Chapter 7 Transportation, Assignment, and Transshipment Problems

1
Chapter 7Transportation, Assignment, and
Transshipment Problems

• Transportation Problem
• Network Representation and LP Formulation
• Transportation Simplex Method
• Assignment Problem
• Network Representation and LP Formulation
• Hungarian Method
• The Transshipment Problem
• Network Representation and LP Formulation

2
Transportation, Assignment, and Transshipment
Problems

• A network model is one which can be represented
  by a set of nodes, a set of arcs, and functions
  (e.g. costs, supplies, demands, etc.) associated
  with the arcs and/or nodes.
• Transportation, assignment, and transshipment
  problems of this chapter, as well as the shortest
  route, minimal spanning tree, and maximal flow
  problems (Chapter 9) and PERT/CPM problems
  (Chapter 10) are all examples of network
  problems.

3
Transportation, Assignment, and Transshipment
Problems

• Each of the three models of this chapter
  (transportation, assignment, and transshipment
  models) can be formulated as linear programs and
  solved by general purpose linear programming
  codes.
• For each of the three models, if the right-hand
  side of the linear programming formulations are
  all integers, the optimal solution will be in
  terms of integer values for the decision
  variables.
• However, there are many computer packages
  (including The Management Scientist) which
  contain separate computer codes for these models
  which take advantage of their network structure.

4
Transportation Problem

• The transportation problem seeks to minimize the
  total shipping costs of transporting goods from m
  origins (each with a supply si) to n destinations
  (each with a demand dj), when the unit shipping
  cost from an origin, i, to a destination, j, is
  cij.
• The network representation for a transportation
  problem with two sources and three destinations
  is given on the next slide.

5
Transportation Problem

• Network Representation

1
d1
c11
1
c12
s1
c13
2
d2
c21
c22
2
s2
c23
3
d3
SOURCES
DESTINATIONS
6
Transportation Problem

• LP Formulation
• The LP formulation in terms of the amounts
  shipped from the origins to the destinations, xij
  , can be written as
  
• Min ??cijxij
• i j
• s.t. ?xij each origin i
  
• j
• ?xij dj for
  each destination j
  
• i
• xij 0 for
  all i and j

7
Transportation Problem

• LP Formulation Special Cases
• The following special-case modifications to the
  linear programming formulation can be made
  
• Minimum shipping guarantee from i to j
• xij Lij
• Maximum route capacity from i to j
• xij
• Unacceptable route
• Remove the corresponding decision variable.

8
Example BBC
Building Brick Company (BBC) has orders for 80
tons of bricks at three suburban locations as
follows Northwood -- 25 tons, Westwood -- 45
tons, and Eastwood -- 10 tons. BBC has two
plants, each of which can produce 50 tons per
week. Delivery cost per ton from each plant to
each suburban location is shown on the next
slide. How should end of week shipments be made
to fill the above orders?
9
Example BBC

• Delivery Cost Per Ton
• Northwood Westwood Eastwood
• Plant 1 24 30
  40
  
• Plant 2 30 40
  42
  

10
Example BBC

• Partial Spreadsheet Showing Problem Data

11
Example BBC

• Partial Spreadsheet Showing Optimal Solution

12
Example BBC

• Optimal Solution
• From To
  Amount Cost
  
• Plant 1 Northwood 5 120
• Plant 1 Westwood 45
  1,350
  
• Plant 2 Northwood 20
  600
  
• Plant 2 Eastwood 10
  420
  
• Total Cost 2,490

13
Example BBC

• Partial Sensitivity Report (first half)

14
Example BBC

• Partial Sensitivity Report (second half)

15
Transportation Simplex Method

• The transportation simplex method requires that
  the sum of the supplies at the origins equal the
  sum of the demands at the destinations.
  
• If the total supply is greater than the total
  demand, a dummy destination is added with demand
  equal to the excess supply, and shipping costs
  from all origins are zero. (If total supply is
  less than total demand, a dummy origin is
  added.)
• When solving a transportation problem by its
  special purpose algorithm, unacceptable shipping
  routes are given a cost of M (a large number).

16
Transportation Simplex Method

• A transportation tableau is given below. Each
  cell represents a shipping route (which is an arc
  on the network and a decision variable in the LP
  formulation), and the unit shipping costs are
  given in an upper right hand box in the cell.

