To improve tax compliance18 the Texas Comptroller’s staff regularly audits at corporate home offices the records of out-of-state corporations doing business in Texas. Texas is considering the opening of a series of small offices near these corporate locations to reduce the travel costs now associated with such out-of-state audits.

The following table shows the fixed cost (in thousands of dollars) of operating such offices at 5 sites i, the number of audits required in each of 5 states j, and the travel cost (in thousands of dollars) per audit performed in each state from a base at any of the proposed office sites.

Tax Site Fixed Cost Cost to Audit of Corporate Location:

1 2 3 4 5 1 160 0 0.4 0.8 0.4 0.8 2 49 0.7 0 0.8 0.4 0.4 3 246 0.6 0.4 0 0.5 0.4 4 86 0.6 0.4 0.9 0 0.4 5 100 0.9 0.4 0.7 0.4 0 Audits 200 100 300 100 200 We seek a minimum total cost auditing plan.

(a) Briefly explain why appropriate decision variables for an optimization model of this problem are xi, j!fraction of audits at j done from i yi! e 1 if office i is opened 0 otherwise

(b) Explain why the yi must be modeled as discrete.

(c) Assign suitable symbolic names to the constants in the foregoing table: the fixed cost of office i, the travel cost for audits done at j from i, and the number of audits at j.

(d) Formulate an objective function minimizing the sum of fixed office operating cost plus travel costs to audit sites. (Hint:
The number of audits done at i from j is xi, j times the total number required at j.)

(e) Formulate a system of 5 main constraints requiring that 100% of audits at each j be performed.

(f) Formulate a system of 25 main constraints specifying that no part of the audits at any j can be done from i unless an office is opened at i.
(g) Complete your model with systems of variable-type constraints for the x and y decision variables.
(h) Is your model best classified as an LP, an NLP, an ILP, or an INLP, and is it singleor multiobjective? Explain.
(i) Enter and solve your model with class optimization software.
(j) Code your model in AMPL (paralleling Table 2.7).