A NETFORM APPROACH FOR COMBINATORIAL OPTIMIZATION PROBLEMS

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A NETFORM APPROACH FOR COMBINATORIAL OPTIMIZATION
PROBLEMS
FRED GLOVER GARY KOCHENBERGER MEHDI
AMINI Hearin Center for Enterprise
Sciences, School of Business Administration,
University of Mississippi. Marketing SCM
Department, Fogelman College of Business
Economics, The University of Memphis.
JUNE 3-5, 2002 MAS V Military Personnel
Research Science Workshop ONR GRANT
N00014-00-1-0917
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A NETFROM APPROACH FOR COMBINATORIAL OPTIMIZATION
PROBLEMS

• OVERVIEW
• Research Motivations and Objectives
• Discrete Generalized NETFORM Modeling Approach
• Constraint Satisfiability Problem (CSP)
• Assignment/Detailing
• Generalized Assignment Problem (GAP)
• Fixed Charge Transportation Problem (FCTP)
• Other MIP Optimization Problems
• Generalized Discrete NETFORM Solution Approach
• Basic BB Method
• Dynamic BB Method
• Computational Results
• Conclusions Future Research Directions

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A NETFROM APPROACH FOR COMBINATORIAL OPTIMIZATION
PROBLEMS
MOTIVATIONS OBJECTIVES
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A NETFROM APPROACH FOR COMBINATORIAL OPTIMIZATION
PROBLEMSMOTIVATIONS

• Netform Defined Network flow-based (or
  network-related) problem formulation. Glover
  etal. (1992)
• Advantages
• Visual modeling capabilities improved
• Communication between modelers and users
• Model fidelity
• Sensitivity analysis
• Implementation of results
• Solution methods capabilities improved
• Approaching large-scale real-world problems
  routinely
• Solution time 50 to 200 times faster than
  general LP method

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A NETFROM APPROACH FOR COMBINATORIAL OPTIMIZATION
PROBLEMSMOTIVATIONS OBJECTIVES

• Harvesting proven solution efficiency of the
  generalized NETFORM approach in solving discrete
  optimization problems.

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A NETFROM APPROACH FOR COMBINATORIAL OPTIMIZATION
PROBLEMSOBJECTIVES

• Further emphasis on the potential of NETFORM
  approach as a unified modeling technique for
  continuous discrete optimization problems
• Development and testing of a new Discrete
  Generalized NETFORM approach for discrete
  optimization problems

DYNAMIC BRANCH AND BOUND METHOD
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A NETFROM APPROACH FOR COMBINATORIAL OPTIMIZATION
PROBLEMS
Discrete Generalized NETFORM Modeling Approach
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A NETFROM APPROACH FOR COMBINATORIAL OPTIMIZATION
PROBLEMS DISCRETE OPTIMIZATION PROBLEMS OF
INTEREST

• Constraint Satisfiability Problem (CSP)
• No embedded network structure
• Generalized Assignment Problem (GAP)
• Full embedded network structure with side
  constraints
• Fixed-Charge Transportation Problem (FCTP)
• Full embedded generalized network structure

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CONSTRAINT SATISFIABILITY PROBLEM NETFROM
MODELING APPROACH POTENTIALS

• NETFORM NOTATION
• NETFROM MODELING APPROACH POTENTIALS
• Any linear constrained optimization problem
  including binary, pure, and mixed integer
  programming problems can be modeled as a NETFORM.

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CONSTRAINT SATISFIABILITY PROBLEM NETFROM
MODELING APPROACH POTENTIALS

• EXAMPLE A DSICRETE ZER-ONE INTEGER PROGRAMMING
• PROBLEM Glover(1992)

A BINARY IP
Minimize 3x1 4x2 7x3 5x4
Subject to 1
2x1 4x3 - 4x4 ? 2 2
-1x1 10x2 - 2x3 - 6x4
? 7 3 4x1 1x2
4x4 5 4
1x3 1x4 ? 1
x1 , x2 , x3 , x4 0 or 1
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CONSTRAINT SATISFIABILITY PROBLEM NETFROM
MODELING APPROACH POTENTIALS

• EXAMPLE A DSICRETE ZER-ONE INTEGER PROGRAMMING
• PROBLEM Glover(1992)

