Network Design in the Supply Chain
Exercise Solutions
1. SC consulting, a supply chain consulting consulting firm, has to decide on the the loc locati ation of its ho home me off offices. ces. The heir ir clie client nts s are are primarily located in the 16 states as given in the following table. There are four potential sites for home offices: Los Angeles, Tulsa, Tulsa, Denver, Denver, and Seattle. Seattle. The annual fixed cost cost of locating an office in Los Angeles is $165,428, Tulsa is $131,230, Denver is $140,000, and Seattle is $ 145,000. The expected number of trips to each state and the travel costs from each potential site are shown in table . Table: Travel cost and number of trips for Sc consulting
State
Tota l # of trips
Cost from LA
Cost from Tulsa
15
Washington
40
0
2 50
15
Oregon California
35 10 0
0
25
75
Nevada
40
0
Montana
25
5
25
00
0
1
50
00
50
75
00 15
Colorado
65
0
40
5
North Dakota
30
0
00
25
30
0
Kansas
40
0
25
50
75
00
3 1 50
1 75
25
Nebraska
2 50 2
2 00
0
1
1
30 20
2 00
00
25
30
South Dakota
50
1
12
New Mexico
1
1
2
Arizona
1 25
1
1
0
1 50
1
1
30
25
25
75
1
1 25
75
1 25
1
2
15
Utah
50
00
15
75 1
2
17
50
00
00
0
25 2
2
10
Wyoming
00
50
Cost from Seattle
2
2
15
Idaho
Cost From Denver
2 00
1 25
1
2 00
1
2
00
25
50
75
75
00
25
3
25
Oklahoma
55
0
1 25
25
3 00
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Each consultant is expected to take at most 25 trips each year. (a) if there are no restriction on the number of consultants at a site and the goal is to minimize costs, where should the home office be located and how many consultants should be assigned to each office? What is the annual cost in terms of the facility and travel? Solution The objective of this model is to decide optimal locations of home offices, and number of trips from each home office, so as to minimize the overall network cost. The overall network cost is a combination of fixed costs of setting up home offices and the total trip costs. There are two constraint sets in the model. The first constraint set requires that a specified number of trips be completed to each state j and the second constraint set prevents trips from a home office i unless it is open. Also, note that there is no capacity restriction at each of the home offices. While a feasible solution can be achieved by locating a single home office for all trips to all states, it is easy to see that this might not save on trip costs, since trip rates vary between home offices and states. We need to identify better ways to plan trips from different home offices to different states so that the trip costs are at a minimum. Thus, we need an optimization model to handle this.
Optimization model: n m D j K i f i cij yi xij
= 4: possible home office locations. = 16: number of states. = Annual trips needed to state j = number of trips that can be handled from a home office As explained, in this model there is no restriction = Annualized fixed cost of setting up a home office = Cost of a trip from home office i to state j = 1 if home office i is open, 0 otherwise = Number of trips from home office i to state j. It should be integral and nonnegative n
Min
∑
n
m
f i yi + ∑∑ cij xij
i =1
i =1 j =1
Subject to n
∑ xij = D j
for j = 1,...,m
(5.1)
i =1 m
∑x ij ≤K iy i for i
= 1,...,n (5.2)
j=1
yi ∈ {0,1} for i = 1,...n
(5.3)
Please note that (5.2) is not active in this model since K is as large as needed. However, it will be used in answering (b).
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SYMBOL
INPUT
D j
Annual trips needed to state j
cij
Transportation cost from office i to state j
f i xij
fixed cost of setting up office i number of consultants from office i to state j.
obj. objective function 5.1 demand constraints (Sheet SC consulting in workbook exercise5.1.xls)
CELL E7:E22 G7:G22,I7:I22, K7:K22,M7:M22 G26,I26,K26,M26 F7:F22,H7:H22, J7:J22,L7:L22 M31 N7:N22
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With this we solve the model to obtain the following results:
State
Tota l # of trips
Trip s from LA
Cost from LA
40

0
Trip s from Tulsa
Cost from Tulsa
15
Washington

California
35 10 0
Idaho
50

50
0
75

00
25

0

00
Nevada
40
0
0

00
Montana
25

5

75
Wyoming
50

0

75
Utah
30

0

50
50
0
75

00
Colorado
65

0

25
New Mexico
40

5

25
North Dakota
30

0

00
South Dakota
20
0
0

75
Nebraska
30

0
Kansas
40

0
Oklahoma
55 67 5

0
10
4
10
50

25

25

25 5
0
1 2
1
5
25

50
1 2
1
5
25

50

00
1 00
3
1
1
2
00

00

50
25

50
75

00

00

00
1
2
6 5
2
4 0
2
3
3 0
1
1 50
2
2
1
2
0
25
00

25

50
75
5
75

00
5
25

25

00

10

50

25
25
3 0
25
1
1
2 5
25
2
1
3
5
1
19 0
25
0
1
30

1
1
30
75
1
2
12
5
1
1
15
25 3
1
1
5
Total Office Cost

1
15
Cost of Trips
00
2
15
# of Consultants Fixed Cost of office

Cost from Seattle
4 0
2
2
17
# of trips
00
2 15
Trips from Seattle
2 
2
0
Arizona
Cost From Denver
2 
15
Oregon
Trips from Denver
1
3
2
1
8
5
10
5
165,42 8 15,25 0 180,67 8
131,23 0 6,2 50 137,48 0
140,00 0 20,75 0 160,75 0
145,00 0 9,8 75 154,87 5
The number of consultants is calculated based on the constraint of 25 trips per consultant. As trips to Kansas cost the same from Tulsa or Denver there are many other solutions possible by distributing the trips to Kansas between these two offices.
The task : Solution to the problem is given to you. You have to do it with Solver & Excel. Explain each step in the class. I.
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