Supply
D3
D2
D1
20
30
15
50
S1
35
40
30
30
S2
Demand
10
45
25
17
Transportation Simplex Method

• The transportation problem is solved in two
  phases
  
• Phase I -- Obtaining an initial feasible
  solution
  
• Phase II -- Moving toward optimality
• Phase I The Minimum-Cost Procedure can be used
  to establish an initial basic feasible solution
  without doing numerous iterations of the simplex
  method.
• Phase II The Stepping Stone Method - using the
  MODI Method for evaluating the reduced costs -
  may be used to move from the initial feasible
  solution to the optimal one.

18
Transportation Simplex Method

• Phase I - Minimum-Cost Method
• Step 1 Select the cell with the least cost.
  Assign to this cell the minimum of its remaining
  row supply or remaining column demand.
  
• Step 2 Decrease the row and column
  availabilities by this amount and remove from
  consideration all other cells in the row or
  column with zero availability/demand. (If both
  are simultaneously reduced to 0, assign an
  allocation of 0 to any other unoccupied cell in
  the row or column before deleting both.) GO TO
  STEP 1.

19
Transportation Simplex Method

• Phase II - Stepping Stone Method
• Step 1 For each unoccupied cell, calculate the
  reduced cost by the MODI method described on an
  upcoming slide.
  
• Select the unoccupied cell with the most
  negative reduced cost. (For maximization
  problems select the unoccupied cell with the
  largest reduced cost.) If none, STOP.

20
Transportation Simplex Method

• Phase II - Stepping Stone Method (continued)
• Step 2 For this unoccupied cell generate a
  stepping stone path by forming a closed loop with
  this cell and occupied cells by drawing
  connecting alternating horizontal and vertical
  lines between them.
• Determine the minimum allocation where a
  subtraction is to be made along this path.

21
Transportation Simplex Method

• Phase II - Stepping Stone Method (continued)
• Step 3 Add this allocation to all cells where
  additions are to be made, and subtract this
  allocation to all cells where subtractions are to
  be made along the stepping stone path.
• (Note An occupied cell on the stepping
  stone path now becomes 0 (unoccupied). If more
  than one cell becomes 0, make only one
  unoccupied make the others occupied with 0's.)
  
• GO TO STEP 1.

22
Transportation Simplex Method

• MODI Method (for obtaining reduced costs)
• Associate a number, ui, with each row and vj
  with each column.
  
• Step 1 Set u1 0.
• Step 2 Calculate the remaining ui's and vj's by
  solving the relationship cij ui vj for
  occupied cells.
  
• Step 3 For unoccupied cells (i,j), the reduced
  cost cij - ui - vj.

23
Example BBC

• Initial Transportation Tableau
• Since total supply 100 and total demand 80,
  a dummy destination is created with demand of 20
  and 0 unit costs.
  

Supply
Dummy
Eastwood
Westwood
Northwood
40
0
30
24
50
Plant 1
42
0
40
30
50
Plant 2
Demand
20
10
45
25
24
Example BBC

• Phase I Minimum-Cost Procedure
• Iteration 1 Tie for least cost (0), arbitrarily
  select x14. Allocate 20. Reduce s1 by 20 to 30
  and delete the Dummy column.
  
• Iteration 2 Of the remaining cells the least
  cost is 24 for x11. Allocate 25. Reduce s1 by
  25 to 5 and eliminate the Northwood column.

25
Example BBC

• Phase I Minimum-Cost Procedure (continued)
• Iteration 3 Of the remaining cells the least
  cost is 30 for x12. Allocate 5. Reduce the
  Westwood column to 40 and eliminate the Plant 1
  row.
• Iteration 4 Since there is only one row with
  two cells left, make the final allocations of 40
  and 10 to x22 and x23, respectively.

26
Example BBC

• Phase II Iteration 1
• MODI Method
• 1. Set u1 0
• 2. Since u1 vj c1j for occupied cells in
  row 1, then
  
• v1 24, v2 30, v4 0.
• 3. Since ui v2 ci2 for occupied cells in
  column 2, then u2 30 40, hence u2 10.
  
• 4. Since u2 vj c2j for occupied cells in
  row 2, then
  
• 10 v3 42, hence v3 32.

27
Example BBC

• Phase II Iteration 1
• MODI Method (continued)
• Calculate the reduced costs (circled numbers on
  the next slide) by cij - ui vj.
  
