Reference no: EM132376852
Business Optimisation Assignment -
Problem 1 - A supplier of construction material is starting up business in Eastern Europe, where a number of production plants will be built.
The supply chain director of the company needs to come up with a suggested supply network design, including the number of plants to build, as well as the locations of these plants.
Seven possible locations for the plants are identified. These are Bratislava, Lvov, Krakow, Sibiu, Kharkiv, Pleven, and Nizhniy. Not all sites need to be used.
The market is divided into 12 different regions. We name these regions from A to L. The estimated yearly demand (in 1000 tons per year) for these market regions are given in the following table:
Region
|
A
|
B
|
C
|
D
|
E
|
F
|
G
|
H
|
I
|
J
|
K
|
L
|
|
200
|
150
|
225
|
120
|
155
|
180
|
170
|
230
|
230
|
190
|
265
|
300
|
The potential production capacities for the different production sites are restricted by the local availability of raw materials and are as follows (in 1000 tons per year):
|
Bratislava
|
370
|
|
Lvov
|
200
|
|
Krakow
|
300
|
|
Sibiu
|
670
|
|
Kharkiv
|
750
|
|
Pleven
|
450
|
|
Nizhniy
|
300
|
The variable cost (cost per ton) is the sum of raw material, variable production cost, and transportation cost from a given production plant to a given market region.
The variable costs (in EUR per ton):
|
From \ to
|
Region
|
|
A
|
B
|
C
|
D
|
E
|
F
|
G
|
H
|
I
|
J
|
K
|
L
|
|
Bratislava
|
52
|
49
|
49
|
39
|
28
|
30
|
22
|
20
|
27
|
14
|
16
|
18
|
|
Lvov
|
48
|
37
|
38
|
27
|
24
|
28
|
24
|
14
|
29
|
13
|
29
|
25
|
|
Krakow
|
34
|
21
|
20
|
29
|
21
|
20
|
28
|
11
|
15
|
14
|
20
|
13
|
|
Sibiu
|
30
|
15
|
20
|
14
|
12
|
22
|
13
|
20
|
10
|
27
|
11
|
20
|
|
Kharkiv
|
27
|
15
|
15
|
29
|
23
|
11
|
21
|
13
|
29
|
47
|
20
|
47
|
|
Pleven
|
16
|
19
|
21
|
20
|
21
|
19
|
25
|
13
|
37
|
47
|
55
|
59
|
|
Nizhniy
|
19
|
30
|
19
|
10
|
11
|
25
|
28
|
30
|
47
|
50
|
66
|
63
|
a) Create an AMPL model that can be used as a tool to find the cost optimal supply network design for the company. How many factories should be built, at which locations, and how much should be supplied from each factory to each market?
b) Now assume that at each factory, there is also a "fixed" cost that does not depend on the factory's production volume.
The fixed costs are given in the following table (mill. EUR per year):
|
Bratislava
|
16
|
|
Lvov
|
13
|
|
Krakow
|
20
|
|
Sibiu
|
12
|
|
Kharkiv
|
18
|
|
Pleven
|
15
|
|
Nizhniy
|
17
|
The above costs are incurred only if the corresponding factory is built.
Modify your AMPL model to take into account the fixed costs and find the cost optimal network design. Try to avoid non-linearities in your model.
How many factories should be built, at which locations, and how much should be supplied from each factory to each market?
c) Modify your AMPL model to take into account the following "single-sourcing" restriction: Each market should be supplied by only one production plant. Try to avoid non-linearities in your model.
How many factories should be built, at which locations, and how much should be supplied from each factory to each market? What is the additional cost of imposing such a "single-sourcing" policy?
d) Modify your AMPL model from c) so that it allows a solution in which one of the markets is supplied by multiple production plants. Try to avoid non-linearities in your model.
How many factories should now be built, at which locations, and how much should be supplied from each factory to each market? How much are the total costs reduced compared to the solution in c)?
Problem 2 - A company is preparing the introduction of a new product and wants to develop an optimisation model that can help determine the best choice of online advertising channels.
