Introduction to LP - Transshipment Problem

Slide 1 of 26 Slide 1 of 26

Linear Programming: The Transshipment Linear Programming: The Transshipment ProblemProblem

Chapter 2DChapter 2D


IntroductionIntroduction

Linear Programming (LP) problems can be solved Linear Programming (LP) problems can be solved on the computer using dedicated software such on the computer using dedicated software such as WhatsBest!, solver (Excel add-on) and many as WhatsBest!, solver (Excel add-on) and many others.others.

There are special classes of LP problems such as There are special classes of LP problems such as the Transshipment Problem (a special class of the Transshipment Problem (a special class of TP). TP).

Efficient solutions methods exist to solve the Efficient solutions methods exist to solve the Transshipment Problem.Transshipment Problem.


A A network modelnetwork model is one which can be is one which can be represented by a set of nodes, a set of arcs, represented by a set of nodes, a set of arcs, and functions (e.g. costs, supplies, demands, and functions (e.g. costs, supplies, demands, etc.) associated with the arcs and/or nodes.etc.) associated with the arcs and/or nodes.

Transshipment Problem is an example of a Transshipment Problem is an example of a network problem.network problem.

Transshipment ProblemTransshipment Problem



Transshipment problemsTransshipment problems are transportation are transportation problems in which a shipment may move through problems in which a shipment may move through intermediate nodes (transshipment nodes) before intermediate nodes (transshipment nodes) before reaching a particular destination node.reaching a particular destination node.

Transshipment problems can be converted to Transshipment problems can be converted to larger transportation problems and solved by a larger transportation problems and solved by a special transportation program.special transportation program.

Transshipment problems can also be solved by Transshipment problems can also be solved by general purpose linear programming codes.general purpose linear programming codes.

The network representation for a transshipment The network representation for a transshipment problem with two sources, three intermediate problem with two sources, three intermediate nodes, and two destinations is shown on the next nodes, and two destinations is shown on the next slide.slide.



Network RepresentationNetwork Representation

2222

3333

4444

5555

6666

7777

1111cc1313

cc1414

cc2323 cc2424

cc2525

cc1515

ss11

cc3636

cc3737

cc4646

cc4747

cc5656

cc5757

dd11

dd22

INTERMEDIATEINTERMEDIATE NODESNODES

SOURCESSOURCES DESTINATIONSDESTINATIONS

ss22



Linear Programming FormulationLinear Programming Formulation

xxijij represents the shipment from node represents the shipment from node ii to node to node jj

Min Min ccijijxxijij

i ji j

s.t. s.t. xxijij << ssii for each source (origin) for each source (origin) ii j j

xxikik - - xxkjkj = 0 for each intermediate= 0 for each intermediate ii jj node node k k (conservation of flow)(conservation of flow)

xxijij >> ddjj for each destination for each destination jj ii xxijij >> 0 for all 0 for all ii and and j j (nonnegativity)(nonnegativity)


Useful Excel/Solver Functions: Useful Excel/Solver Functions: SumproductSumproduct

SUMPRODUCTSUMPRODUCTMultiplies corresponding components in the given arrays, Multiplies corresponding components in the given arrays,

and returns the sum of those products.and returns the sum of those products. Syntax: SUMPRODUCTSyntax: SUMPRODUCT((array1array1,,array2array2,array3, ,array3,

...)...)• Array1, array2, array3, ... are 2 to 30 arrays Array1, array2, array3, ... are 2 to 30 arrays

whose components you want to multiply and whose components you want to multiply and then add. then add.


Useful Excel/Solver Functions: Useful Excel/Solver Functions: VLOOKUPVLOOKUP

Searches for a value in the leftmost column of a table, and then Searches for a value in the leftmost column of a table, and then returns a value in the same row from a column you specify in returns a value in the same row from a column you specify in the table. Use VLOOKUP instead of HLOOKUP when your the table. Use VLOOKUP instead of HLOOKUP when your comparison values are located in a column to the left of the comparison values are located in a column to the left of the data you want to find.data you want to find.


Useful Excel/Solver Functions: Useful Excel/Solver Functions: VLOOKUPVLOOKUP Syntax: Syntax:

VLOOKUPVLOOKUP((lookup_valuelookup_value,,table_arraytable_array,,col_indecol_index_numx_num,range_lookup),range_lookup)• Lookup_value is the value to be found in the Lookup_value is the value to be found in the

first column of the array.first column of the array.• Table_array is the table of information in Table_array is the table of information in

which data is looked up. which data is looked up. If range_lookup is TRUE, the values in the If range_lookup is TRUE, the values in the

first column of table_array must be placed first column of table_array must be placed in ascending order.in ascending order.