• Unoccupied Cell Reduced
  Cost
  
• (1,3) 40 - 0 -
  32 8
  
• (2,1) 30 - 24
  -10 -4
  
• (2,4) 0 -
  10 - 0 -10

28
Example BBC

• Phase II Iteration 1
• MODI Method (continued)

ui
Dummy
Eastwood
Westwood
Northwood
25
5
8
20
40
0
30
24
0
Plant 1
-4
40
10
-10
42
0
40
30
10
Plant 2
vj
0
32
30
24
29
Example BBC

• Phase II Iteration 1
• Stepping Stone Method
• The stepping stone path for cell (2,4) is
  (2,4), (1,4), (1,2), (2,2). The allocations in
  the subtraction cells are 20 and 40,
  respectively. The minimum is 20, and hence
  reallocate 20 along this path. Thus for the next
  tableau
• x24 0 20 20 (0 is its current
  allocation)
  
• x14 20 - 20 0 (blank for the
  next tableau)
  
• x12 5 20 25
• x22 40 - 20 20
• The other occupied cells remain the
  same.

30
Example BBC

• Phase II - Iteration 2
• MODI Method
• The reduced costs are found by calculating
  the ui's and vj's for this tableau.
  
• 1. Set u1 0.
• 2. Since u1 vj cij for occupied cells in
  row 1, then
  
• v1 24, v2 30.
• 3. Since ui v2 ci2 for occupied cells in
  column 2, then u2 30 40, or u2 10.
  
• 4. Since u2 vj c2j for occupied cells in
  row 2, then
  
• 10 v3 42 or v3 32 and, 10
  v4 0 or v4 -10.

31
Example BBC

• Phase II - Iteration 2
• MODI Method (continued)
• Calculate the reduced costs (circled numbers on
  the next slide) by cij - ui vj.
  
• Unoccupied Cell Reduced
  Cost
  
• (1,3) 40 - 0 -
  32 8
  
• (1,4) 0 - 0 -
  (-10) 10
  
• (2,1) 30 - 10 -
  24 -4
  

32
Example BBC

• Phase II - Iteration 2
• MODI Method (continued)

ui
Dummy
Eastwood
Westwood
Northwood
25
25
8
10
40
0
30
24
0
Plant 1
-4
20
10
20
42
0
40
30
10
Plant 2
vj
-6
36
30
24
33
Example BBC

• Phase II - Iteration 2
• Stepping Stone Method
• The most negative reduced cost is -4
  determined by x21. The stepping stone path for
  this cell is (2,1),(1,1),(1,2),(2,2). The
  allocations in the subtraction cells are 25 and
  20 respectively. Thus, the new solution is
  obtained by reallocating 20 on the stepping stone
  path.

34
Example BBC

• Phase II - Iteration 2
• Stepping Stone Method (continued)
• Thus, for the next tableau
• x21 0 20 20 (0 is its
  current allocation)
  
• x11 25 - 20 5
• x12 25 20 45
• x22 20 - 20 0 (blank
  for the next tableau)
  
• The other occupied cells remain the
  same.
  

35
Example BBC

• Phase II - Iteration 3
• MODI Method
• The reduced costs are found by calculating
  the ui's and vj's for this tableau.
  
• 1. Set u1 0
• 2. Since u1 vj c1j for occupied cells in
  row 1, then
  
• v1 24 and v2 30.
• 3. Since ui v1 ci1 for occupied cells in
  column 2, then u2 24 30 or u2 6.
  
• 4. Since u2 vj c2j for occupied cells in
  row 2, then
  
• 6 v3 42 or v3 36, and 6 v4
  0 or v4 -6.

36
Example BBC

• Phase II - Iteration 3
• MODI Method (continued)
• Calculate the reduced costs (circled numbers on
  the next slide) by cij - ui vj.
  
• Unoccupied Cell Reduced
  Cost
  
• (1,3) 40 - 0 - 36
  4
  
• (1,4) 0 - 0 -
  (-6) 6
  
• (2,2) 40 - 6 -
  30 4
  

37
Example BBC

• Phase II - Iteration 3
• MODI Method (continued)
• Since all the reduced costs are non-negative,
  this is the optimal tableau.

ui
Dummy
Eastwood
Westwood
Northwood
5
45
4
6
40
0
30
24
0
Plant 1
20
4
10
20
42
0
40
30
6
Plant 2
vj
-6
36
30
24
38
Example BBC

• Optimal Solution
• From To
  Amount Cost
  
• Plant 1 Northwood 5
  120
  
• Plant 1 Westwood 45
  1,350
  
• Plant 2 Northwood 20
  600
  
• Plant 2 Eastwood 10
  420
  
• Total Cost 2,490

39
Assignment Problem

• An assignment problem seeks to minimize the total
  cost assignment of m workers to m jobs, given
  that the cost of worker i performing job j is
  cij.
• It assumes all workers are assigned and each job
  is performed.
  