EQUIVALENT NETFORM
0 (0, 1) 2
(2, ?)
x1
1
(0, 1) 3
0 (0, 1) -1
(0, 7)
x2
2
3
0 (0, 1) -4
4 (0, 1) 2
0 (0, 1) 4
(0, 4)
0 (0, 1) -6
0
7 (0, 1) 3
(5, 4)
x3
3
0 (0, 1) 4
5 (0, 1) 4
(1, ?)
x4
4
0 (0, 1) 1
Variable Arcs Variable Nodes
Constraint Arcs Constraint Nodes
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I. CONSTRAINT SATISFIABILITY PROBLEMA
MATHEMATICAL FORMULATION
Let, A A constraint coefficient matrix in
Rmxn x A decision variable vector in Rn b A
right-hand-side vector in Rm D Decision
variables finite value-domain in Rk A CSP can be
defined, Ax ? b x ? D
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CONSTRAINT SATISFIABILITY PROBLEMOPTIMIZATION
PROBLEM AS A SEQUENCE OF CSPs
Let c A cost vector in Rn A A constraint
coefficient matrix in Rmxn x A decision
variable vector in Rn b A right-hand-side
vector in Rm D Decision variables finite
value-domain in Rk z A threshold value for the
objective function
Optimization Problem A CSP
Equivalent Minimize cx cx ? z Subject
to Ax ? b Ax ? b x ? D x ?
D
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CONSTRAINT SATISFIABILITY PROBLEM NETFROM
MODELING APPROACH POTENTIALS

• EXAMPLE A CSP Kochenberger (2000)

A CSP
1 x1 x2
? 1 2 x1
x3 ? 1 3
x2 x3 ?
1 4 x1 x2
? 1 5
x1 - x3 ? 0
x1 , x2 , x3 0 or 1
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CONSTRAINT SATISFIABILITY PROBLEM NETFROM
MODELING APPROACH POTENTIALS

• EXAMPLE A CSP Kochenberger (2000)

NETFORM EQUIVALENT
0 (0, 1) 1
(1, ?)
x1
1
(0, 1) 4
0 (0, 1) 1
(1, ?)
2
0
0 (0, 1) 1
(0, 3)
0 (0, 1) 3
0
x2
0 (0, 1) 1
(1, ?)
3
0 (0, 1) 3
(0, 1)
4
x3
0 (0, 1) -1
(0, ?)
5
Variable Arcs Variable Nodes
Constraint Arcs Constraint Nodes
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II. GENERALIZED ASSIGNMENT PROBLEMA
MATHEMATICAL FORMULATION
Let, J Index set of tasks, J 1, , n, I
Index set of agents, I 1, , m, c A cost
vector in Rmxn x A binary decision variable
vector in Rn a A task-load vector in Rmxn b A
resource vector in Rm A GAP can be defined,
Minimize ?Im1
?jn1 cij xij Subject to ?jn1 aij xij
? bi, For? i ? I, ?im1 xij 1,
For j ? J, xij ? 0, 1,
For? i ? I ? j ? J.
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GENERALIZED ASSIGNMENT PROBLEM A NETFROM MODEL

• EXAMPLE A GAP Kochenberger (2000)

A GAP EXAMPLE
0 Minimize 17x11 21x12 22x13 18x14
24x15 23x21 16x22 21x23 16x24 17x25
16x31 20x32 16x33 25x34 24x35
Subject to 1 x11
x21 x31 1 2 x12
x22 x32 1 3
x13 x23 x33 1 4
x14 x24 x34 1 5
x15 x25 x35 1 6 8x11
15x12 14x13 23x14 8x15 ? 36 7 15x21
7x22 23x23 22x24 11x25 ? 34 8 21x31
20x32 6x33 22x34 24x35 ? 38 10 x11 ,
x12 , x13 , x14, x15, x21 , x22 , x23 , x24 ,
x25, x31, x32 , x33 , x34, x35 0 or 1
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GENERALIZED ASSIGNMENT PROBLEM NETFROM MODELING
APPROACH POTENTIALS

• EXAMPLE A GAP Kochenberger (2000)

NETFORM EQUIVALENT
17 (0, 1) 8
(1, 1)
(1, 36)
1
1
(1, 1)
2
16 (0, 1) 21
(1, 34)
(1, 1)
3
2
(1, 1)
4
(0, 38)
24 (0, 1) 24
3
5
(1, 1)
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III. FIXED-CHARGE TRANSPORTATION PROBLEMA
MATHEMATICAL FORMULATION
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FIXED-CHARGE TRANSPORTATION PROBLEM NETFROM
MODELING APPROACH POTENTIALS
                                                 

 
SUPPLY ORIGINE
DESTINATION DEMAND

VC, FC, LB, UB
S1
D1
20
15
2, 30, 0, 15
3, 10, 0, 10
1, 20, 0, 20
10
D2
3, 30, 0, 15
2, 10, 0, 10
30
25
S2
D3
2, 50, 0, 25
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FIXED-CHARGE TRANSPORTATION PROBLEM NETFROM
MODELING APPROACH POTENTIALS
                                                 