Useful Excel/Solver Functions: Useful Excel/Solver Functions: VLOOKUPVLOOKUP Syntax: Syntax:

VLOOKUPVLOOKUP((lookup_valuelookup_value,,table_arraytable_array,,col_indecol_index_numx_num,,range_lookuprange_lookup))• Col_index_num is the column number in table_array Col_index_num is the column number in table_array

from which the matching value must be returned. A from which the matching value must be returned. A col_index_num of 1 returns the value in the first column col_index_num of 1 returns the value in the first column in table_array; a col_index_num of 2 returns the value in table_array; a col_index_num of 2 returns the value in the second column in table_array, and so on. in the second column in table_array, and so on.

• Range_lookup is a logical value that specifies whether Range_lookup is a logical value that specifies whether you want VLOOKUP to find an exact match or an you want VLOOKUP to find an exact match or an approximate match. If TRUE or omitted, an approximate match. If TRUE or omitted, an approximate match is returned. approximate match is returned.


Useful Excel/Solver Functions: SumifUseful Excel/Solver Functions: Sumif

SUMIF: SUMIF: Adds the cells specified by a given criteria.Adds the cells specified by a given criteria. Syntax: Syntax: SUMIFSUMIF((rangerange,,criteriacriteria,,sum_rangesum_range))

• Range is the range of cells you want Range is the range of cells you want evaluated.evaluated.

• Criteria is the criteria in the form of a number, Criteria is the criteria in the form of a number, expression, or text that defines which cells will expression, or text that defines which cells will be added. be added.

• Sum_range are the actual cells to sum. The Sum_range are the actual cells to sum. The cells in sum_range are summed only if their cells in sum_range are summed only if their corresponding cells in range match the corresponding cells in range match the criteria. If sum_range is omitted, the cells in criteria. If sum_range is omitted, the cells in range are summed.range are summed.


Example 1: Thomas & WashburnExample 1: Thomas & Washburn

Thomas Industries and Washburn Thomas Industries and Washburn Corporation supply three firms (Zrox, Hewes, Corporation supply three firms (Zrox, Hewes, Rockwright) with customized shelving for its Rockwright) with customized shelving for its offices. They both order shelving from the same offices. They both order shelving from the same two manufacturers, Arnold Manufacturers and two manufacturers, Arnold Manufacturers and Supershelf, Inc. Supershelf, Inc.

Currently weekly demands by the users Currently weekly demands by the users are 50 for Zrox, 60 for Hewes, and 40 for are 50 for Zrox, 60 for Hewes, and 40 for Rockwright. Both Arnold and Supershelf can Rockwright. Both Arnold and Supershelf can supply at most 75 units to its customers. supply at most 75 units to its customers.

Additional data is shown on the next slide. Additional data is shown on the next slide.



Because of long standing contracts based on past orders, Because of long standing contracts based on past orders, unit costs from the manufacturers to the suppliers are:unit costs from the manufacturers to the suppliers are:

ThomasThomas WashburnWashburn Arnold 5 8Arnold 5 8 Supershelf 7 4Supershelf 7 4

The cost to install the shelving at the various locations are:The cost to install the shelving at the various locations are:

ZroxZrox HewesHewes RockwrightRockwright Thomas 1 5 8Thomas 1 5 8

Washburn 3 4 4Washburn 3 4 4

Find the quantities to be shipped from each source to each Find the quantities to be shipped from each source to each destination to minimize the total shipping cost. destination to minimize the total shipping cost.




ARNOLD

WASHBURN

ZROX

HEWES

-75-75

-75-75

+50+50

+60+60

+40+40

55

88

77

44

1155

88

33 44

44

ArnoldArnold

SuperSuperShelfShelf

HewesHewes

ZroxZrox

ThomasThomas

Wash-Wash-BurnBurn

Rock-Rock-WrightWright

11

22

33

44

55

66

77

Supply nodes Transshipment nodes

Demand nodes



Linear Programming FormulationLinear Programming Formulation• Decision Variables Defined Decision Variables Defined

xxijij = amount shipped from manufacturer = amount shipped from manufacturer ii to supplier to supplier jj

xxjkjk = amount shipped from supplier = amount shipped from supplier jj to customer to customer kk

where where ii = 1 (Arnold), 2 (Supershelf) = 1 (Arnold), 2 (Supershelf)

jj = 3 (Thomas), 4 (Washburn) = 3 (Thomas), 4 (Washburn)

kk = 5 (Zrox), 6 (Hewes), 7 (Rockwright) = 5 (Zrox), 6 (Hewes), 7 (Rockwright)• Objective Function DefinedObjective Function Defined