• An assignment problem is a special case of a
  transportation problem in which all supplies and
  all demands are equal to 1 hence assignment
  problems may be solved as linear programs.
• The network representation of an assignment
  problem with three workers and three jobs is
  shown on the next slide.

40
Assignment Problem

• Network Representation

c11
1
1
c12
c13
AGENTS
TASKS
c21
c22
2
2
c23
c31
c32
3
3
c33
41
Assignment Problem

• LP Formulation
• Min ??cijxij
• i j
• s.t. ?xij 1
  for each agent i
  
• j
• ?xij 1
  for each task j
  
• i
• xij 0 or 1
  for all i and j
  
• Note A modification to the right-hand side of
  the first constraint set can be made if a worker
  is permitted to work more than 1 job.

42
Assignment Problem

• LP Formulation Special Cases
• Number of agents exceeds the number of tasks
• ?xij
• j
• Number of tasks exceeds the number of agents
• Add enough dummy agents to equalize the
• number of agents and the number of tasks.
• The objective function coefficients for
  these
  
• new variable would be zero.

43
Assignment Problem

• LP Formulation Special Cases (continued)
• The assignment alternatives are evaluated in
  terms of revenue or profit
  
• Solve as a maximization problem.
• An assignment is unacceptable
• Remove the corresponding decision variable.
• An agent is permitted to work a tasks
• ?xij
• j

44
Example Hungry Owner
A contractor pays his subcontractors a fixed
fee plus mileage for work performed. On a given
day the contractor is faced with three electrical
jobs associated with various projects. Given
below are the distances between the
subcontractors and the projects.

Projects Subcontractor A
B C Westside 50 36
16 Federated
28 30 18 Goliath
35 32 20
Universal 25 25 14
How should the contractors be assigned to min
imize total costs?
45
Example Hungry Owner

• Network Representation

50
West.
A
36
16
Subcontractors
Projects
28
30
B
Fed.
18
32
35
C
Gol.
20
25
25
Univ.
14
46
Example Hungry Owner

• Linear Programming Formulation
• Min 50x1136x1216x1328x2130x2218x23
• 35x3132x3220x3325x4125x4214x43
• s.t. x11x12x13
• x21x22x23
• x31x32x33
• x41x42x43
• x11x21x31x41 1
• x12x22x32x42 1
• x13x23x33x43 1
• xij 0 or 1 for all i and j

Agents
Tasks
47
Hungarian Method

• The Hungarian method solves minimization
  assignment problems with m workers and m jobs.
  
• Special considerations can include
• number of workers does not equal the number of
  jobs -- add dummy workers or jobs with 0
  assignment costs as needed
  
• worker i cannot do job j -- assign cij M
• maximization objective -- create an opportunity
  loss matrix subtracting all profits for each job
  from the maximum profit for that job before
  beginning the Hungarian method

48
Hungarian Method

• Step 1 For each row, subtract the minimum
  number in that row from all numbers in that row.
  
• Step 2 For each column, subtract the minimum
  number in that column from all numbers in that
  column.
  
• Step 3 Draw the minimum number of lines to
  cover all zeroes. If this number m, STOP -- an
  assignment can be made.
  
• Step 4 Subtract d (the minimum uncovered
  number) from uncovered numbers. Add d to numbers
  covered by two lines. Numbers covered by one
  line remain the same. Then, GO TO STEP 3.

49
Hungarian Method

• Finding the Minimum Number of Lines and
  Determining the Optimal Solution
  
• Step 1 Find a row or column with only one
  unlined zero and circle it. (If all rows/columns
  have two or more unlined zeroes choose an
  arbitrary zero.)
• Step 2 If the circle is in a row with one zero,
  draw a line through its column. If the circle is
  in a column with one zero, draw a line through
  its row. One approach, when all rows and columns
  have two or more zeroes, is to draw a line
  through one with the most zeroes, breaking ties
  arbitrarily.
• Step 3 Repeat step 2 until all circles are
  lined. If this minimum number of lines equals m,
  the circles provide the optimal assignment.

