B.2 Transshipment and Shortest Route
Lesson Topics
Transshipment (4) Problems are Transportation
Problems extended so that a shipment may move
through intermediate nodes (transshipment nodes)
before reaching a particular destination node.
Transshipment Problems with
Transshipment Origins (1) are Transshipment
Problems where goods from one origin may move
through other origins before reaching a destination.
Shortest Route (1) Problems are Transshipment
Problems where there is one origin, one destination,
one unit supply, and one unit demand, and where that
unit is indivisible, as in driving through cities to work.
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Review Questions
B.2 Transshipment and Shortest Route
Review Questions
Transshipment
Question. The Northside and Southside facilities of
Green Landscapes supply two stores (Albertsons,
Best Buy) with synthetic king palm trees for their
landscaping. They both order trees from the same
two tree nurseries, Long Beach Organic Inc. and Greenhouse Gas
Nurseries Inc.
Currently, yearly demands by the users are 25 for Albertsons, and 35 for
Best Buy. Long Beach Organic can supply up to 40 units to its customers,
and Greenhouse Gas Nurseries can supply up to 50 units to its customers.
Because of long-standing contracts based on past orders, unit costs from
the nurseries to the suppliers are:
Long Beach Organic
Greenhouse Gas
Green Landscapes N
5
7
Green Landscapes S
8
4
The costs to install the trees at the various locations are:
Green Landscapes N
Green Landscapes S
Albertsons
1
3
Best Buy
5
4
Formulate a linear program for satisfying those yearly demands at
minimum cost. Compute an optimum using a computer.
Tip: Your written answer should define the decision variables, formulate the
objective and constraints, and solve for the optimum. --- You will not earn
full credit if you just solve for the optimum; you must also define the
decision variables, and formulate the objective and constraints.
2
B.2 Transshipment and Shortest Route
Review Questions
Answer to Question: Define decision variables:
xij = amount shipped from nursery i to supplier j
xjk = amount shipped from supplier j to customer k
where i = 1 (Long Beach Organic), 2 (Greenhouse Gas )
j = 3 (Green Landscapes N), 4 (Green Landscapes S)
k = 5 (Albertsons), 6 (Best Buy)
Define objective function: Minimize total costs.
Min 5x13 + 8x14 + 7x23 + 4x24 + 1x35 + 5x36 + 3x45 + 4x46
Constrain amount out of Long Beach Organic:
x13 + x14 < 40
Constrain amount out of Greenhouse Gas :
x23 + x24 < 50
Constrain amount through Green Landscapes N: x13 + x23 - x35 - x36 = 0
Constrain amount through Green Landscapes S: x14 + x24 - x45 - x46 = 0
Constrain amount into Albertsons:
x35 + x45 = 25
Constrain amount into Best Buy:
x36 + x46 = 35
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B.2 Transshipment and Shortest Route
4
Review Questions
B.2 Transshipment and Shortest Route
Review Questions
Transshipment
Question. The Northside and Southside facilities of
Green Lanscapes supply three firms (Albertsons, Best
Buy, Cookie Cutters) with palm trees for their
landscaping. They both order trees from the same
two tree nurseries, Long Beach Organic Inc. and
Greenhouse Gas Nurseries Inc.
Currently, yearly demands by the users are 30 for Altbertsons, 40 for Best
Buy, and 55 for Cookie Cutters. Both Long Beach Organic and
Greenhouse Gas Nurseries can supply up to 50 units to its customers.
Because of long-standing contracts based on past orders, unit costs from
the nurseries to the suppliers are:
Long Beach Organic
Greenhouse Gas
Green Lanscapes N
2
4
Green Lanscapes S
3
5
The costs to install the trees at the various locations are:
Green Lanscapes N
Green Lanscapes S
Albertsons
2
3
Best Buy
3
4
Cookie Cutters
4
4
Formulate the problem of satisfying those yearly demands at minimum
cost. But you need not compute an optimum.
Tip: Your written answer should define the decision variables, and
formulate the objective and constraints.
5
B.2 Transshipment and Shortest Route
Review Questions
Answer to Question: Define decision variables:
xij = amount shipped from nursery i to supplier j
xjk = amount shipped from supplier j to customer k
where i = 1 (Long Beach Organic), 2 (Greenhouse Gas )
j = 3 (Green Landscapes N), 4 (Green Landscapes S)
k = 5 (Albertsons), 6 (Best Buy), 7 (Cookie Cutters)
Define objective function: Minimize total costs.
Min 2x13 + 3x14 + 4x23 + 5x24 + 2x35 + 3x36 + 4x37 + 3x45 + 4x46 + 4x47
Constrain amount out of Long Beach Organic:
x13 + x14 < 50
Constrain amount out of Greenhouse Gas:
x23 + x24 < 50
Constrain amount through Green Landscapes N:
x13 + x23 - x35 - x36 - x37 = 0
Constrain amount through Green Landscapes S:
x14 + x24 - x45 - x46 - x47 = 0
Constrain amount into Albertsons:
x35 + x45 = 30
Constrain amount into Best Buy:
x36 + x46 = 40
Constrain amount into Cookie Cutters:
x37 + x47 = 55
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B.2 Transshipment and Shortest Route
Review Questions
Transshipment
Question.
