OPM 330 University of Central Oklahoma Distribution and Network Models Quiz the final is on Wednesday july 22nd 9am-11:30am saudi arabia time. I need the t

OPM 330 University of Central Oklahoma Distribution and Network Models Quiz the final is on Wednesday july 22nd 9am-11:30am saudi arabia time. I need the tutor to be online the whole time to help me solve the final. I also have the text book if needed. The final includes two chapters (attached below). I need to score high pleas .
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SLIDES BY
John Loucks
St. Edward’s Univ.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 1
Chapter 10
Distribution and Network Models
Supply Chain Models
• Transportation Problem
• Transshipment Problem
• Shortest-Route Problem
Assignment Problem
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 2
Transportation, Transshipment,
and Assignment 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, transshipment, assignment,
shortest-route, and maximal flow problems of this
chapter as well as the PERT/CPM problems (in
Chapter 13) are all examples of network problems.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 3
Transportation, Transshipment,
and Assignment Problems
Each of the problems of this chapter can be
formulated as linear programs and solved by
general purpose linear programming codes.
For each of the problems, 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 that
contain separate computer codes for these problems
which take advantage of their network structure.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 4
Supply Chain Models
A supply chain describes the set of all
interconnected resources involved in producing
and distributing a product.
In general, supply chains are designed to satisfy
customer demand for a product at minimum cost.
Those that control the supply chain must make
decisions such as where to produce a product, how
much should be produced, and where it should be
sent.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 5
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.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 6
Transportation Problem
Network Representation
s1
s2
1
c11
Sources
c23
d1
2
d2
3
d3
c12
c13
c21
2
1
c22
Destinations
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 7
Transportation Problem
Linear Programming Formulation
Using the notation:
xij = number of units shipped from
origin i to destination j
cij = cost per unit of shipping from
origin i to destination j
si = supply or capacity in units at origin i
dj = demand in units at destination j
continued
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 8
Transportation Problem
Linear Programming Formulation (continued)
Min
m
n
?? c x
i =1 j =1
n
?x
j =1
ij
m
åx
ij
ij ij
? si
i =1, 2,
= dj
j =1,2,… ,n
,m
Supply
Demand
i=1
xij > 0 for all i and j
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 9
Transportation Problem
LP Formulation Special Cases
• Total supply exceeds total demand:
No modification of LP formulation is necessary.
• Total demand exceeds total supply:
Add a dummy origin with supply equal to the
shortage amount. Assign a zero shipping cost
per unit. The amount “shipped” from the
dummy origin (in the solution) will not actually
be shipped.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 10
Transportation Problem
LP Formulation Special Cases (continued)
• The objective is maximizing profit or revenue:
Solve as a maximization problem.
• Minimum shipping guarantee from i to j:
xij > Lij
• Maximum route capacity from i to j:
xij < Lij • Unacceptable route: Remove the corresponding decision variable. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 11 Transportation Problem: Example #1 Acme Block Company has orders for 80 tons of concrete blocks at three suburban locations as follows: Northwood -- 25 tons, Westwood -- 45 tons, and Eastwood -- 10 tons. Acme 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? © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 12 Transportation Problem: Example #1 Delivery Cost Per Ton Plant 1 Plant 2 Northwood 24 30 Westwood 30 40 Eastwood 40 42 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 13 Transportation Problem: Example #1 Optimal Solution Variable x11 x12 x13 x21 x22 x23 From Plant 1 Plant 1 Plant 1 Plant 2 Plant 2 Plant 2 To Amount Cost Northwood 5 120 Westwood 45 1,350 Eastwood 0 0 Northwood 20 600 Westwood 0 0 Eastwood 10 420 Total Cost = $2,490 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 14 Transportation Problem: Example #2 The Navy has 9,000 pounds of material in Albany, Georgia that it wishes to ship to three installations: San Diego, Norfolk, and Pensacola. They require 4,000, 2,500, and 2,500 pounds, respectively. Government regulations require equal distribution of shipping among the three carriers. