Route Planning

Texas Transfer Corp (TTC) Case

1

Linear programmingExample: Woodcarving, Inc.

Manufactures two types of wooden toysSoldiers sell for $27 and cost $24 to produceTrains sell for $21 and cost $19 to produceRequires two types of skilled laborFinishing (100 available hours each week)Carpentry (80 available hours each week)Soldiers require 2 hours of finishing labor and 1 hour of carpentry laborTrains require 1 hour of finishing labor and 1 hour of carpentry laborDemand for trains is unlimited, but no more than 40 soldiers are bought each weekObjective: maximize weekly profit

Example: Woodcarving, Inc. (continued)

Decision Variables:X=number of soldiers to produce each week

Y=number of trains to produce each week

Objective Function:Maximize the total profit of soldier and train sales

Max 3X+2Y

Sell Price Production cost

Soldier $27 $24

Train $21 $19

→ Profit $3

→ Profit $2

Example: Woodcarving, Inc. (continued)

Constraints:Labor hour constraints:

Total finishing hours must be 100 hours:

2X+Y 100 (1)

Total carpentry hours must be 80 hours:

X+Y 80 (2)

Demand constraint: do not produce more than 40 soldiers: X 40 (3)

Finishing Carpentry

Soldier 2 1

Train 1 1

Total available hours 100 80

Linear programming problem

Decision variables• What decision must we make (e.g. production levels)?

Objective function•What is our measure of success? •What are we trying to maximize (e.g. profits) or minimize (e.g. cost)?

Constraints•What limits our success (e.g. available resources)?

Relationships (equations) are linear

40

80

1002 s.t.

23 Max,

X

YX

YX

YXYX

FeasibleRegion

Graphical solution

X 40

2X + Y 100

X + Y 80

100

10000

Y = # trains

X = # soldiers

X=0, Y=80, profit=160

X=20, Y=60, profit=180 (optimal solution)

X=40, Y=20, profit=160

X=40, Y=0, profit=120

40

80

1002 s.t.

23 Max,

X

YX

YX

YXYX

Use Excel Solver: Data-Solver

Objective Function

Decision Variables

Constraints

The solution

Easy rider toys

Easy Rider Toys (ERT) manufactures and markets toy vehicles, including cars and trucksCurrently ERT has10,000 cars and 12,000 trucks in inventory.Inventories can be sold in two different bundles:The Racer Set consists of 7 cars and 2 trucks, and is sold for $34.99. The Construction Set consists of 3 cars and 12 trucks, and is sold for $43.99.

How many sets of each bundle ERT should produce to maximize the profit?

Easy rider formulation

X = # of Racer Sets to produceY = # of Construction Sets to produce

Problem formulation:

Cars Trucks Price

Racer Set 7 2 $34.99

Construction Set 3 12 $43.99

Available Inventory 10,000 12,000

Transportation example

Philadelphia and Washington D.C. are supply nodesBuffalo, Pittsburgh, and Harrisburg are demand nodesTotal supply = total demand

Washington D.C. 150

Harrisburg 100$30

Philadelphia 100

$50

$20

$25

Buffalo 50

Pittsburgh 100

$30 $15$70

1

2

3

4

5

Graphical representation of network flow problems

(5,10) (4,9)

(3,6)(1,3)

(4,8)4 6 2

5

4

Arc (i, j)

Cost of sending one unit from i to j

Minimum and Maximumallowable flows on arc (i,j)

i j

Translating a graph to an LP

One decision variable Xi,j for each arc (i,j)

(the amount of flow to send on that arc)Two constraints for each arc (i,j)

Xi,j upper bound for flow on arc (i,j)

Xi,j lower bound for flow on arc (i,j)

One constraint for each node (flow-balance contr.)

flow into node i ═ flow out of node i

iji

ji

iih

ih XX

node leaving, arcs all

,

node entering, arcs all

,

LP formulation

Objective function: Minimize the overall flow cost

,

all arcs ,in the network

cost to send one unit ofmin

flow along arc

s.t:

For each arc :

Upper bound for flow on arc ( )

Lower bound for flow on arc ( )

For each no

i ji j

i, j

i, j

Xi, j

(i, j)

X i, j

X i, j

, ,

Flow into node i out of node i

de :

h i i jall h all j

Flow

i

X X

Decision variables

One decision variable for every arcInterpretation: How much flow (i.e. how many cars) should we send along that arc?

Washington D.C. 150

Harrisburg 100

Philadelphia 100Buffalo 50

Pittsburgh 100

1

2

3

4

5X23

X13X34

X35

X45 X54 X12

Objective function

Minimize total transportation costs

70×X12 + 30×X13 + 50×X23 + 20×X34 + 25×X35 + 30×X45 + 15×X54

Interpretation: We will ship at the lowest cost possible while still satisfying demand.

Constraints

One constraint for every nodeOutflow = InflowPhiladelphia: X12 + X13 = 100Washington: X23 = 150 + X12Harrisburg: X13 + X23 = 100 + X34 + X35Buffalo: X34 + X54 = 50 + X45Pittsburgh: X45 + X35 = 100 + X54

The shortest path problem

Texas Transfer Corp. is an express package delivery company

Services 10 cities in Texas

Must plan fastest routes

The shortest path problem

The flow limits on origin and destination arcs are (1,1)

The flow limits on all other arcs are (0,1)

Assign a flow of 1 along an arc if it is used in the route; otherwise the flow is 0.

Minimize the overall flow cost (distance).

Decision variables

One decision variable for each arc in the network

Example:

X12: A flow from City 1 to City 2

X12 is either 0 or 1

User input variables

Oi = 1 if the origin is city i, 0 otherwiseExample: O1 = 1, O2 = … = O10 = 0

Dj = 1 if the destination is city j, 0 otherwiseExample: D10 = 1, D1 = … = D9 = 0

Travel time

City 1 → City 2

122 / 51 + 0 = 2.39

22

Distance / Avg. speed + Delay

23

Min 2.39X12 + 6.08X13 + … + 3.41X10_9

Excel function:

=SUMPRODUCT( , )

Objective function

Column Column

Constraints

Flow-balance constraints

City Constraint

(1) Amarillo O1 + X21 + X31 = D1 + X12 + X13

(2) Lubbock O2 + X12 + X42 + X52 = D2 + X21 + X24 + X25

(3) Fort Worth O3 + X13 + X53 + X83 + X93 = D3 + X31 + X35 + X38 + X39

(4) El Paso O4 + X24 + X54 + X64 = D4 + X42 + X45 + X46

(5) Abilene O5 + X25 + X35 + X45 + X65 = D5 + X52 + X53 + X54 + X56

(6) San Angelo O6 + X46 + X56 + X76 = D6 + X64 + X65 + X67

(7) San Antonio O7 + X67 + X87 + X97 + X10_7 = D7 + X76 + X78 + X79 + X7_10

(8) Austin O8 + X38 + X78 = D8 + X83 + X87

(9) Houston O9 + X39 + X79 + X10_9 = D9 + X93 + X97 + X9_10

(10) Corpus Christi

O10 + X7_10 + X9_10 = D10 + X10_7 + X10_9

Flow-balance constraints

Example: City 1

O1 + X21 + X31 = D1 + X12 + X13

Excel Function: SUMIF

O1 + SUMIF( , , )

D1 + SUMIF( , , )

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Column 1 Column

Column 1 Column

Other constraints

We also include non-negativity constraints.

We can ignore the upper bounds on the decision variables (e.g. X21≤1) since they are redundant.

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Results

Shortest route:

1→3→9→10

Travel time:

13.81 hours

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