The total market has been divided into 14 market segments and there are 20 different advertising channels available to reach the various segments. Advertising cost per channel is shown in the following table:
|
Costs
|
|
Ch. 1
|
12
|
|
Ch. 2
|
45
|
|
Ch. 3
|
86
|
|
Ch. 4
|
56
|
|
Ch. 5
|
72
|
|
Ch. 6
|
73
|
|
Ch. 7
|
82
|
|
Ch. 8
|
40
|
|
Ch. 9
|
25
|
|
Ch. 10
|
53
|
|
Ch. 11
|
28
|
|
Ch. 12
|
27
|
|
Ch. 13
|
53
|
|
Ch. 14
|
77
|
|
Ch. 15
|
64
|
|
Ch. 16
|
33
|
|
Ch. 17
|
75
|
|
Ch. 18
|
84
|
|
Ch. 19
|
75
|
|
Ch. 20
|
83
|
The following table shows which channels can reach which segments of the market (1 means that the channel reaches the market):
|
S. 1
|
S. 2
|
S. 3
|
S. 4
|
S. 5
|
S. 6
|
S. 7
|
S. 8
|
S. 9
|
S. 10
|
S. 11
|
S. 12
|
S. 13
|
S. 14
|
|
Ch. 1
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
|
Ch. 2
|
1
|
0
|
0
|
0
|
0
|
1
|
0
|
1
|
0
|
0
|
0
|
1
|
0
|
0
|
|
Ch. 3
|
0
|
0
|
0
|
0
|
1
|
0
|
0
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
|
Ch. 4
|
0
|
0
|
0
|
0
|
0
|
1
|
1
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
|
Ch. 5
|
0
|
0
|
1
|
1
|
1
|
0
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
|
Ch. 6
|
1
|
0
|
0
|
0
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
|
Ch. 7
|
1
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
0
|
0
|
0
|
|
Ch. 8
|
0
|
0
|
0
|
1
|
0
|
1
|
0
|
0
|
0
|
0
|
0
|
1
|
0
|
1
|
|
Ch. 9
|
0
|
0
|
0
|
0
|
1
|
1
|
0
|
0
|
0
|
0
|
1
|
1
|
0
|
0
|
|
Ch. 10
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
1
|
0
|
0
|
1
|
0
|
0
|
|
Ch. 11
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
0
|
|
Ch. 12
|
0
|
0
|
0
|
0
|
0
|
1
|
1
|
1
|
0
|
1
|
0
|
0
|
1
|
0
|
|
Ch. 13
|
0
|
0
|
1
|
0
|
0
|
1
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
|
Ch. 14
|
0
|
0
|
1
|
0
|
0
|
0
|
1
|
0
|
0
|
0
|
1
|
0
|
0
|
0
|
|
Ch. 15
|
0
|
1
|
0
|
0
|
0
|
0
|
0
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
|
Ch. 16
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
1
|
0
|
0
|
|
Ch. 17
|
0
|
1
|
0
|
0
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
|
Ch. 18
|
0
|
0
|
0
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
0
|
|
Ch. 19
|
0
|
0
|
0
|
0
|
0
|
0
|
1
|
1
|
0
|
0
|
0
|
0
|
0
|
1
|
|
Ch. 20
|
0
|
1
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
0
|
In the following, try to avoid non-linearities in your models.
a) Create an AMPL model that chooses the optimal mix of advertising channels in such a way that total costs are minimized and all market segments are covered by at least one advertising channel.
b) The company believes that the effects of channel 7 and 8 reinforce each other so that if one of these channels is chosen than the other channel must also be chosen. In other words, we want to forbid solutions where channel 7 is chosen and channel 8 is not chosen, and vice versa. How much does this requirement increase the total costs compared with the solution in a)?
c) Disregard the information given in b), but assume the following:
If at least one of channel 11 and channel 12 is chosen, then at least one of channel 16 and channel 17 must be chosen.
How much does this requirement increase the total costs compared with the solution in a)?
d) Disregard the information given in b) and c), but assume the following: If both channel 12 and channel 13 are chosen, then also channel 18 must be chosen. How much does this requirement increase the total costs compared with the solution in a)?
e) In the solution in a), some of the market segment are covered more than once (that is, they are covered by more than one channel). Based on the model in a), add constraints that ensure that at least six market segments are covered more than once.
f) The following table shows expected revenue per segment.
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
13
|
14
|
|
571
|
384
|
773
|
844
|
934
|
113
|
313
|
190
|
629
|
965
|
924
|
505
|
973
|
779
|
Assume that the revenue per segment is achieved if the segment is covered at least once.
Assume that there are not multiple revenues per segment.
Modify the model in a) so that total revenues are maximized, given a total advertising budget of 200.
Problem 3 - In maritime logistics, the ability to increase the average size of ships (vessels) is advantageous both of economic and environmental reasons. Larger ships have both a lower cost and a lower CO2 emission per ton transported, as long as the cargo capacity is fully utilized. The ship/cargo size is however often restricted by limited storage capacities in ports, as in the following case.