Minimize Overall Shipping Costs: Minimize Overall Shipping Costs:

Min 5Min 5xx1313 + 8 + 8xx1414 + 7 + 7xx2323 + 4 + 4xx2424 + 1 + 1xx3535 + 5 + 5xx3636 + + 88xx3737

+ 3+ 3xx45 45 + 4+ 4xx4646 + 4 + 4xx4747



Constraints DefinedConstraints Defined

Amount Out of Arnold: Amount Out of Arnold: xx1313 + + xx1414 << 75 75

Amount Out of Supershelf: Amount Out of Supershelf: xx2323 + + xx2424 << 75 75

Amount Through Thomas: Amount Through Thomas: xx1313 + + xx2323 - - xx3535 - - xx3636 - - xx3737 = = 0 0

Amount Through Washburn: Amount Through Washburn: xx1414 + + xx2424 - - xx4545 - - xx4646 - - xx4747 = = 0 0

Amount Into Zrox: Amount Into Zrox: xx3535 + + xx4545 >> 50 50

Amount Into Hewes: Amount Into Hewes: xx3636 + + xx4646 >> 60 60

Amount Into Rockwright: Amount Into Rockwright: xx3737 + + xx4747 >> 40 40

Non-negativity of Variables: Non-negativity of Variables: xxijij >> 0, for all 0, for all ii and and jj..


Example 1: Thomas & Washburn problem Example 1: Thomas & Washburn problem via LPvia LP

The solver formulation is:The solver formulation is:

Ship From To Unit Cost0 1 Arnold 3 Thomas $50 1 Arnold 4 Washburn $80 2 SuperShelf 3 Thomas $70 2 SuperShelf 4 Washburn $40 3 Thomas 5 Zrox $10 3 Thomas 6 Hewes $50 3 Thomas 7 Rock-Wright $80 4 Washburn 5 Zrox $30 4 Washburn 6 Hewes $40 4 Washburn 7 Rock-Wright $4

The Transshipment Problem

Nodes Net Flow Supply/Demand1 Arnold 0 -752 SuperShelf 0 -753 Thomas 0 04 Washburn 0 05 Zrox 0 506 Hewes 0 607 Rock-Wright 0 40

Total Transportation Cost $0


Example 1: Thomas & Washburn problem Example 1: Thomas & Washburn problem via LPvia LP

The solver solution is:The solver solution is:

Ship From To Unit Cost75 1 Arnold 3 Thomas $50 1 Arnold 4 Washburn $80 2 SuperShelf 3 Thomas $775 2 SuperShelf 4 Washburn $450 3 Thomas 5 Zrox $125 3 Thomas 6 Hewes $50 3 Thomas 7 Rock-Wright $80 4 Washburn 5 Zrox $335 4 Washburn 6 Hewes $440 4 Washburn 7 Rock-Wright $4

The Transshipment Problem

Supply (-) or Nodes Net Flow Demand (+)

1 Arnold -75 -752 SuperShelf -75 -753 Thomas 0 04 Washburn 0 05 Zrox 50 506 Hewes 60 607 Rock-Wright 40 40

Total Transportation Cost $1,150


Example 2: The Bavarian Motors Company Example 2: The Bavarian Motors Company (BMC)(BMC)

BMC manufactures (very) expensive sports luxury cars in BMC manufactures (very) expensive sports luxury cars in Germany and ships them for sale in the US to Newark and Germany and ships them for sale in the US to Newark and Jacksonville. From these entry ports the cars can be Jacksonville. From these entry ports the cars can be transported to distributors located in Boston, Columbus, transported to distributors located in Boston, Columbus, Atlanta, Richmond and Mobile. There are 200 cars Atlanta, Richmond and Mobile. There are 200 cars available for shipment in Newark and 300 in Jacksonville. available for shipment in Newark and 300 in Jacksonville. The distributors ordered the following quantities of cars:The distributors ordered the following quantities of cars:• Boston: 100Boston: 100• Columbus: 60Columbus: 60• Atlanta: 170Atlanta: 170• Richmond: 80Richmond: 80• Mobile: 70. Mobile: 70.

The shipping costs per unit for are shown in the next slide. The shipping costs per unit for are shown in the next slide. You are retained (for a large consulting fee) to determine You are retained (for a large consulting fee) to determine shipping quantities to minimize the total shipping cost. shipping quantities to minimize the total shipping cost.