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 15 Transportation Problem: Example #2 The shipping costs per pound for truck, railroad, and airplane transit are shown on the next slide. Formulate and solve a linear program to determine the shipping arrangements (mode, destination, and quantity) that will minimize the total shipping cost. © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 16 Transportation Problem: Example #2 Mode Truck Railroad Airplane Destination San Diego Norfolk Pensacola $12 20 30 $6 11 26 $5 9 28 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 17 Transportation Problem: Example #2 Define the Decision Variables We want to determine the pounds of material, xij , to be shipped by mode i to destination j. The following table summarizes the decision variables: Truck Railroad Airplane San Diego Norfolk Pensacola x11 x12 x13 x21 x22 x23 x31 x32 x33 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 18 Transportation Problem: Example #2 Define the Objective Function Minimize the total shipping cost. Min: (shipping cost per pound for each mode per destination pairing) x (number of pounds shipped by mode per destination pairing). Min: 12x11 + 6x12 + 5x13 + 20x21 + 11x22 + 9x23 + 30x31 + 26x32 + 28x33 © 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slide 19 Transportation Problem: Example #2 Define the Constraints Equal use of transportation modes: (1) x11 + x12 + x13 = 3000 (truck) (2) x21 + x22 + x23 = 3000 (railroad) (3) x31 + x32 + x33 = 3000 (airplane) Destination material requirements: (4) x11 + x21 + x31 = 4000 (San Diego) (5) x12 + x22 + x32 = 2500 (Norfolk) (6) x13 + x23 + x33 = 2500 (Pensacola) Non-negativity of variables: xij > 0, i = 1, 2, 3 and j = 1, 2, 3
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 20
Transportation Problem: Example #2
Computer Output
Objective Function Value = 142000.000
Variable
Value
Reduced Cost
x11
1000.000
0.000
x12
2000.000
0.000
x13
0.000
1.000
x21
0.000
3.000
x22
500.000
0.000
x23
2500.000
0.000
x31
3000.000
0.000
x32
0.000
2.000
x33
0.000
6.000
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 21
Transportation Problem: Example #2
Solution Summary
• San Diego will receive 1000 lbs. by truck
and 3000 lbs. by airplane.
• Norfolk will receive 2000 lbs. by truck
and 500 lbs. by railroad.
• Pensacola will receive 2500 lbs. by railroad.
• The total shipping cost will be $142,000.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 22
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.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 23
Transshipment Problem
Network Representation
s1
c15
Supply
s2
3
c13
1
c37
c14
Sources
c25
6
c46
c47
4
c23
2
c36
c56
c24
5
Demand
7
c57
d1
d2
Destinations
Intermediate Nodes
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 24
Transshipment Problem
Linear Programming Formulation
Using the notation:
xij = number of units shipped from node i to node j
cij = cost per unit of shipping from node i to node j
si = supply at origin node i
dj = demand at destination node j
continued
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 25
Transshipment Problem
Linear Programming Formulation (continued)
Min
?
c ij xij
?
xij ?
?
xij ?
all arcs
s.t.
arcs out
ij
? si
Origin nodes i
?x
ij
=0
Transhipment nodes
arcs in
arcs out
?x
arcs in
?x
arcs in
ij
?
?
xij = d j Destination nodes j
arcs out
xij > 0 for all i and j
continued
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 26
Transshipment Problem
LP Formulation Special Cases
• Total supply not equal to total demand
• Maximization objective function
• Route capacities or route minimums
• Unacceptable routes
The LP model modifications required here are
identical to those required for the special cases in
the transportation problem.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 27
Transshipment Problem: Example
The Northside and Southside facilities of Riyadh
Industries supply three firms (F1, F2, F3) with
customized shelving for its offices. They both order
shelving from the same two manufacturers, Alsudairi
Manufacturers and Almutlak, Inc.
Currently weekly demands by the users are 50
for F1, 60 for F2, and 40 for F3. Both Alsudairi and
Almutlak can supply at most 75 units to its
customers.