50
Example Hungry Owner

• Initial Tableau Setup
• Since the Hungarian algorithm requires that
  there be the same number of rows as columns, add
  a Dummy column so that the first tableau is
  
• A B
  C Dummy
  
• Westside 50 36 16 0
• Federated 28 30 18
  0
  
• Goliath 35 32 20
  0
  
• Universal 25 25 14
  0

51
Example Hungry Owner

• Step 1 Subtract minimum number in each row from
  all numbers in that row. Since each row has a
  zero, we would simply generate the same matrix
  above.
• Step 2 Subtract the minimum number in each
  column from all numbers in the column. For A it
  is 25, for B it is 25, for C it is 14, for Dummy
  it is 0. This yields
• A B
  C Dummy
  
• Westside 25 11 2
  0
  
• Federated 3 5
  4 0
  
• Goliath 10 7
  6 0
  
• Universal 0 0
  0 0

52
Example Hungry Owner

• Step 3 Draw the minimum number of lines to
  cover all zeroes. Although one can "eyeball"
  this minimum, use the following algorithm. If a
  "remaining" row has only one zero, draw a line
  through the column. If a remaining column has
  only one zero in it, draw a line through the row.
  
• A B C Dummy
• Westside 25 11 2 0
• Federated 3 5 4 0
• Goliath 10 7 6
  0
  
• Universal 0 0 0 0

53
Example Hungry Owner

• Step 4 The minimum uncovered number is 2.
  Subtract 2 from uncovered numbers add 2 to all
  numbers covered by two lines. This gives
  
• A B C
  Dummy
  
• Westside 23 9 0 0
  
  
• Federated 1 3 2
  0
  
• Goliath 8 5 4
  0
  
• Universal 0 0 0
  2

54
Example Hungry Owner

• Step 3 Draw the minimum number of lines to
  cover all zeroes.
  
• A B C Dummy
• Westside 23 9 0
  0
  
• Federated 1 3 2
  0
  
• Goliath 8 5 4
  0
  
• Universal 0 0 0
  2

55
Example Hungry Owner

• Step 4 The minimum uncovered number is 1.
  Subtract 1 from uncovered numbers. Add 1 to
  numbers covered by two lines. This gives
  
• A B C Dummy
• Westside 23 9 0
  1
  
• Federated 0 2 1
  0
  
• Goliath 7 4 3
  0
  
• Universal 0 0 0
  3

56
Example Hungry Owner

• Step 3 The minimum number of lines to cover all
  0's is four. Thus, there is a minimum-cost
  assignment of 0's with this tableau. The optimal
  assignment is
• Subcontractor Project
  Distance
  
• Westside C 16
• Federated A
  28
  
• Goliath (unassigned)
• Universal B 25
• Total Distance 69 miles

57
Transshipment Problem

• Transshipment problems are transportation
  problems in which a shipment may move through
  intermediate nodes (transshipment nodes)before
  reaching a particular destination node.
• Transshipment problems can be converted to larger
  transportation problems and solved by a special
  transportation program.
  
• Transshipment problems can also be solved by
  general purpose linear programming codes.
  
• The network representation for a transshipment
  problem with two sources, three intermediate
  nodes, and two destinations is shown on the next
  slide.

58
Transshipment Problem

• Network Representation

c36
3
c13
c37
6
1
s1
d1
c14
c46
c15
4
c47
Demand
Supply
c23
c56
c24
7
2
d2
s2
c25
5
c57
INTERMEDIATE NODES
SOURCES
DESTINATIONS
59
Transshipment Problem

• Linear Programming Formulation
• xij represents the shipment from node i to node
  j
  
• Min ??cijxij
• i j
• s.t. ?xij for each origin i
  
• j
• ?xik - ?xkj 0 for
  each intermediate
  
• i j
  node k
  
• ?xij dj
  for each destination j
  
• i
• xij 0
  for all i and j

60
Example Transshipping

• Thomas Industries and Washburn Corporation
  supply three firms (Zrox, Hewes, Rockwright) with
  customized shelving for its offices. They both
  order shelving from the same two manufacturers,
  Arnold Manufacturers and Supershelf, Inc.
• Currently weekly demands by the users are 50
  for Zrox, 60 for Hewes, and 40 for Rockwright.
  Both Arnold and Supershelf can supply at most 75
  units to its customers.
• Additional data is shown on the next slide.

