CASE 8.1 Shipping Wood to Market
Alabama Atlantic is a lumber company that has three sources of wood and five markets to be
supplied. The annual availability of wood at sources 1, 2, and 3 is 15, 20, and 15 million board
feet, respectively. The amount that can be sold annually at markets 1, 2, 3, 4, and 5 is 11, 12, 9,
10, and 8 million board feet, respectively.
In the past the company has shipped the wood by train. However, because shipping costs have
been increasing, the alternative of using ships to make some of the deliveries is being investigated.
This alternative would require the company to invest in some ships. Except for these investment
costs, the shipping costs in thousands of dollars per million board feet by rail and by water (when
feasible) would be the following for each route:
Unit cost by Rail Market ($1,000s)
Unit Cost by Ship Market ($1,000’s)
Sources
1
2
3
4
5
1
2
3
4
5
1
61
72
45
55
66
31
38
24
0
35
2
69
78
60
49
56
36
43
28
24
31
3
59
66
63
61
47
0
33
36
32
26
The capital investment (in thousands of dollars) in ships required for each million board feet to be
transported annually by ship along each route is given as follows:
Investment For Ships
Sources 1
2
3
4
5
1
275
303
238
0
285
2
293
318
270
250
265
3
0
283
275
268
240
Considering the expected useful life of the ships and the time value of money, the equivalent
uniform annual cost of these investments is one-tenth the amount given in the table. The objective
is to determine the overall shipping plan that minimizes the total equivalent uniform annual cost
(including shipping costs).
You are the head of the OR team that has been assigned the task of determining this shipping plan
for each of the following three options.
Option 1: Continue shipping exclusively by rail.
Option 2: Switch to shipping exclusively by water (except where only rail is feasible).
Option 3: Ship by either rail or water, depending on which is less expensive for the particular route.
Present your results for each option. Compare.
Finally, consider the fact that these results are based on current shipping and investment costs, so
the decision on the option to adopt now should take into account management’s projection of how
these costs are likely to change in the future. For each option, describe a scenario of future cost
changes that would justify adopting that option now.
Solution:
Introduction:
The Given Case of “Shipping wood to Market” is a Transportation problem. Where, Alabama
Atlantic, a lumber company which has three sources of wood and five markets to be supplied. The
annual availability of wood at sources 1, 2, and 3 is 15, 20, and 15 million board feet, respectively.
Whereas, the amount that can be sold annually at markets 1, 2, 3, 4, and 5 is 11, 12, 9, 10, and 8
million board feet, respectively.
The company so far has used Rails to supply woods. However due to increase in shipping cost,
Atlantic is looking for alternatives to minimize cost, requiring company to invest in ships.
As a head of OR Team, we will be evaluating the independent cost of shipping exclusively through
rail, cost of switching to shipment, and cost of making use of rail or water transportation,
depending on which is less expensive for the particular route.
For Minimizing cost of shipment, VAM model has been used in evaluating three different
alternatives. All the alternatives have been processed using QM for Windows V4 software.
Option 1:
Objective: Minimum cost of shipping Exclusively by Train.
Initial Problem:
Unit cost by Rail Market ($1,000s)
Sources
1
2
3
4
5
Supply
1
61
72
45
55
66
15000
2
69
78
60
49
56
20000
3
59
66
63
61
47
15000
Requirement
11000
12000
9000
10000
8000
50000
There are 3 different sources of wood and 5 different markets. The annual availability of wood at
sources 1, 2, and 3 is 15, 20, and 15 million board feet, respectively. Whereas, the amount that can
be sold annually at markets 1, 2, 3, 4, and 5 is 11, 12, 9, 10, and 8 million board feet, respectively.
All the cost has been presented in thousands.
Initial Solution:
Initial Solution calculated using VAM model suggested that, the cost of shipment from, railways
is 2.834 Billion.
Cost =2834000
Source 1
Source 2
Source 3
Destination Destination Destination Destination Destination
1
2
3
4
5
6000
-2
9000
-14
-15
5000
5000
-7
10000 ( -3 )
-2
7000
-22
-24
8000
However it is not an optimal solution. Hence further iteration is required,
Iterations:
Iteration 1
Source 1
Source 2
Destination
1
Destination
2
Destination
3
6000
5000
-2
5000
9000
-7
Destination
4
Destination
5
-14
10000 (-3)
-15
Source 3
Iteration 2
Source 1
Source 2
Source 3
Iteration 3
Source 1
Source 2
Source 3
-2
7000
-22
-24
8000
6000
5000
-5
-3
12000
9000
-7
-19
-14
10000
-21
-18
5000
3000
6000
2000
3000
-4
-2
12000
9000
-7
-20
-14
10000
-22
-18
8000
-1
(-1)
Minimum cost of shipment from the railways have been calculated by using QM for Windows V4
software and VAM model. It requires 3 Iteration to achieve the optimal solution. After 3 Iterations
we receive our optimal solution.