The Northside and Southside facilities of Green
Lanscapes supply three firms (Albertsons, Best Buy,
Cookie Cutters) with palm trees for their landscaping.
They both order trees from the same two tree nurseries, Long Beach
Organic Inc. and Greenhouse Gas Nurseries Inc.
Currently, yearly demands by the users are 30 for Altbertsons, 40 for Best
Buy, and 50 for Cookie Cutters. Both Long Beach Organic and
Greenhouse Gas Nurseries can supply up to 80 units to its customers.
Because of long-standing contracts based on past orders, unit costs from
the nurseries to the suppliers are:
Green Lanscapes N Green Lanscapes S
Long Beach Organic
5
8
Greenhouse Gas
7
4
The costs to install the trees at the various locations are:
Albertsons Best Buy Cookie Cutters
Green Lanscapes N
1
5
8
Green Lanscapes S
3
4
4
Formulate the problem of satisfying those yearly demands at minimum
cost. Compute an optimum using a computer.
Tip: Your written answer should define the decision variables, formulate the
objective and constraints, and solve for the optimum. --- You will not earn
full credit if you just solve for the optimum; you must also define the
decision variables, and formulate the objective and constraints.
7
B.2 Transshipment and Shortest Route
Review Questions
Answer to Question: Define decision variables:
xij = amount shipped from nursery i to supplier j
xjk = amount shipped from supplier j to customer k
where i = 1 (Long Beach Organic), 2 (Greenhouse Gas )
j = 3 (Green Lanscapes N), 4 (Green Lanscapes S)
k = 5 (Albertsons), 6 (Best Buy), 7 (Cookie Cutters)
Define objective function: Minimize total costs.
Min 5x13 + 8x14 + 7x23 + 4x24 + 1x35 + 5x36 + 8x37 + 3x45 + 4x46 + 4x47
Constrain amount out of Long Beach Organic:
x13 + x14 < 80
Constrain amount out of Greenhouse Gas:
x23 + x24 < 80
Constrain amount through Green Lanscapes N:
x13 + x23 - x35 - x36 - x37 = 0
Constrain amount through Green Lanscapes S:
x14 + x24 - x45 - x46 - x47 = 0
Constrain amount into Albertsons:
x35 + x45 = 30
Constrain amount into Best Buy:
x36 + x46 = 40
Constrain amount into Cookie Cutters:
x37 + x47 = 50
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B.2 Transshipment and Shortest Route
Review Questions
Interpretation:
Given indicies
i = 1 (Long Beach Organic), 2 (Greenhouse Gas)
j = 3 (Green Lanscapes N), 4 (Green Lanscapes S)
k = 5 (Albertsons), 6 (Best Buy), 7 (Cookie Cutters)
40 trees are shipped from Long Beach Organic to Green Lanscapes N,
80 trees are shipped from Greenhouse Gas to Green Lanscapes S,
30 trees are shipped from Green Lanscapes N to Albertsons,
10 trees are shipped from Green Lanscapes N to Best Buy,
30 trees are shipped from Green Lanscapes S to Best Buy,
50 trees are shipped from Green Lanscapes S to Cookie Cutters.
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B.2 Transshipment and Shortest Route
Review Questions
Transshipment
Question. The west-coast and east-coast divisions of
Maxwell House supplies three groceries (Albertsons,
Ralphs, Vons) with coffee. Maxwell House, in turn,
gets it coffee from Brazil and Columbia.
Currently, yearly demands by the users are 30 for Albertsons, 40 for
Ralphs, and 50 for Vons. Brazil can supply up to 60 units to Maxwell
House, and Columbia can supply up to 70 units to Maxwell House.
Unit transportation costs from Brazil and Columbia to the divisions of
Maxwell House are:
Maxwell House West Maxwell House East
Brazil
5
8
Columbia
7
4
Unit transportation costs from the divisions of Maxwell House to the
groceries are:
Albertsons
Ralphs
Vons
Maxwell House West
1
5
8
Maxwell House East
3
4
4
Formulate the problem of satisfying those yearly demands at minimum
cost. But you need not compute an optimum.
Tip: Your written answer should define the decision variables, and
formulate the objective and constraints.
10
B.2 Transshipment and Shortest Route
Review Questions
Answer to Question: Define decision variables:
xij = amount shipped from i to j
xjk = amount shipped from j to k
where i = 1 (Brazil), 2 (Columbia)
j = 3 (Maxwell House West), 4 (Maxwell House East)
k = 5 (Albertsons), 6 (Ralphs), 7 (Vons)
Define objective function: Minimize total costs.
