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 $150 for Gatineau. Region 1 of eastern Ontario requires 80 units per week, region 2 requires 70 units per week, and region 3 requires 40 units per week. The costs (including production and shipping costs) of sending one unit from a plant to a region are shown in Table below. To ($) From Region 1 Region 2 Region 3 Ottawa 20 40 50 Kingston 48 15 26 Kemptville 26 35 18 Gatineau 24 50 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

1.

Obejctive function = Fixed cost of opening each warehouse + Transportation cost from each warehouse to each region

Obejctive function = (400*xo + 500*xki + 300*xke + 150*xg) + (uo1*20 + uo2*40 + uo3*50) + (uki1*48 + uki2*15 + uki3*26) + (uke1*26 + uke2*35 + uke3*18) + (ug1*24+ ug2*50 + ug3*35)

where xi and uij are binary and integer respectively.

Constraints:

Regional Demand Constraints:

uo1 +uki1 + uke1 + ug1 = 80 (Region 1 Demand)

uo2 +uki2 + uke2 + ug2 = 70 (Region 2 Demand)

uo13 +uki3 + uke3 + ug3 = 40 (Region 3 Demand)

Capacity Constraints:

Each warehouse has a maximum capacity of 100.

(uo1 + uo2 + uo3) <= 100

(uki1 + uki2 + uki3) <= 100

(uke1 + uke2 + uke3) <= 100

(ug1 + ug2 + ug3) <= 100

Ottawa-Kingston:

xki >= xo   (Kingston can be open when Ottawa is closed)

xki + xo + xke + xg <= 2 (Maximum 2 open warehouses)

xki + xg >= 1 (Assuming both Gatineau and Kingston can be open)

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