a third type of optimal solution which is dierent from those optimal solutions obtained by minimizing either total cost or total dollars al- located. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Assignment; Transportation; Modelling; Optimization; Network analysis 1. Introduction Assignment, transportation and transshipment problems belong to a broad class of problems known as network ¯ow problems [1,2]. Typically they involve minimizing total cost subject to cer- tain supply and demand node constraints once the individual costs for each task are speci®ed. Often, however, the individual costs for all the arcs comprising the project cannot be estimated a pri- ori. Instead the allocated dollars that can be bud- geted to the individual arcs are what is known. Those allocated dollars can, for example, be esti- mated a priori based on a percentage of the ex- pected gain to be achieved if a speci®c person/ machine/service is assigned, or speci®c goods are transported, from supply node i to demand node j. In such instances, clearly, it would be useful to establish a fundamental relation between the in- dividual cost along arc ij and the allocated dollars for the same arc. Then minimizing total cost for all the tasks can be reduced to a problem involving the same but now with the allocated dollars as coecients. The key variables in such a relationship be- tween actual cost and allocated dollars for task ij should depend only on the characteristics of the European Journal of Operational Research 112 (1999) 570±581 * Corresponding author. E-mail: cpilot@bell.mma.edu. 0377-2217/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 9 7 ) 0 0 3 9 5 - 0