Skip to content

6. Depot Location

Ardous

Description

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. 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. The two sorts of cost are quite different: set-up costs are capital costs which may usually be written off over several years, and transport costs are operating costs. A detailed discussion of how to combine these two costs is beyond the scope of this scenario โ€” we assume here that they have been put on some comparable basis, perhaps by taking the costs over a year.

There are 12 sites available for the construction of new depots and 12 sales centers need to receive deliveries from these depots.

The following Table 6.1 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 tonnes according to Table 6.3) from depot 1 is $60000/30t, i.e. $2000/t. Certain deliveries that are impossible are marked with the infinity symbol \(โˆž\).

Table 6.1: Delivery costs for satisfying entire demand of customers

Customer
Depot 1 2 3 4 5 6 7 8 9 10 11 12
1 100 80 50 50 60 100 120 90 60 70 65 110
2 120 90 60 70 65 110 140 110 80 80 75 130
3 140 110 80 80 75 130 160 125 100 100 80 150
4 160 125 100 100 80 150 190 150 130 โˆž โˆž โˆž
5 190 150 130 โˆž โˆž โˆž 200 180 150 โˆž โˆž โˆž
6 200 180 150 โˆž โˆž โˆž 100 80 50 50 60 100
7 100 80 50 50 60 100 120 90 60 70 65 110
8 120 90 60 70 65 110 140 110 80 80 75 130
9 140 110 80 80 75 130 160 125 100 100 80 150
10 160 125 100 100 80 150 190 150 130 โˆž โˆž โˆž
11 190 150 130 โˆž โˆž โˆž 200 180 150 โˆž โˆž โˆž
12 200 180 150 โˆž โˆž โˆž 100 80 50 50 60 100

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 6.2

Table 6.2: Fix costs and capacity limits of the depot locations

Depot 1 2 3 4 5 6 7 8 9 10 11 12
Cost(k$) 3500 9000 10000 4000 3000 9000 9000 3000 4000 10000 9000 3500
Capacity(t) 300 250 100 180 275 300 200 220 270 250 230 180

The quantities demanded by the sales centers (customers), are summarized in the following table.

Table 6.3: Demand data

Customer 1 2 3 4 5 6 7 8 9 10 11 12
Demand(t) 120 80 75 100 110 100 90 60 30 150 95 120

In every case, the demand of a customer needs to be satisfied but a sales center may be delivered to from several depots.

Which depots should be opened to minimize the total cost of construction and of delivery, whilst satisfying all demands?

๐Ÿš€ Test your code

Try to solve the problem below in the code editor before reading the solution.

Results

View Result 
Total Minimum Cost = $18103

Depot Transactions

flowchart TD
    D1["Depot 1"]
    D2["Depot 2"]
    D3["Depot 3"]
    D4["Depot 4"]
    D5["Depot 5"]
    D6["Depot 6"]
    D7["Depot 7"]
    D8["Depot 8"]
    D9["Depot 9"]
    D10["Depot 10"]
    D11["Depot 11"]
    D12["Depot 12"]
    C1["Customer 1"]
    C2["Customer 2"]
    C3["Customer 3"]
    C4["Customer 4"]
    C5["Customer 5"]
    C6["Customer 6"]
    C7["Customer 7"]
    C8["Customer 8"]
    C9["Customer 9"]
    C10["Customer 10"]
    C11["Customer 11"]
    C12["Customer 12"]

    D1 e1@==>|5.0 tons| C2
    e1@{ animate: true, animation: slow }
    D1 e2@==>|75.0 tons| C3
    e2@{ animate: true, animation: slow }
    D1 e3@==>|100.0 tons| C4
    e3@{ animate: true, animation: slow }
    D1 e4@==>|120.0 tons| C12
    e4@{ animate: true, animation: slow }
    D5 e5@==>|120.0 tons| C1
    e5@{ animate: true, animation: slow }
    D5 e6@==>|40.0 tons| C2
    e6@{ animate: true, animation: slow }
    D8 e7@==>|35.0 tons| C2
    e7@{ animate: true, animation: slow }
    D8 e8@==>|100.0 tons| C6
    e8@{ animate: true, animation: slow }
    D8 e9@==>|85.0 tons| C10
    e9@{ animate: true, animation: slow }
    D9 e10@==>|110.0 tons| C5
    e10@{ animate: true, animation: slow }
    D9 e11@==>|65.0 tons| C10
    e11@{ animate: true, animation: slow }
    D9 e12@==>|95.0 tons| C11
    e12@{ animate: true, animation: slow }
    D12 e13@==>|90.0 tons| C7
    e13@{ animate: true, animation: slow }
    D12 e14@==>|60.0 tons| C8
    e14@{ animate: true, animation: slow }
    D12 e15@==>|30.0 tons| C9
    e15@{ animate: true, animation: slow }
Press "Alt" / "Option" to enable Pan & Zoom

Answer ๐Ÿ”’

Akhil Singh Chauhan Akhil Singh Chauhan
Creator

Comments