A company is considering opening warehouses in four cities: Ottawa, Kingston, Ke
ID: 367467 • Letter: A
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
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Yi = 1 when the warehouse i must be opened
Yi = 0 when the warehouse i must NOT be opened
X ij = Quantity of the items to be shipped from ware house i to region j
Costs involved:
F i = Fixed cost of maintain the ware house i when kept opened
C i j = cost per unit for the ware house i + the shipping cost from warehouse i to region j
Constants engaged:
D j = demand of region j that is supposed to be supplied from the warehouses
The generalized moel is:
Minimize Sigma i = 1 to m Sigma j = 1 to n C i j * X i j + Sigma i = 1 to m F i * Y i
Subject to constraints:
Sigma i = 1 to m X i j = D j for j from 1 to n
Sigma j = 1 to n X i j - Y i * (Sigma j = 1 to n D j ) <= 0 for i from 1 to m
X i j >= 0 for i from 1 to m and j from 1 to n
Y i = either zero or one for i ranging from 1 to m
Now we can apply the above model to the data in our question:
C i j
Ware House
Region 1
R2
R3
Fi $
Ottawa
20
40
50
400
Kingston
48
15
26
500
Kemptville
26
35
18
300
Gatineau
24
50
35
150
Weekly Demand
80
70
40
If y1 = 1 then y2 must be = 1
Y1, Y2, Y3, and Y4 must have value of 1 for at most 2
Y4 = 1 or Y2 = 1
m = 4, n = 3
Minimize cost
Minimize
Z = C11 * X 11 + C 12 * X 12 + C13 X 13
+ C 21 X 21 + C 22 X 22 + C 23 X 23
+ C 31 X 31 + C 32 X 32 + C 33 X 33
+ C 41 X 41 + C 42 X 42 + C 43 X 43
+ F1Y1 + F2Y2 + F3Y3 + F4 Y4
F1Y1 <= 400
F2Y2 <= 500
F3Y3 <= 300
F4Y4 <= 150
Subject to constraints:
D1 = demand by region 1, D2 = demand by region 2, D3 = demand by region 3;
X 11 + X 21 + X 31 + X 41 = D 1 = 80
X 12 + X 22 + X 32 + X 42 = D 2 = 70
X 11 + X 21 + X 31 + X 41 = D 3 = 40
and
X 11 - Y 1 *( D1 + D2 + D3 ) <= 0
X 21 - Y 2 *( D1 + D2 + D3 ) <= 0
X 31 - Y 3 *( D1 + D2 + D3 ) <= 0
X 41 - Y 4 *( D1 + D2 + D3 ) <= 0
D1 + D2 + D3 = 80 + 70 + 40 = 190
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C i j
Ware House
Region 1
R2
R3
Fi $
Ottawa
20
40
50
400
Kingston
48
15
26
500
Kemptville
26
35
18
300
Gatineau
24
50
35
150
Weekly Demand
80
70
40
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