Linear Programming (LP)

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Linear Programming (LP)
An important topic of Deterministic Operations
Research

• Agenda
• Modeling problems
• Examples of models and some classical problems
• Graphical interpretation of LP
• Solving LP by Simplex using MS Excel
• Some theoretical ideas behind LP and Simplex

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Example 1 Product Mix Problem
Fertilizer manufacturing company, 2 types of
fertilizer Type A high phosphorus Type B low
phosphorus
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Product Mix Problem Modeling
Step 1. The decision variables Daily production
of Type A x tons Type B y tons
Step 2. The objective function (maximize
profit) z 15x 10y
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Product Mix Problem Modeling..
Step 3. The constraints Limited supply of raw
materials per day Urea 2x y
1500 Potash x y 1200 Rock Phosphate x
500
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Product Mix Problem Complete model
Maximize z( x, y) 15 x 10y subject to 2x y
1500 x y 1200 x 500 x 0, y
0
Interesting Aspects Linearity, Inequalities
Feasible solutions (0, 0), (1, 1), Infeasible
solutions (600, 500),
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Example 2. Blending Problem
Three types of petrol (minimum Octane rating 85,
90, 95) Four types of oils (Octane rating 68,
86, 91, 99) Blending oils ? petrol, with
proportional Octane rating Objective best
product mix how much of each petrol, oil to sell
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Example 2. Blending Problem, the data
Raw oil OcR Available amount (barrels/day) Cost/barrel Sale price
1 68 4000 31.02 36.85
2 86 5050 33.15 36.85
3 91 7100 36.35 38.95
4 99 4300 38.75 38.95
Petrol Type Min OcR Selling Price Demand (barrels/day)
1 (Premium) 95 45.15 10,000
2 (Super) 90 42.95 No limit
3 (Regular) 85 40.99 15,000
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Blending Problem Modeling
Step 1. The decision variables xij
barrels/day of oil i ( i 1, 2, 3, or 4)
to make petrol j (j 1, 2, or 3)
Total premium petrol per day x11 x21 x31
x41
68x11 86x21 91x31 99x41 - 95(x11 x21
x31 x41) 0.
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Blending Problem Modeling..
Step 2. The objective function Maximize
profit ?? Maximize revenue
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Blending Problem Modeling...
Step 3. The constraints
(a) The OcR constraints 68x11 86x21
91x31 99x41 - 95(x11 x21 x31 x41) 0
68x12 86x22 91x32 99x42 - 90(x12
x22 x32 x42) 0 68x13 86x23
91x33 99x43 - 85(x13 x23 x33 x43) 0
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Blending Problem Modeling....
Step 3. The constraints..
(b) Cant use more oil than we have x11 x12
x13 4000 x21 x22 x23 5050 x31 x32
x33 7100 x41 x42 x43 4300
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Blending Problem Modeling..
Step 3. The constraints...
(c) The demand constraints x11 x21 x31
x41 10,000 x13 x23 x33 x43 15,000 (d)
Allowed values of variables xij 0 for i 1,
2, 3, 4, and j 1, 2, 3.
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Blending Problem complete model
Maximize 45.15(x11 x21 x31 x41)
42.95(x12 x22 x32 x42) 40.99(x13 x23
x33 x43) 36.85(4000 (x11 x12 x13))
36.85 (5050 (x21 x22 x23)) 38.95 (7100
(x31 x32 x33)) 38.95 (4300 (x41 x42
x43)) Subject to 68x11 86x21 91x31 99x41
- 95(x11 x21 x31 x41) 0 68x12 86x22
91x32 99x42 - 90(x12 x22 x32 x42)
0 68x13 86x23 91x33 99x43 - 85(x13 x23
x33 x43) 0 x11 x12 x13 4000 x21 x22
x23 5050 x31 x32 x33 7100 x41 x42
x43 4300 x11 x21 x31 x41 10,000 x13
x23 x33 x43 15,000 xij 0 for I 1, 2,
3, 4, and j 1, 2, 3.
Octane rating
Supply
Demand
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Example 3 Transportation problem
Background Company has several factories
(sinks), and several
suppliers (sources) Objective Minimize the
cost of transportation
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Example 3. Transportation problem, the data
transportation cost per ton transportation cost per ton transportation cost per ton
mine capacity/day plant 1 plant 2 plant 3
Mine 1 800 11 8 2
Mine 2 300 7 5 4
daily ore requirement at each plant daily ore requirement at each plant daily ore requirement at each plant daily ore requirement at each plant daily ore requirement at each plant
400 500 200
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Transportation problem the model
Step 1. The decision variables xij amount of
ore shipped from mine i to plant j per day.
Step 2 The objective function Minimize the
transportation costs Minimize 11x11 8x12
2x13 7x21 5x22 4x23
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Transportation problem the model..
Step 3. The constraints (a) Shipment from each
mine less than daily production x11 x12 x13
800 capacity of mine 1 x21 x22 x23
300 capacity of mine 2 (b) Demand of each
plant must be met x11 x21 400 demand at
plant 1 x12 x22 500 demand at plant
2 x13 x23 200 demand at plant 3 (c)
Decision variables cant be negative xij 0,
for all i 1, 2, j 1, 2, 3.
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Transportation problem historical note
