A company is considering opening warehouses in four cities: Ottawa, Kingston, Ke

Question

A company is considering opening warehouses in four cities: Ottawa, Kingston, Kemptville, and Gatineau to cover eastern Ontario. Each warehouse can ship 100 units per week. The weekly fixed cost of keeping each warehouse open is $400 for Ottawa, $500 for Kingston, $300 for Kemptville, and $150 for Gatineau. Region1 of eastern Ontario requires 80 units per week, region 2 requires 70 units per week, and region 3 requires 40 units per week. The costs (including production and shipping costs) of sending one unit from a plant to a region are shown in Table below From Ottawa Kingston Kemptville Gatineau Region 1 20 48 26 24 To (S) Region 2 40 15 35 50 Region 3 50 26 18 35 We want to meet weekly demands at minimum cost, subject to the preceding information and the following restrictions: 1. If the Ottawa warehouse is opened, then the Kingston warehouse must be opened 2. At most two warehouses can be opened 3. Either the Gatineau or the Kingston warehouse must be opened. Formulate an Integer/Binary program that can be used to minimize the weekly costs of meeting demand. (Do not solve)

Explanation / Answer

ILP model

Decision variables: Xij = Quantity shipped from warehouse i to region j

Yi = 1, if warehouse i is kept open

Objective: Minimize Z = Yi*Fi + Cij*Xij , where Fi is the fixed cost of keeping warehouse i open and Cij is the cost of shipping one unit from warehouse i to region j

Minimize Z = 400Y1 + 500Y2 + 300Y3 + 150Y4 + 20X11 + 40X12 + 50X13 + 48X21 + 15X22 + 26X23 + 26X31 + 35X32 + 18X33 + 24X41 + 50X42 + 35X43

s.t.

100Y1 - (X11+X12+X13) >= 0

100Y2 - (X21+X22+X23) >= 0

100Y3 - (X31+X32+X33) >= 0

100Y4 - (X41+X42+X43) >= 0

X11+X21+X31+X41 = 80

X12+X22+X32+X42 = 70

X13+X23+X33+X43 = 40

Xij >= 0

Yi = 0,1

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