61
Example Transshipping
Because of long standing contracts based on
past orders, unit costs from the manufacturers to
the suppliers are
Thomas Washburn
Arnold 5 8
Supershelf 7
4 The costs to install the shelving at t
he various locations are
Zrox Hewes Rockwright
Thomas 1 5
8 Washburn 3 4
4
62
Example Transshipping

• Network Representation

Zrox
ZROX
50
1
5
Thomas
Arnold
ARNOLD
75
5
8
8
Hewes
60
HEWES
3
7
Super Shelf
Wash- Burn
4
WASH BURN
75
4
4
Rock- Wright
40
63
Example Transshipping

• Linear Programming Formulation
• Decision Variables Defined
• xij amount shipped from manufacturer i to
  supplier j
  
• xjk amount shipped from supplier j to
  customer k
  
• where i 1 (Arnold), 2
  (Supershelf)
  
• j 3 (Thomas), 4 (Washburn)
• k 5 (Zrox), 6 (Hewes), 7
  (Rockwright)
  
• Objective Function Defined
• Minimize Overall Shipping Costs
• Min 5x13 8x14 7x23 4x24 1x35 5x36
  8x37
  
• 3x45 4x46 4x47

64
Example Transshipping

• Constraints Defined
• Amount Out of Arnold x13 x14 75
  
• Amount Out of Supershelf x23 x24
• Amount Through Thomas x13 x23 - x35 -
  x36 - x37 0
  
• Amount Through Washburn x14 x24 - x45 - x46
  - x47 0
  
• Amount Into Zrox x35 x45
  50
  
• Amount Into Hewes x36 x46
  60
  
• Amount Into Rockwright x37 x47 40
• Non-negativity of Variables xij 0, for all
  i and j.

65
Example Transshipping

• Optimal Solution (from The Management Scientist
  )
  
• Objective Function Value
  1150.000
  
• Variable Value
  Reduced Costs
  
• X13 75.000
  0.000
  
• X14
  0.000 2.000
  
• X23
  0.000 4.000
  
• X24
  75.000 0.000
  
• X35
  50.000 0.000
  
• X36
  25.000 0.000
  
• X37
  0.000 3.000
  
• X45
  0.000 3.000
  
• X46
  35.000 0.000
  
• X47
  40.000 0.000

66
Example Transshipping

• Optimal Solution

Zrox
ZROX
50
50
75
1
5
Thomas
Arnold
ARNOLD
75
5
25
8
8
Hewes
60
35
HEWES
3
4
7
Super Shelf
Wash- Burn
40
WASH BURN
75
4
4
75
Rock- Wright
40
67
Example Transshipping

• Optimal Solution (continued)
• Constraint Slack/Surplus Dual
  Prices
  
• 1 0.000
  0.000
  
• 2 0.000
  2.000
  
• 3 0.000
  -5.000
  
• 4 0.000
  -6.000
  
• 5 0.000
  -6.000
  
• 6 0.000
  -10.000
  
• 7 0.000
  -10.000

68
Example Transshipping

• Optimal Solution (continued)
• OBJECTIVE COEFFICIENT RANGES
• Variable Lower Limit Current Value
  Upper Limit
  
• X13 3.000
  5.000 7.000
  
• X14 6.000
  8.000 No Limit
  
• X23 3.000
  7.000 No Limit
  
• X24 No Limit
  4.000 6.000
  
• X35 No Limit
  1.000 4.000
  
• X36 3.000
  5.000 7.000
  
• X37 5.000
  8.000 No Limit
  
• X45 0.000
  3.000 No Limit
  
• X46 2.000
  4.000 6.000
  
• X47 No Limit
  4.000 7.000

69
Example Transshipping

• Optimal Solution (continued)
• RIGHT HAND SIDE RANGES
• Constraint Lower Limit Current Value
  Upper Limit
  
• 1 75.000
  75.000 No Limit
  
• 2 75.000
  75.000 100.000
  
• 3 -75.000
  0.000 0.000
  
• 4 -25.000
  0.000 0.000
  
• 5 0.000
  50.000 50.000
  
• 6 35.000
  60.000 60.000
  
• 7 15.000
  40.000 40.000

70
The End of Chapter 7