Min 5x13 + 8x14 + 7x23 + 4x24 + 1x35 + 5x36 + 8x37 + 3x45 + 4x46 + 4x47
Constrain amount out of Brazil:
x13 + x14 < 60
Constrain amount out of Columbia:
x23 + x24 < 70
Constrain amount through Maxwell House W.:
x13 + x23 - x35 - x36 - x37 = 0
Constrain amount through Maxwell House E.:
x14 + x24 - x45 - x46 - x47 = 0
Constrain amount into Albertsons:
x35 + x45 = 30
Constrain amount into Ralphs:
x36 + x46 = 40
Constrain amount into Vons:
x37 + x47 = 50
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B.2 Transshipment and Shortest Route
Review Questions
Transshipment with Transshipment Origins
Question. U Haul has rental truck lots in 5 cities in
California
i = 1 (San Diego),
i = 2 (Los Angeles),
i = 3 (Santa Barbara),
i = 4 (San Lois Obispo),
i = 5 (San Francisco).
Suppose San Diego has a surplus of 3 trucks (it has 3 more trucks than it
needs), Santa Barbara has a surplus of 2 trucks (it has 2 more trucks than
it needs), and San Francisco has a deficit of 4 trucks (it needs 4 more
trucks than it has).
Suppose you calculate the following costs per trucks of transporting trucks
between the cities:
• transporting between 1 and 2 (that is, either 1 to 2, or 2 to 1) costs $2
• transporting between 1 and 3 costs $3
• transporting between 1 and 4 costs $4
• transporting between 1 and 5 costs $5
• transporting between 2 and 3 costs $2
• transporting between 2 and 4 costs $3
• transporting between 2 and 5 costs $4
• transporting between 3 and 4 costs $2
• transporting between 3 and 5 costs $3
• transporting between 4 and 5 costs $2
Formulate the problem of how to move trucks between cities to satisfy San
Francisco’s deficit and without creating a deficit in any other city. But you
need not compute an optimum.
Tip: Your written answer should define the decision variables, and
formulate the objective and constraints.
12
B.2 Transshipment and Shortest Route
Review Questions
Answer to Question: Define decision variables:
xij = amount of trucks moved from City i to City j
Define objective function: Minimize total costs.
Min 2(x12+x21) + 3(x13+x31) + 4(x14+x41) + 5(x15+x51) + 2(x23+x32)
+ 3(x24+x42) + 4(x25+x52) + 2(x34+x43) + 3(x35+x53) + 2(x45+x54)
Constrain trucks out of City 1:
x12 + x13 + x14 + x15 < 3 + x21 + x31 + x41 + x51
Constrain trucks out of City 2:
x21 + x23 + x24 + x25 < x12 + x32 + x42 + x52
Constrain trucks out of City 3:
x31 + x32 + x34 + x35 < 2 + x13 + x23 + x43 + x53
Constrain trucks out of City 4:
x41 + x42 + x43 + x45 < x14 + x24 + x34 + x54
Constrain trucks out of City 5:
x51 + x52 + x53 + x54 < -4 + x15 + x25 + x35 + x45
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B.2 Transshipment and Shortest Route
Review Questions
Shortest Route
Question. Susan Winslow has an important business
meeting in Paducah this evening. She has a number
of alternate routes by which she can travel from the
company headquarters in Lewisburg to Paducah. The
network of alternate routes and their total costs appear below.
For example, Route A connects node 1 to node 2, Route J connects node 3
to node 6, and Route I connects node 4 to node 5.
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B.2 Transshipment and Shortest Route
Review Questions
Formulate a linear programming problem to minimize total transportation
costs. But you need not compute an optimum.
Tip: Your written answer should define the decision variables, and
formulate the objective and constraints.
15
B.2 Transshipment and Shortest Route
Review Questions
Answer to Question:
Define indices: Nodes 1 (origin), 2, …, 5, 6 (destination)
Define decision variables:
xij = 1 if the route from node i to node j is on the shortest route
Define objective function: Minimize total transportation costs.
Min 80x12 + 40x13 + 80x14 + 130x15 + 180x16 + 60x25 + 100x26 + 30x34 + 90x35 + 120x36 + 30x43 +
50x45 + 90x46 + 60x52 + 90x53 + 50x54 + 30x56
Node flow-conservation constraints:
x12 + x13 + x14 + x15 + x16 = 1 (origin)
– x12 + x25 + x26 – x52 = 0 (node 2)
– x13 + x34 + x35 + x36 – x43 – x53 = 0 (node 3)
– x14 – x34 + x43 + x45 + x46 – x54 = 0 (node 4)
– x15 – x25 – x35 – x45 + x52 + x53 + x54 + x56 = 0 (node 5)
x16 + x26 + x36 + x46 + x56 = 1 (destination)
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