A port terminal contains 32 storage tanks of varying sizes. 20 different products are stored in the tanks. The products have different densities (in tons per m3) and demands, as shown in the following table:
|
|
Density
|
Demand
|
|
Product 01
|
1.46
|
2633
|
|
Product 02
|
1.49
|
2355
|
|
Product 03
|
1.27
|
1035
|
|
Product 04
|
1.45
|
2538
|
|
Product 05
|
1.66
|
2981
|
|
Product 06
|
1.79
|
437
|
|
Product 07
|
1.28
|
459
|
|
Product 08
|
1.72
|
1479
|
|
Product 09
|
1.90
|
1810
|
|
Product 10
|
1.62
|
774
|
|
Product 11
|
1.82
|
1760
|
|
Product 12
|
1.48
|
148
|
|
Product 13
|
1.30
|
1864
|
|
Product 14
|
1.40
|
1344
|
|
Product 15
|
1.90
|
792
|
|
Product 16
|
1.77
|
728
|
|
Product 17
|
1.87
|
552
|
|
Product 18
|
1.54
|
144
|
|
Product 19
|
1.22
|
1063
|
|
Product 20
|
.50
|
672
|
For each product, the maximum time between replenishments (deliveries by ship) is determined by the products's storage capacity and its daily demand.
The current allocation is as follows:
|
Tank
|
Volum (m3)
|
Product
|
|
Tank 01
|
2500
|
Product 13
|
|
Tank 02
|
3800
|
Product 15
|
|
Tank 03
|
3400
|
Product 8
|
|
Tank 04
|
1600
|
Product 18
|
|
Tank 05
|
1800
|
Product 3
|
|
Tank 06
|
6600
|
Product 2
|
|
Tank 07
|
5200
|
Product 1
|
|
Tank 08
|
2200
|
Product 3
|
|
Tank 09
|
4000
|
Product 7
|
|
Tank 10
|
4300
|
Product 1
|
|
Tank 11
|
5000
|
Product 4
|
|
Tank 12
|
4000
|
Product 11
|
|
Tank 13
|
2500
|
Product 9
|
|
Tank 14
|
2100
|
Product 17
|
|
Tank 15
|
4200
|
Product 19
|
|
Tank 16
|
5600
|
Product 16
|
|
Tank 17
|
5500
|
Product 14
|
|
Tank 18
|
5800
|
Product 12
|
|
Tank 19
|
7000
|
Product 10
|
|
Tank 20
|
7700
|
Product 14
|
|
Tank 21
|
3400
|
Product 8
|
|
Tank 22
|
5200
|
Product 5
|
|
Tank 23
|
4600
|
Product 13
|
|
Tank 24
|
6600
|
Product 4
|
|
Tank 25
|
7000
|
Product 5
|
|
Tank 26
|
5600
|
Product 20
|
|
Tank 27
|
3000
|
Product 19
|
|
Tank 28
|
2600
|
Product 16
|
|
Tank 29
|
2600
|
Product 13
|
|
Tank 30
|
4800
|
Product 9
|
|
Tank 31
|
6700
|
Product 6
|
|
Tank 32
|
1200
|
Product 3
|
In the current setup, product 4 has been allocated Tank 11 (5000 m3) and Tank 24 (6600 m3), which gives a total capacity of 11600 m3.
Since Product 4 has a density of 1,45 tons per m3 its capacity in tons is 16820 tons.
Product 4 has a daily demand of 2538 tons, so that 16820 tons covers 16820 / 2538 = 6,63 days.
Hence, there needs to be a transport of Product 4 at least every 6,63 days on average. The cycle time for Product 4 is said to be 6,63 days.
The bottleneck product in the current setup is Product 11, which has been allocated Tank 12. Capacity in tons = 4000 × 1,82 = 7280 tons. Demand for Product 11 is 1760 tons per day. Hence, Product 11 must be replenished at least every 7280 / 1760 = 4,14 days on average. The cycle time for Product 11 is 4,14 days. Since this is the lowest cycle time for all products.
It determines the maximum time between deliveries for the whole terminal. In this case, the terminal needs a replenishment on average every 4,14 days. This limits the total quantity that can be delivered and hence the maximum vessel size that can be used efficiently.
To minimize costs and emissions we want to maximize the time between replenishments, that is, maximize the minimum cycle time.
In the following, try to avoid non-linearities in your models.
a) Assume first that that the company pays a fixed fee per storage tank used (independent of tank size) and hence wants to simply minimize the number of storage tanks used, not explicitly taking into account demand or trying to calculate cycle times. Assume just that each product should have a minimum of 6500 tons storage capacity. Formulate an AMPL model and find the solution that minimizes the number of tanks used. How many tanks are needed?
b) Now consider the situation explained in the introduction. Assume that all tanks are used. The only goal is to maximize the time between deliveries to the terminal, that is, maximize the minimum cycle time (the cycle time for the bottleneck product). The optimal solution to this problem will help utilizing large ships with low costs and low emissions per ton transported. Formulate and solve an AMPL model to find the optimal solution. Which product is the bottleneck product in your solution and what is the minimum cycle time?
Attachment:- Business Optimisation Assignment Files.rar