operations-research_188 - 9.1 Illustrative Applications 355...

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9.1 Illustrative Applications 355 Street A 2 Street B 1 1f--------------"""""------------k.3 Street E ::.:: d) Cl.l =: (/) Street C 0 5 ::t:: ..... OJ v Cl.l Cl.l ... ... en en Street D 0) 8 s l- :r '- 'e :r- of ets , . . :;.: ' FIGURE 9.1 Street Map of the U of A Campus X7 + Xg ~ 1 (Street D) X6 + X7 ~1 (Street E) X2 + X6 ~1 (Street F) Xl + X6 ~l (Street G) X4 + X7 ~1 (Street H) X2 + X4 ~ 1 (Street 1) Xs + Xs ~ 1 (Street J) X3 + Xs ~ 1 (Street K) Xj = (0, l),j = 1,2, ... ,8 The optimum solution of the problem requires installing four telephones at intersections 1,2,5, and 7. Remarks. In the strict sense, set-covering problems are characterized by (1) the variables Xj, j = 1,2, ... , n, are binary, (2) the left-hand-side coefficients of the constraints are 0 or 1, (3) the right-hand side of each constraint is of the form (~ 1), and (4) the objective function minimizes CIXl + C2X2 + ... + CnX m where Cj > 0 for all j = 1,2, ... , n. In the present example, Cj = 1 for all j. If Cj represents the installation cost in location j, then these coefficients may assume values other than 1. Variations of the set-covering problem include additional side conditions, as some of the situ- ations in Problem Set 9.lb

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