Kantorovich in USSR in the 1930s, Koopmans in
1940s
Dantzig in 1950s ? Simplex method
Kantorovich and Koopmans, Nobel prize
(Economics) in 1975
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The Geometry of Linear Programs
Line in 2D ax by c
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The Geometry of Linear Programs
Plane in 3D ax by cz d
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The Geometry of Linear Programs
Hyper-plane in n-Dimensions a1x1 a2x2
anxn c
??
2-D Half spaces
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The Geometry of LP Product Mix revisited
max z( x, y) 15 x 10y ST 2x y 1500 x y
1200 x 500 x 0, y 0
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The Geometry of LP Product Mix revisited
max z( x, y) 15 x 10y ST 2x y 1500 x y
1200 x 500 x 0, y 0
Try point x 0, y 0
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Summary
1. LP formulations are very common in modern
industry 2. Beautiful connection between Algebra
and Geometry 3. Geometry not useful for gt 3
variables 4. Practical problems 1000s of
variables (see next slide) 5. Need Algebraic
method !
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Some real world examples of LP
Military patient evacuation problem The US Air
Force Military Airlift Command (MAC) has a
patient evacuation problem that can be modeled as
an LP. They use this model to determine the flow
of patients moved by air from an area of conflict
to army bases and hospitals. The objective is to
minimize the time that patients are in the air
transport system. The constraints are - all
patients that need transporting must be
transported - limits on the size and composition
of hospitals, capacity of air fleet, air-lift
points MAC have generated a series of problems
based on the number of time periods (days). A 50
day problem consists of an LP with 79,000
constraints and 267,000 variables.
This LP can be solved (using a fast computer) in
approximately 10 Hours
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Some real world examples of LP..
Military logistics planning The US Department
of Defense Joint Chiefs of Staff have a logistics
planning problem that models the feasibility of
supporting military operations during a crisis.
The problem is to determine if different
materials (called movement requirements) can be
transported overseas within strict time windows.
The LP includes capacities at embarkation and
debarkation ports, capacities of the various
aircraft and ships that carry the movement
requirements and penalties for missing delivery
dates. A typical problem of this type may
consider 15 time periods, 12 ports of
embarkation, 7 ports of debarkation and 9
different types of vehicle for 20,000 movement
requirements. This resulted in an LP with 20,500
constraints and 520,000 variables.
This LP can be solved in approximately 75 minutes
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Some real world examples of LP...
Airline crew scheduling (American
Airlines) Within a fixed airline schedule (the
schedule changing twice a year typically) each
flight in the schedule can be broken down into a
series of flight legs. A flight leg comprises a
takeoff from a specific airport at a specific
time to the subsequent landing at another airport
at a specific time. For example a flight from HK
? Bangkok ? Phuket has two legs. A key point is
that these flight legs may be flown by different
crews. For crew scheduling, aircraft types have
been pre-assigned (not all crews can fly all
types). For a given aircraft type and a given
time period (the schedule repeats over a 1 week
period) we must ensure that all flight legs for a
particular aircraft type can have a crew
assigned. Note here that by crew we mean not only
the pilots/flight crew but also the cabin service
staff, typically these work together as a team
and are kept together over a schedule. There are
restrictions on how many hours the crews (pilots
and others) can work. A potential crew schedule
is a series of flight legs that satisfies these
restrictions. All such potential crew schedules
can have a cost assigned to them. Usually a crew
schedule ends up with the crew returning to their
home base, e.g. A-D and D-A in crew schedule 1
above. A crew schedule such as 2 above (A-B and
B-C) typically includes as part of its associated
cost the cost of returning the crew (as
passengers) to their base. Such carrying of crew
as passengers (on their own airline or on another
airline) is called deadheading. For our American
Airlines problem the company has a database with
12 million potential crew schedules. The
objective is to select the combination of
schedules (out of the 12 million) which shall
minimize costs. The constraints are to ensure
that all flight legs have a crew assigned to
them, and work restrictions are violated. One
case of this problem was formulated as an LP,
with 12 million variables, and 750
constraints. Note a small percentage
improvement of the schedule ? ten's of millions
of dollars!
This LP could be solved in approximately 27
minutes using a software called OSL
next How to solve LPs using MS Excel