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The z& are variables introduced to model the objective function and are not required to be integer or non-negative because they will be automatically be so in any optimum solution provided the x~ are integer. That this formulation has a weak linear programming relaxation can be shown for the special case where all b, = b. In this case, note that in order for the problem to have a feasible integer solution it must be that Then, each x~ = IlL and equal to z& = IlL is a feasible solution to the linear program with objective value Leu.

L. , "Alternative Approaches to Modeling", TIMS/ORSA Joint National Meeting, Los Angeles, April 1986. , "An Introduction to Structured Modeling", Management Science 33 [1987], 547-588. E. Gomory, "A Linear Programming Approach to the Cutting Stock Problem", Operations Research 9 [1961], 849-859. , and K. Spielberg, "Propagation, Penalty Improvement, and Use of Logical Inequalities", Mathematics of Operations Research 25 [1977], 157-171. [I I] Jensen, D. , and E. L. Johnson, "Preprocessing and Implication Graphs for Mixed-Integer Programming Problems", [in preparation] [12] Johnson, E.

2 TOT (Blank ennies indicate possible lra/lsactions) Figure 4: A Social Accounting Matrix and the Underlying Network Model One problem encountered in using these tables is the following. The elements in the table are usually calculated from different agents in an economy and are often obtained through sampling procedures. The total expenditure (or income) of the agents, however, is normally available through government sources and is in general more accurate and up-to-date. The model may thus be formulated as follows: given a SAM whose elements are out-of-date, compute updated values for these elements that satisfy a prespecified set of row and column sums.