Integer Linear Programming (ILP)

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Integer Linear Programming (ILP)

• Prof KG Satheesh Kumar
• Asian School of Business

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Types of ILP Models

• ILP
• A linear program in which some or all variables
  are restricted to integer values.
• Types
• All-integer LP or a pure ILP
• Mixed-Integer LP
• 0-1 integer LP

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An All-Integer IP or Pure ILP

• Maximise 2 x1 3 x2
• Subject to
• 3 x1 2 x2 12
• ¼ x1 1 x2 4
• 1 x1 2 x2 6
• x1, x2 0 and integer

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Illustration of All-Integer LP

• Sweeny Eastborne Reality is considering
  investing in townhouses (T) and apartments(A).
  Determine the number of Ts and As to be
  purchased. (Integers)
• Funds available 2 million.
• Cost 282k per T and 400k per A
• Numbers available 5 Ts and any number of As.
• Management time available 140 h/mo
• Time needed 4 hrs/mo for T and 40 h/mo for A.
• Contribution 10k for T and 15k for A.

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Sweeny Eastborne ILP Model

• Maximise 10T 15A
• Subject to 282T 400A 2000
• 4T 40A 140
• T
  5
• T, A 0 and
  integer

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Sweeny Eastborne Relaxed LP Rounded Solution
If we relax the integer restriction, the optimum
solution is T2.48, A 3.25, Z 73.57 This is
not acceptable because townhouses and apartments
cannot be purchased in fractions!
Rounding down gives integer solution with T 2,
A 3 and Z 65 Such rounding down may sometimes
yield an optimum solution, but one cannot be sure!
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Sweeny Eastborne ILPOptimal Integer Solution
The optimal solution is T 4, A 2, Z 70 and
not the rounded-down solution which gives Z 65.
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ILP Algorithms

• The ILP algorithms are based on exploiting the
  tremendous computational success of LP. The
  strategy involves three steps
• Relax the ILP Remove integer restrictions
  replace any binary variable y with continuous
  range 0 ? y ? 1.
• Solve the relaxed LP as a regular LP.
• Starting with the relaxed optimum, add
  constraints that iteratively modify the solution
  space to satisfy the integer requirements.

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BB and Cutting Plane Methods

• The two commonly used methods are
• Branch and bound method
• Cutting Plane Method
• Neither method is consistently effective but
  BB is far more successful.

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Branch-and-Bound (BB)

• Developed in 1960 by A Land and G Doig
• Relax the integer restrictions in the problem and
  solve it as a regular LP. Lets call this LP0 (to
  imply node-zero LP)
• Test if integer requirements are met. Else branch
  to get sub-problems LP1 and LP2.

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Branching

• If LP0 (in general LPi) fails to yield integer
  solution, branch on any variable that fails to
  meet this requirement. The process of branching
  is illustrated below.
• If LPi yields x1 3.5 and x1 is taken as the
  branching variable, we get two sub-problems,
  LPi1 LPi (x1? 3) and LPi2 LPi (x1? 4).
• Note In mixed integer problems, a continuous
  variable is never selected for branching.

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Bounding / Fathoming

• Select LP1 (in general LPi) and solve. Three
  conditions arise.
• Infeasible solution, declare fathomed (no further
  investigation of LPi).
• Integer solution. If it is superior to the
  current best solution update the current best.
  Declare fathomed.
• Non-integer solution. If it is inferior to the
  current best, declare fathomed. Else branch again.

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Best Bound

• In maximisation, the solution to a sub-problem is
  superior if it raises the current lower bound.
• In minimisation, the solution to a sub-problem is
  superior if it lowers the current upper bound.
• When all sub-problems have been fathomed, stop.
  The current bound is the best bound.