From To Unit Cost1 Newark 2 Boston $301 Newark 4 Richmond $402 Boston 3 Columbus $503 Columbus 5 Atlanta $355 Atlanta 3 Columbus $405 Atlanta 4 Richmond $305 Atlanta 6 Mobile $356 Mobile 5 Atlanta $257 Jacksonville 4 Richmond $507 Jacksonville 5 Atlanta $457 Jacksonville 6 Mobile $50




ARNOLD

WASHBURN

ZROX

HEWES

-200-200

-300-300

8080

170170

7070

$30$30

$40$40

$50$50

$35$35

Newark Newark (1)(1)

JacksonvilleJacksonville (7)(7)

AtlantaAtlanta (5)(5)

Richmond Richmond (4)(4)

BostonBoston (2)(2)

ColumbusColumbus (3)(3)

MobileMobile (6)(6)Supply nodes

100100

6060

$50$50

$40$40

$30$30

$35$35

$25$25

$50$50

$45$45

$50$50



Linear Programming Formulation:Linear Programming Formulation:

• Decision Variables Defined Decision Variables Defined

xxijij = amount shipped from source = amount shipped from source ii to destination to destination j j

(i, j = 1 through 7)(i, j = 1 through 7)

• Objective Function DefinedObjective Function Defined

Minimize Overall Shipping Costs: Minimize Overall Shipping Costs:

Min 30Min 30xx1212 + 40 + 40xx1414 + 50 + 50xx2323 + 35 + 35xx3535 + 40 + 40xx5353 + + 3030xx5454 + + 3535xx5656 + 25 + 25xx65 65 + 50+ 50xx7474 + 45 + 45xx7575+ + 5050xx7676



Constraints DefinedConstraints Defined

Flow constraint for node 1:Flow constraint for node 1: --xx1212 - - xx1414 >> -200 -200

Flow constraint for node 2:Flow constraint for node 2: xx1212 - - xx2323 >> 100 100

Flow constraint for node 3: Flow constraint for node 3: xx2323 + + xx5353 - - xx3535 >> 60 60

Flow constraint for node 4:Flow constraint for node 4: xx1414 + + xx5454 + + xx7474 >> 80 80

Flow constraint for node 5:Flow constraint for node 5:

xx3535 + + xx6565 + + xx7575 - - xx5353 - - xx5454 - - xx5656 >> 170 170

Flow constraint for node 6:Flow constraint for node 6: xx5656 + + xx7676 - - xx6565 >> 70 70

Flow constraint for node 7:Flow constraint for node 7: --xx7474 - - xx7575 - - xx7676 >> -300 -300

Non-negativity of variables: Non-negativity of variables: xxijij >> 0, for all 0, for all ii and and jj..



The solver formulation is:The solver formulation is:

Ship From To Unit Cost Nodes Net Flow Supply/Demand0 1 Newark 2 Boston $30 1 Newark 0 -2000 1 Newark 4 Richmond $40 2 Boston 0 1000 2 Boston 3 Columbus $50 3 Columbus 0 600 3 Columbus 5 Atlanta $35 4 Richmond 0 800 5 Atlanta 3 Columbus $40 5 Atlanta 0 1700 5 Atlanta 4 Richmond $30 6 Mobile 0 700 5 Atlanta 6 Mobile $35 7 Jacksonville 0 -3000 6 Mobile 5 Atlanta $250 7 Jacksonville 4 Richmond $500 7 Jacksonville 5 Atlanta $450 7 Jacksonville 6 Mobile $50

Total Transportation Cost $0

Bavarian Motor Company



The solver solution is:The solver solution is:

Ship From To Unit Cost Nodes Net Flow Supply/Demand120 1 Newark 2 Boston $30 1 Newark -200 -20080 1 Newark 4 Richmond $40 2 Boston 100 10020 2 Boston 3 Columbus $50 3 Columbus 60 600 3 Columbus 5 Atlanta $35 4 Richmond 80 8040 5 Atlanta 3 Columbus $40 5 Atlanta 170 1700 5 Atlanta 4 Richmond $30 6 Mobile 70 700 5 Atlanta 6 Mobile $35 7 Jacksonville -280 -3000 6 Mobile 5 Atlanta $250 7 Jacksonville 4 Richmond $50

210 7 Jacksonville 5 Atlanta $4570 7 Jacksonville 6 Mobile $50

Total Transportation Cost $22,350

Bavarian Motor Company


The End of Chapter 2DThe End of Chapter 2D

Introduction to LP - Transshipment Problem

Documents

range range

transshipment problem

range of cells

lookup range

column of table

range sumifrange

network problem

j nonnegativity slide