Optimal Solution:
Optimal solution value =
$2816000
Source 1
Source 2
Source 3
Destination Destination Destination Destination Destination
1
2
3
4
5
6000
9000
2000
10000
8000
3000
12000
The Optimum cost of shipping goods from Railways is $2.816 Billion as the cost can no longer be
minimized any further.
Shipping List:
From
Source 1
Source 2
To
Destination
1
Destination
3
Destination
1
Destination
4
Destination
5
Shipment
Cost per
unit
Shipment
cost
6000
61
366000
9000
45
405000
2000
69
138000
10000
49
490000
8000
56
448000
Destination
1
Destination
2
Source 3
3000
59
177000
12000
66
792000
As shown in the above figure, the optimum cost of shipping from railways is $2.816 Billion. This
is obtained by allocating 6000 units of woods to Destination 1 and 9000 units to Destination 3
from source 1. Similarly, 2000 units of wood has been Transported to Destination 1, 10000 units
to Destination 4 and 8000 units to Destination 5 from Source 2. Further on 3000 units of wood has
been shipped to Destination 1 and 12000 units of wood to Destination 2 from source 3.
Justification of Using Railways:
1. Railway is best means of shipment for land locked location,
2. No further investment is necessary as required infrastructure has already been developed.
3.
Option: 2
Objective: To determine the overall shipping plan that minimizes the total equivalent uniform
annual cost, including shipping costs.
In the past the company has shipped the wood by train. However, because shipping costs have
been increasing, the alternative of using ships to make some of the deliveries is being investigated.
This alternative would require the company to invest in some ships. The capital investment (in
thousands of dollars) in ships required for each million board feet to be transported annually by
ship along each route is given as follows:
Sources
1
2
3
1
275
293
0
Investment For Ships in 1000’s
2
3
4
303
238
0
318
270
250
283
275
268
5
285
265
240
However, after considering the expected useful life of the ships and the time value of money, the
equivalent uniform annual cost of these investments turns out to be one-tenth the amount given in
the table.
Sources
1
2
3
Investment For Ships per year
1
2
3
27.5
30.3
23.8
29.3
31.8
27
0
28.3
27.5
4
0
25
26.8
5
28.5
26.5
24
Fig: Per year cost of investment
Initial Problem: Total Shipping Cost
Unit Cost by Ship Market ($1,000’s)
Sources
1
2
3
4
5 Supply
1
58.5
68.3
47.8
55
63.5
15000
2
65.3
74.8
55
49
57.5
20000
3
59
61.3
63.5
58.8
50
15000
Requirement
11000
12000
9000
10000
8000
50000
Fig: Shipment cost after adding Initial Cash Outlay
While calculating Total Shipping cost, Unit cost by ship to market has been added with Initial cash
outlay after considering the time value of money. However, shipping goods to all market is not
completely possible through sea ways. Hence, Railways have to be considered for markets where
goods cannot be shipped through seas. Hence, Railways cost of transportation has been considered
at places where we cannot ship.
In this case, it is not possible to ship goods from source 3 to destination 1 and from source 1 to
destination 4. This route has been considered as not feasible zone of shipment from sea ways and
hence railways must be used to reach those destinations. Hence, while calculating optimum cost
of transportation through seas cost of railways have been used for those places where shipping
through seas is not feasible.
Initial Solution:
Cost
=2,800,800
Source 1
Source 2
Source 3
Destination Destination Destination Destination Destination
1
2
3
4
5
6000
-0.3
9000
-12.8
-6.8
5000
5000
-0.4
10000 ( -6 )
-7.2
7000
-22.4
-23.3
8000
The initial cost of shipping woods from sea ways calculated using VAM model is $2.8 Billion.
However it is not an optimal solution. Hence further iteration is required,
Iterations:
Iteration 1
Source 1
Source 2
Source 3
Iteration 2
Source 1
Source 2
Source 3
Destination
1
Destination
2
Destination
3
Destination
Destination 4 5
6000
5000
-7.2
-0.3
5000
7000
9000
-0.4
-22.4
-12.8
10000
-23.3
-6.8
-6
8000
6000
5000
-1.2
-6.3
-6
12000
9000
-0.4
-16.4
-12.8
10000
-17
-12.8
5000
3000
Minimum cost of shipment from the seaways have been calculated by using QM for Windows V4
software and VAM model. It requires 2 Iteration to achieve the optimal solution. After 2 Iterations
we receive our optimal solution.
Optimum Solution:
Optimal solution value =
$2770800
Source 1
Source 2
Source 3
Destination Destination Destination Destination
Destination
1
2
3
4
5
6000
9000
5000
10000
5000
12000
3000
The Optimum cost of shipping goods from Railways is $2.77 Billion as the cost can no longer be
minimized any further.
Shipping List:
From
Source 1
Source 2
Source 3
To
Destination
1
Destination
3
Destination
1
Destination
4
Destination
5
Destination
2
Destination
5
Shipment
Cost per
unit
Shipment
cost
6000
58.5
351000
9000
47.8
430200
5000
65.3
326500
10000
49
490000
5000
57.5
287500
12000
61.3
735600
3000
50
150000
As shown in the above figure, the optimum cost of shipping from railways is $2.77 Billion. This
is obtained by allocating 6000 units of wood to destination 1 and 9000 units of wood to destination
3 from source 1. Similarly, 5000 units of wood is supplied to destination 1, 10000 units of wood
to destination 4 and 5000 units of wood to destination 5 from source 2. At last, source 3 is used to
supply 12000 units of woods to destination 2 and 3000 units of wood to destination 5.
