Lodi, Andrea

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Almae Matris Studiorum Campus

Example 1A Health Agency has to decide about opening new ambulatories to serve aset of m users (i = 1, . . . ,m). A preliminary investigation suggests a set of npotential locations. Each users can be served by an ambulatory on location j(j = 1, . . . , n) if the location is not “too far” from the user’s home. Thus, weare given a matrix A whose entry aij has value 1 if user i can be served by anambulatory open in location j and 0 otherwise. The cost of opening an ambu-latory on location j is equal to cj . Moreover, the ambulatory in location j canserve at most Mj users. Finally, for planning reasons, the Health Agency wantsto open at least P ambulatories, serve all users and minimize the overall cost.Integer Linear Programming Model:xij :={1 if user i is served by ambulatory j0 otherwisei = 1, . . . ,m; j = 1, . . . , n;yj :={1 if ambulatory j is open0 otherwisej = 1, . . . , n;minn∑j=1cjyj (1a)n∑j=1aijxij = 1 ∀i = 1, . . . ,m (1b)m∑i=1xij ≤Mjyj ∀j = 1, . . . , n (1c)n∑j=1yj ≥ P (1d)yj ∈ {0, 1} ∀j = 1, . . . , n (1e)xij ∈ {0, 1} ∀i = 1, . . . ,m; ∀j = 1, . . . , n (1f)1Example 2The Ministry of Home Security must decide concerning the opening of newFire Stations so as to serve q neighborhoods (i = 1, . . . , q) in Bologna. A preli-minary investigation led to the consider a set of s potential areas j, j = 1, . . . , swhere a Fire Station can be built. Each neighborhood i can be served by aFire Station on area j if that is not “too far”. Then, we are given a binarymatrix A whose entry aij is 1 if neighborhood i can be served by a Fire Stationon area j and 0 otherwise. The opening cost for the Fire Station in area j isrj . For security reasons, each neighborhood must be potentially served by atleast 2 (open) Fire Stations, and at least B Fire Stations must be open. Theconstraints must be satisfied by minimizing the overall opening cost.Integer Linear Programming Model:yj :={1 if Fire Station j is open0 otherwisej = 1, . . . , s;mins∑j=1rjyj (2a)s∑j=1aijyj ≥ 2 ∀i = 1, . . . , q (2b)s∑j=1yj ≥ B (2c)yj ∈ {0, 1} ∀j = 1, . . . , s (2d)(2e)2Example 3A pet shop has n water pools to be used to show a set of m tropical fishes.Each pool j has a cost of cj and a volume Vj . The shop owner wants to minimi-ze the overall cost of the pools used by considering that, for space reasons, theshop can use at most K pools simultaneously. It is given a graph G = (V,E)whose generic edge aih defines the incompatibility between fishes i and j, respec-tively, of being put in the same pool (because of the natural nutrition chain).Moreover, we are given a binary matrix B whose entry bij is equal to 1 if fishi can be assigned to pool j and 0 otherwise. Finally, each fish i needs a livingspace of vi, thus the sum of the living space of the fishes assigned to the samepool j cannot be higher than the volume of the volume Vj .Integer Linear Programming Model:yj :={1 if pool j is used0 otherwisej = 1, . . . , n;xij :={1 if fish i is assigned to pool j0 otherwisei = 1, . . . ,m; , j = 1, . . . , n;minn∑j=1cjyj (3a)n∑j=1yj ≤ K (3b)n∑j=1bijxij = 1 ∀i = 1, . . . ,m (3c)m∑i=1vixij ≤ Vjyj ∀j = 1, . . . , n (3d)xij + xhj ≤ yj ∀(i, h) ∈ A, j = 1, . . . , n (3e)xij , yj ∈ {0, 1} ∀i = 1, . . . ,m, j = 1, . . . , n (3f)Discuss alternatives and/or possible improvements to the above model.3

Three examples of ILP models

http://campus.unibo.it/127615/1/Examples.pdf

Three examples of ILP models

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