A large company wishes to open new depots to deliver to its sales centers. Every new...

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A large company wishes to open new depots to deliver to its sales centers. Every new set-up of a depot has a fixed cost (one time construction cost of the depot). Goods are delivered from a depot to the sales centers close to the site. Every delivery has a cost that depends on the distance covered. There are 9 sites available for the construction of new depots and 9 sales centers need to receive deliveries from these depots. Table 1 shows total demand of each customer in tons, and Table 2 gives the costs (in thousand $) of satisfying the entire demand of each customer (sales center) from a depot (not the unit costs). So, for instance, the cost per unit of supplying customer 9 (who has a total demand of 30 tons according to Table 1) from depot 1 is $60000/30ton, i.e. $2000/ton. Certain deliveries that are impossible are marked with the infinity symbol oo. Table 1: Demand data. Customer 1 2 3 4 5 6 7 8 9 Demand (ton) 120 80 75 100 110 100 90 60 30 Table 2: Delivery costs for satisfying entire demand of customers. Customers 4 5 6 7 8 9 50 120 90 60 3 100 80 50 120 90 60 110 80 125 100 100 70 65 140 110 80 140 80 75 160 125 100 190 150 130 60 100 110 130 80 150 00 00 ∞ 60 100 120 90 60 70 65 110 140 110 80 150 130 ∞ 200 180 150 100 80 50 50 180 150 ∞ 80 50 90 60 140 110 80 100 120 80 75 130 160 125 100 Depot 1 2 1 2 3 4 5 6 7 8 9 160 190 200 In addition, for every depot we have the following information: the fixed cost for constructing the depot that needs to be included into the objective function and its capacity limit, all listed in Table 3. Table 3: Fix costs and capacity limits of the depot locations. 4 3 7 8 9 2 9000 10000 250 100 4000 3000 9000 9000 3000 4000 180 275 300 200 220 270 Depot Cost (in thousand $) 3500 Capacity (ton) 300 In every case, the demand of a customer needs to be satisfied but a sales center may be delivered to from several depots. (1) Which depots should be opened and how should we assign customer demands to the opened depots to minimize the total cost of construction and of delivery, whilst satisfying all demands? (your models must be algebraic). A large company wishes to open new depots to deliver to its sales centers. Every new set-up of a depot has a fixed cost (one time construction cost of the depot). Goods are delivered from a depot to the sales centers close to the site. Every delivery has a cost that depends on the distance covered. There are 9 sites available for the construction of new depots and 9 sales centers need to receive deliveries from these depots. Table 1 shows total demand of each customer in tons, and Table 2 gives the costs (in thousand $) of satisfying the entire demand of each customer (sales center) from a depot (not the unit costs). So, for instance, the cost per unit of supplying customer 9 (who has a total demand of 30 tons according to Table 1) from depot 1 is $60000/30ton, i.e. $2000/ton. Certain deliveries that are impossible are marked with the infinity symbol oo. Table 1: Demand data. Customer 1 2 3 4 5 6 7 8 9 Demand (ton) 120 80 75 100 110 100 90 60 30 Table 2: Delivery costs for satisfying entire demand of customers. Customers 4 5 6 7 8 9 50 120 90 60 3 100 80 50 120 90 60 110 80 125 100 100 70 65 140 110 80 140 80 75 160 125 100 190 150 130 60 100 110 130 80 150 00 00 ∞ 60 100 120 90 60 70 65 110 140 110 80 150 130 ∞ 200 180 150 100 80 50 50 180 150 ∞ 80 50 90 60 140 110 80 100 120 80 75 130 160 125 100 Depot 1 2 1 2 3 4 5 6 7 8 9 160 190 200 In addition, for every depot we have the following information: the fixed cost for constructing the depot that needs to be included into the objective function and its capacity limit, all listed in Table 3. Table 3: Fix costs and capacity limits of the depot locations. 4 3 7 8 9 2 9000 10000 250 100 4000 3000 9000 9000 3000 4000 180 275 300 200 220 270 Depot Cost (in thousand $) 3500 Capacity (ton) 300 In every case, the demand of a customer needs to be satisfied but a sales center may be delivered to from several depots. (1) Which depots should be opened and how should we assign customer demands to the opened depots to minimize the total cost of construction and of delivery, whilst satisfying all demands? (your models must be algebraic).

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