How to draw a network model?
How to draw a network model for a distributon problem?
A water company ships bottled water by truck from two factories to three cities. Information about the supply at each factory and the demand at each city is given below,in tonnes per day. It is possible to ship fractions of a tonne,that is,the manufacturer is not restricted to integer-valued shipping quantities.
Factory:
A
B
Supply (tonnes):
50
70
City:
1
2
3
Demand (tonnes):
30
60
40
Shipping Costs in dollars per tonne are given
Factory A:
TO City 1:7
TO City 2:8
TO City 3:6
Factory B:
to City 1:9
to City 2:3
to City 3:4
I was wondering how to draw the network model for this distribution problem.
I googled my question and nothing really came up,so if you can offer any help,thank you!!!
I am not sure how it may be solved as a network modeling problem. But I think you can definitely solve it as a linear programming (optimization) problem.
Let a1,a2,a3 be the respective amount shipped to Cities 1,2,3 from factory A and
b1,b2,b3 be the respective amount shipped to Cities 1,2,3
Then we want to minimize the total shipping costs,
i.e.,
7 a1 + 8 a2 + 6 a3 + 9 b1 + 3 b2 + 4 b3
subject to the constraints that
a1 + a2 + a3 <= 50 (the maximum that factory A can produce)
b1 + b2 + b3 <= 70 (the maximum that factory B can produce)
and
a1 + b1 >= 30 (enough to satisfy the demand of City 1)
a2 + b2 >= 60 (enough to satisfy the demand of City 2)
a3 + b3 >= 40 (enough to satisfy the demand of City 3)
we also need to make sure a1,a2,a3,b1,b2,b3 are non-negative also.
Solving this linear programming problem (you can use any software to solve it. Personally,I like cvx,a matlab optimization package by Stanford):
cvx_begin
variable a(3)
variable b(3)
minimize(7*a(1) + 8 *a(2) + 6 * a(3) + 9 * b(1) + 3 * b(2) + 4 * b(3));
subject to
a(1) + a(2) + a(3) <= 50;
b(1) + b(2) + b(3) <= 70;
a(1) + b(1) >= 30;
a(2) + b(2) >= 60;
a(3) + b(3) >= 40;
a >= 0;
b >= 0;
cvx_end
btw,I just realize that your problem is ill-posed. You total supply (50+70=120) is smaller than your total demand (30+60+40=130). I guess you may want to double check your question. Assuming that you can afford a larger supply (let say double). We have
cvx_begin
variable a(3)
variable b(3)
minimize(7*a(1) + 8 *a(2) + 6 * a(3) + 9 * b(1) + 3 * b(2) + 4 * b(3));
subject to
a(1) + a(2) + a(3) <= 100;% double the allowed supply
b(1) + b(2) + b(3) <= 140;
a(1) + b(1) >= 30;
a(2) + b(2) >= 60;
a(3) + b(3) >= 40;
a >= 0;
b >= 0;
cvx_end
This gives us a1=30,a2=0,a3=0 and b1=0,b2=60,b3=40. In other words,city 1 should get water from factor A and cities 2 and 3 should get water from factor B entirely.
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