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BB Tree for Eastborne
LP0
T 2.48, A 3.25,
Z 73.57
Non-integer, non-inferior to current best, branch
on T
LP1 LP0 T 2 T 2, A
3.3, Z 69.5 Non-integer, cant give
better solution than LP5, fathomed
LP2 LP0 T 3 T 3, A
2.89 Z 73.28 Non-integer, non-inferior to
current best, branch on A
LP3 LP0 T 3 A 2 T 4.26, A 2,
Z 72.55 Non-integer, non-inferior to current
best, branch on T
LP4 LP0 T 3 A 3 Infeasible, fathomed
LP5 LP0 T ? 3,4 A 2 T 4, A 2,
Z 70 Integer, Lower (best) bound
LP6 LP0 T 5 A 2
T 5, A 1.48, Z 72.13
Cant give better solution than LP5,
fathomed
Note Z is a multiple of 5 and hence only Z 75
can be better than z 70
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LP0 T 2.48, A 3.25, Z 73.57
Non-integer, non-inferior to current best, branch
on T
LP1 LP0 T 2 T 2, A 3.3, Z 69.5
Non-integer, cant give better solution
than LP5, fathomed
LP2 LP0 T 3
T 3, A 2.89 Z 73.28 Non-integer,
non-inferior to current best, branch on A
LP3 LP0 T 3 A 2 T 4.26, A 2,
Z 72.55 Non-integer, non-inferior to current
best, branch on T
LP4 LP0 T 3 A 3 Infeasible, fathomed
LP6 LP0 T 5 A 2 T
5, A 1.48, Z 72.13 Cant give better
solution than LP5, fathomed
LP5 LP0 T ? 3,4 A 2 T 4, A 2,
Z 70 Integer, Lower (best) bound
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• See more illustrations of Branch-and-Bound
  algorithm in the Excel sheet

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0-1 Integer LP

• 0 -1 decision variables are used in problems
  where an Yes-No decision is to be taken regarding
  multiple choices.
• If the variable is 1, the corresponding choice
  is selected if the variable is 0, it is not
  selected.

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0-1 ILP Capital Budgeting
Five projects are being evaluated over a three
year planning horizon. The table gives the
expected returns for each project and associated
yearly expenditures. Which project should be
selected over the 3-year horizon?
Expenditure in million /year Expenditure in million /year Expenditure in million /year Returns, million
Project Year 1 Year 2 Year 3 Returns, million
1 5 1 8 20
2 4 7 10 40
3 3 9 2 20
4 7 4 1 15
5 8 6 10 30
Available funds, million 25 25 25
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0-1 ILP Model

• Maximise 20 P1 40 P2 20 P3 15 P4 30 P5
• subject to
• 5 P1 4 P2 3 P3 7 P4
  8 P5 lt 25
• 1 P1 7 P2 9 P3 4 P4
  6 P5 lt 25
• 8 P1 10 P2 2 P3 1 P4
  10 P5 lt 25
• P1,
  P2, P3, P4, P5 0,1

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• The Ice Cold refrigerator Company is considering
  investing in several projects that have varying
  capital requirements over the next 4 years. Faced
  with limited capital each year, management would
  like to select the most profitable projects. The
  estimated net present value for each project, the
  capital requirements, and the available capacity
  over the four year period are shown

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Projects
Plant expansion Warehouse expansion New Machinery New product research Total capital available
Present value 90000 40000 10000 37000
Yr1 15000 10000 10000 15000 40000
Yr2 20000 15000 10000 50000
Yr3 20000 20000 10000 40000
yr4 15000 5000 4000 10000 35000
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Fixed Cost

• Three raw materials are used for a company to
  produce three products a fuel additive, a
  solvent base, a carpet cleaning. The profit
  contribution are 40 per ton for the fuel
  additive, 30 per ton for the solvent base, and
  50 per ton for the cleaning fluid. Each ton of
  fuel additive is a blend of 0.4 tons of material
  1 and 0.6 tons of material 3. Each ton of
  solvent base is a blend of 0.5 of material 1, 0.2
  of material 2, 0.3 of material 3, Each ton of
  carpet cleaning fluid is a blend of 0.6 of
  material 1, 0.1 of material 2, 0.3 of material 3.