Additional data is shown on the next slide.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 28
Transshipment Problem: Example
Because of long standing contracts based on
past orders, unit costs from the manufacturers to the
suppliers are:
Riyadh N
Alsudairi
5
Almutlak
7
Riyadh S
8
4
The costs to install the shelving at the various
locations are:
Riyadh N
Riyadh S
F1
1
3
F2
5
4
F3
8
4
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 29
Transshipment Problem: Example
Network Representation
ZROX
75
Alsud
ARNOLD
5
airi
Riyadh
N
8
75
4
50
F2
HEWES
60
F3
40
5
8
3
7
Almut
lak
1
F1
Riyadh
WASH
BURN
S
4
4
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 30
Transshipment Problem: Example
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 (Alsudairi), 2 (Almutlak)
j = 3 (Riyadh N), 4 (Riyadh S)
k = 5 (F1), 6 (F2), 7 (F3)
• Objective Function Defined
Minimize Overall Shipping Costs:
Min 5×13 + 8×14 + 7×23 + 4×24 + 1×35 + 5×36 + 8×37
+ 3×45 + 4×46 + 4×47
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 31
Transshipment Problem: Example
Constraints Defined
Amount Out of Alsudairi:
Amount Out of Almutlak:
Amount Through Riyadh N:
Amount Through Riyadh S:
Amount Into F1:
Amount Into F2:
Amount Into F3:
x13 + x14 < 75 (Origin) x23 + x24 < 75 (Origin) x13 + x23 - x35 - x36 - x37 = 0 (Tran) x14 + x24 - x45 - x46 - x47 = 0 (Tran) x35 + x45 = 50 (Destination) x36 + x46 = 60 (Destination) x37 + x47 = 40 (Destination) Non-negativity of Variables: xij > 0, for all i and j.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 32
Transshipment Problem: Example
Computer Output
Objective Function Value =
1150.000
Variable
Value
Reduced Cost
X13
X14
X23
X24
X35
X36
X37
X45
X46
X47
75.000
0.000
0.000
75.000
50.000
25.000
0.000
0.000
35.000
40.000
0.000
2.000
4.000
0.000
0.000
0.000
3.000
3.000
0.000
0.000
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 33
Transshipment Problem: Example
Solution
ZROX
75
Alsud
ARNOLD
airi
5
75
Riyadh
N
8
75
4
50
F2
HEWES
60
F3
40
5
8
3 4
7
Almut
lak
1
F1
Riyadh
WASH
BURN
S
4
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 34
Shortest-Route Problem
The shortest-route problem is concerned with
finding the shortest path in a network from one
node (or set of nodes) to another node (or set of
nodes).
If all arcs in the network have nonnegative values
then a labeling algorithm can be used to find the
shortest paths from a particular node to all other
nodes in the network.
The criterion to be minimized in the shortest-route
problem is not limited to distance even though the
term “shortest” is used in describing the procedure.
Other criteria include time and cost. (Neither time
nor cost are necessarily linearly related to distance.)
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 35
Shortest-Route Problem
Linear Programming Formulation
Using the notation:
xij =
1 if the arc from node i to node j
is on the shortest route
0 otherwise
cij = distance, time, or cost associated
with the arc from node i to node j
continued
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 36
Shortest-Route Problem
Linear Programming Formulation (continued)
Min
?
c ij xij
?
xij
?
xij ?
all arcs
s.t.
=1
Origin node i
arcs out
arcs out
?x
ij
=0
Transhipment nodes
?x
ij
=1
Destination node j
arcs in
arcs in
xij = 0 or 1
for all i and j
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 37
Example: Shortest Route
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 respective travel time,
ticket cost, and transport mode appear on the next two
slides.
If Susan earns a wage of $15 per hour, what route
should she take to minimize the total travel cost?
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 38
Example: Shortest Route
Network Representation
F
2
A
K
B
C
1
Lewisburg
5
D
E
L
G
3
J
6
I
H
4
M
Paducah
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 39
Example: Shortest Route
Network Representation
F
2
A
K
B
C
1
Lewisburg
5
D
E
L
G
3
J
6
I
H
4
M
Paducah
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 40
Example: Shortest Route
Route
A
B
C
D
E
F
G
H
I
J
K
L
M
Transport
Mode
Train
Plane
Bus
Taxi
Train
Bus
Bus
Taxi
Train
Bus
Taxi
Train
Bus
Time
(hours)
4
1
2
6
3 1/3
3
4 2/3
1
2 1/3
6 1/3
3 1/3
1 1/3
4 2/3
Ticket
Cost
$ 20
$115
$ 10
$ 90
$ 30
$ 15
$ 20
$ 15
$ 15
$ 25
$ 50
$ 10
$ 20
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Slide 41
Example: Shortest Route
Route
A(1,2)
B (1,5)
C (1,3)
D (1,6)
E (1,4)
F (2,5)
G (3,5)
H (3,4)
I (5,4)
J (3,6)
K (6,2)
L (5,6)
M (4,6)
Transport
Mode
Train
Plane
Bus
Taxi
Train
Bus
Bus
Taxi
Train
Bus
Taxi
Train
Bus
Time
(hours)
4
1
2
6
3 1/3
3
4 2/3
1
2 1/3
6 1/3
3 1/3
1 1/3
4 2/3
Time
Cost
$60
$15
$30
$90
$50
$45
$70
$15
$35
$95
$50
$20
$70
Ticket
Cost
$ 20
$115
$ 10
$ 90
$ 30
$ 15
$ 20
$ 15
$ 15
$ 25
$ 50
$ 10
$ 20
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied
or duplicated, or posted to a publicly accessible website, in whole or in part.
Total
Cost
$ 80
$130
$ 40
$180
$ 80
$ 60
$ 90
$ 30
$ 50
$120
$100
$ 30
$ 90
Slide 42
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