Justification of Using Waterways:
1. Even after adding present value of initial cost outlay to shipping cost, Optimum cost of
shipping through seaways is lower than that of Railways. Hence, waterways must be used.
2. After 10 years, the initial investment should no longer be added; hence the cost of shipment
will further decrease.
3.
Option: 3
Objective: Minimize Total cost, by selecting alternative that is less expensive for the particular
route from either Railways or Seaways.
Initial Problem:
The objective of this transportation problem is to minimize Total cost of Transportation by using
both the means of Shipping. Hence, by comparing shipment cost of delivering goods from various
sources to various destinations, through both means, we have selected those cost, that is minimum.
Unit Cost by means where cost is minimum ($1,000’s)
1
2
3
4
5 Supply
58.5(S)
68.3(S)
45(R)
55(R)
63.5(S)
15000
65.3(S)
74.8(S)
55(S)
49(S)
56(S)
20000
59(R)
61.3(S)
63(R)
58.8(S)
47(R)
15000
Sources
1
2
3
Requiremen
t
11000
12000
9000
10000
8000
50000
Initial Solution:
Cost
=2751600
Source 1
Source 2
Source 3
Destination Destination Destination Destination Destination
1
2
3
4
5
6000
-0.3
9000
-12.8
-9.8
5000
5000
-3.2
10000
-4.5
-7.2
7000
-24.7
-23.3
8000
The initial cost of shipping woods from both ways calculated using VAM model is $2.752 Billion.
However it is not an optimal solution. Hence further iteration is required.
Iterations:
Iteration 1
Source 1
Source 2
Source 3
Iteration 2
Source 1
Source 2
Destination
1
Destination
2
Destination
3
Destination
4
Destination
5
6000
5000
-7.2
-0.3
5000
7000
9000
-3.2
-24.7
-12.8
-9.8
10000 (-4.5)
-23.3
8000
6000
5000
-4.8
-4.5
9000
-3.2
-12.8
10000
-14.3
5000
Source 3
-2.7
12000
-20.2
-18.8
3000
Minimum cost of shipment from both ways has been calculated by using QM for Windows V4
software and VAM model. It requires 2 Iteration to achieve the optimal solution. After 2 Iterations
we receive our optimal solution.
Optimum Solution:
Optimal solution value =
$2729100
Source 1
Source 2
Source 3
Destination Destination Destination Destination Destination
1
2
3
4
5
6000
9000
5000
10000
5000
12000
3000
The Optimum cost of shipping goods both ways is $2.729 Billion as the cost can no longer be
minimized any further.
Shipping List:
From
Source 1
Source 2
Source 3
To
Destination
1
Destination
3
Destination
1
Destination
4
Destination
5
Destination
2
Destination
5
Shipment
Cost per
unit
Shipment
cost
6000
58.5
351000
9000
45
405000
5000
65.3
326500
10000
49
490000
5000
56
280000
12000
61.3
735600
3000
47
141000
As shown in the above figure, the optimum cost of shipping from railways is $2.77 Billion. This
is obtained by allocating 6000 units of wood to destination 1 and 9000 units of wood to destination
3 from source 1. Similarly, 5000 units of wood is supplied to destination 1, 10000 units of wood
to destination 4 and 5000 units of wood to destination 5 from source 2. At last, source 3 is used to
supply 12000 units of woods to destination 2 and 3000 units of wood to destination 5.
Even though allocation made, while calculating Transportation cost using both ways is similar to
Sea ways, Cost is further reduced by use of railways while shipping goods from Source 1 to
Destination 3, similarly from Source 2 to destination 5 and Source 3 to destination 5. This further
reduces cost by 41.7 Million.
Justification of Using both ways:
1. The Ultimate objective of Transportation model is to minimize cost, using both ways
results in minimum Transportation cost. Hence, mixture of both railways and seas ways
must be used to minimize cost.
2. It is wise to use this model for next 10 years, until the day, the initial cash outlay for
investing in shipment is fully recovered.
Conclusion:
Means of Transportation
Optimum Cost
Railways
2,816,000,000
Sea ways
2,770,800,000
45,200,000
Both
2,729,100,000
41,700,000
Reduction in cost
-
In the nutshell it can be concluded that, it is wise for Alabama Atlantic to make use of both means
of shipment for transporting woods from its 3 different sources to 5 different destinations, as
collectively both means minimizes the total cost of transportation. If that is not possible Alabama
Atlantic should use seaways rather than Railways, as it minimizes cost of shipment. Making Use
of Sea ways instead of Railways helps Alabama Atlantic save 45.2 Million dollars. Similarly using
both means help Alabama Atlantic save 86.9 Million dollars that they are spending right now,
while using railways.