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• F tons of fuel additive used
• S tons of solvent base used
• C tons of carpet cleaning used.
• Max 40F30S50C
• s.t. 0.4F0.5S0.6Clt20
• 0.2S0.1Clt5
• 0.6F0.3S0.3Clt21
• F,S,Cgt0

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product Set up cost Maximum Production
Fuel additive 200 50 tons
Solvent base 50 25 tons
Carpet cleaning fluid 400 40 tons

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• SF 1 if fuel additive is produced ,0 else
• SS 1 if fuel additive is produced ,0 else
• SC 1 if fuel additive is produced ,0 else
• Max 40F30S50C-200SF-50SS-400SC (net profit)
• s.t .
• Flt50SF or F-50SF lt0 Maximum F
• Slt25SS or S-25SS lt0 Maximum S
• Clt40SC or C-40SC lt0 Maximum C

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Distribution Systems Design

• The Martin-Beck Company operates a plant in
• St. Louis with an annual capacity of 30,000
  units. Product is shipped to regional centers
  located in Boston, Atlanta, Houston. Bcos of an
  anticipated increase in demand, Martin-Beck plans
  to increase the capacity by constructing a new
  plant in one or more of the following cities
  Detroit, Toledo, Denver or Kansas City. The
  estimated annual fixed cost and the annual
  capacity for the 4 proposed plants is as follows
  

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Proposed plant Annual fixed cost Annual Capacity
Detroit 175000 10000
Toledo 300000 20000
Denver 375000 30000
Kansas city 500000 40000
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Annual Demand of the Distribution Centers
Distribution Centers Annual Demand
Boston 30000
Atlanta 20000
Houston 20000
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Plant size Boston Atlanta Houston
Detroit 5 2 3
Toledo 4 3 2
Denver 9 7 5
Kansas city 10 4 2
St.louis 8 4 3
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• Max
• 5x112x123x134x213x224x239x317x325x3310x41
  4x422x438x514x523x53175y1300y2375y3500y4
• y11 if a plant is constructed in Detroit, 0
  else y21 if a plant is constructed
  in Toledo, 0 else
• y31 if a plant is constructed in Denver, 0
  else y41 if a plant is constructed in
  Kansas city, 0 else

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• s.t.
• x11x12x13lt10y1 or x11x12x13-10y1lt0
• x21x22x23lt20y1 or x21x22x23-20y1
• x31x32x33lt30y1 or x31x32x33-30y1
• x41x42x43lt40y1 or x41x42x43-40y1
• x51x52x53lt30
• x11x21x31x41x5130
• x12x22x32x42x5220
• x13x23x33x43x5320

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Branch Location

• The long-range planning department for the Ohio
  Trust Company is considering expanding its
  operation into a 20-county region in NE Ohio.
  Currently, Ohio Trust does not have a principal
  place of business in any of the 20 counties.
  According to the banking laws in Ohio, if a bank
  establishes a principal place of business (PPB)
  in any county, branch banks can be estd in that
  county and in any adjacent county. However, to
  establish a new PPB, Ohio Trust must either
  obtain approval for a new bank from the states
  superintendent of banks or purchase an existing
  bank.

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  COUNTIES IN THE OHIO TRUST EXPANSION REGION COUNTIES IN THE OHIO TRUST EXPANSION REGION
  COUNTIES UNDER CONSIDERATION ADJACENT COUNTIES
1 Ashtabula 2,12,16
2 Lake 1,3,12
3 Cuyahoga 2,4,9,10,12,13
4 Lorain 3,5,7,9
5 Huron 4,6,7
6 Richland 5,7,17
7 Ashland 4,5,6,8,9,17,18
8 Wayne 7,9,10,11,18
9 Medina 3,4,7,8,10
10 Summit 3,8,9,11,12,13
11 Stark 8,10,13,14,15,18,19,20
12 Geauga 1,2,3,10,13,16
13 portage 3,10,11,12,15,16
14 Columbians 11,15,20
15 Mahoning 11,13,14,16
16 Trumbull 1,12,13,15
17 Knox 6,7,18
18 Holmes 7,8,11,17,19
19 Tuscarawas 11,18,20
20 Carroll 11,14,19
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• Decision variables
• Let xi 1 if a PBB is estd in county i 0
  otherwise
• Objective Function
• Minimise Z x1x2x3.,.,.x20
• Subject to
• x1 x2 x12 x16 gt 1
  (Ashtabula)
• x1 x2 x3 x12 gt 1
  (Lake)
• 
  .,.,,.,.,.,(20 constraints)
• Non-negativity
• Xi 0,1
• i 1,2,.,., 20

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• See more illustrations of
• 0-1 ILP
• and solution using
• Excel Solver

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Reference

• Hamdy A Taha, Operations Research An
  Introduction Pearson
• Anderson, Sweeny and Williams, An Introduction
  to Management Science Thomson
• Thank you for listening