# Extending CPM for Multiform Project Time - Cost Curves

*by*Walter L. Meyer,

Louis R. Shaffer,

**Serial Information**:

*Journal of the Construction Division*, 1965, Vol. 91, Issue 1, Pg. 45-68

**Document Type:**Journal Paper

**Abstract:**The time-cost phase of CPM is extended with Integer Linear Programming techniques. Using the mathematical models developed, it is possible to determine the minimum project cost for each feasible project duration irrespective of the types of duration-cost relationships associated with the operations in the project. The types of operation duration-cost relationships that are considered are: (1) Non-increasing, bounded, piece-wise linear, continuous, and non-convex; (2) bounded but defined only at discrete points; and (3) bounded but discontinuous. An example project composed of nine operations is included to demonstrate that the formulations describing these relationships can be combined in any number of ways to describe any conceivable operation, duration-cost relationship. The formulations of the example are solved using a large scale computer, and the results are presented and analyzed. The conclusions indicate that while the models developed are sound and completely general, the required computer solution process needs a considerable degree of development. Present (1965) computer programs are capable of solving only projects with less than 50 operations.

**Subject Headings:**Critical path method | Linear functions | Construction costs | Computer models | Computer programming | Mathematical models | Computer analysis

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