SUPPLY ORIGINE

DESTINATION DEMAND
AMULT,C, LB, UB,
BMULT -UB,VCUBFC, 0, 1, UB,
-1,-VC,LB,UB,1
 
-1,-2,0,15,1
20
-1,0,0,15,1
1
9
15
3
-15,60,0,1,15,
4
5
10
10
6
7
30
25
-25,100,0,1,25,
2
11
8
-1,-2,0,25,1
-10,0,25,1
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A NETFROM APPROACH FOR COMBINATORIAL OPTIMIZATION
PROBLEMS
Discrete Generalized NETFORM Solution Approach
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NETFROM SOLUTION APPROACHES POTENTIALS

• Current NETFORM solution approaches can solve
• Pure NETFORM as much as 100 to 200 times faster
  than the most sophisticated general-purpose
  linear programming codes.
• Generalized NETFORM as much as 50 times faster
  than the most sophisticated general-purpose
  linear programming codes.
• To date, there is no discrete generalized NETFORM
  algorithm and/or code available!

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A NEW DISCRETE NETFROM SOLUTION APPROACHEA
DYNAMIC BRANCH-AND-BOUND METHOD Glover 2002

• Consider a basic branch-and-bound procedure with
• Branching method
• Tentative Branch Exploration (TBE)
• Basic idea Postpone taking a branch until more
  information is generated about its consequences.
• Backtracking method
• Dynamic Retrospective Choices (DRC)
• Basic idea Applying information at a deeper
  level of the BB tree, and employing it to
  re-structure the tree!

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A NEW DISCRETE NETFROM SOLUTION APPROACHEA
DYNAMIC BRANCH-AND-BOUND METHOD Glover 2002

• Branching method Tentative Branch Exploration
  (TBE)

. . .
s
Down-Branch
Up-Branch
u
t
Infeasible!
Feasible!
Variable Selection Rule Select the variable,
arc, with the largest worst and best
penalties difference. Select the branch with
best penalty.






. . .
Calculate Penalties for Up Down tentative
Branches classify them as the best Worst
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A NEW DISCRETE NETFROM SOLUTION APPROACHEA
DYNAMIC BRANCH-AND-BOUND METHOD Glover 2002

• Backtracking method First-Depth Rule

s
Down-Branch
Up-Branch
u
t
Infeasible!
Feasible!
v
Feasible!
z
w
Feasible!
x
y
Infeasible!
Infeasible!
BACKTRACK!
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A NEW DISCRETE NETFROM SOLUTION APPROACHEA
DYNAMIC BRANCH-AND-BOUND METHOD Glover 2002

• Backtracking method Dynamic Retrospective
  Choices (DRC)

Feasible!
s
Up-Branch (Forced)
Selected branch! Re-structure the BB tree!
Down-Branch
u
t
Infeasible!
Feasible!
Down-Branch (Chosen)
True Reduced Cost
v
Feasible!

Down-Branch (Chosen)
True reduced Cost

w
Feasible!
Branch selection Rule The branch with the
worst true reduced cost.
x
y
Infeasible!
Infeasible!
BACKTRACK!
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A NEW DISCRETE NETFROM SOLUTION APPROACHEA
DYNAMIC BRANCH-AND-BOUND METHOD Glover 2002

• Backtracking method Dynamic Retrospective
  Choices (DRC)

Feasible!
s
Up-Branch (Forced)
Down-Branch
v
t
Infeasible!
Feasible!
Down-Branch (Chosen)
w
Feasible!
z
Re-structured BB Tree!
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A NEW DISCRETE NETFROM SOLUTION APPROACHEA
DYNAMIC BRANCH-AND-BOUND METHOD

• The basic implementation issue Managing discrete
  arcs flows, bounds, and primal feasibility
  throughout the solution process
• A DUAL-ARC CONSTRUCTION

A DISCRETE ARC-PAIR
P
C
0, U
Basic
F
L
Nonbasic
S
B
0
0, 0
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A NEW DISCRETE NETFROM SOLUTION APPROACHEA
DYNAMIC BRANCH-AND-BOUND METHOD

• A DUAL-ARC CONSTRUCTION EXAMPLE

A DISCRETE ARC-PAIR WITH FRACTIONAL FLOW
Basic
NB_at_0
Basic
A ROUND UP BRANCH CHOICE IS EXECUTED
Pre-Post-Opt Conversion Forcing fractional
flow, 0.80 to 1.
NB_at_0
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A NEW DISCRETE NETFROM SOLUTION APPROACHEA
DYNAMIC BRANCH-AND-BOUND METHOD

• A DUAL-ARC CONSTRUCTION EXAMPLE

Basic/NB_at_UB
ROUNDUP BRANCH FEASIBLE!
NB_at_0
Basic/NB_at_0
POST-OPT CONVERSION
NB_at_0
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A NEW DISCRETE NETFROM SOLUTION APPROACHEA
DYNAMIC BRANCH-AND-BOUND METHOD

• A DUAL-ARC CONSTRUCTION EXAMPLE

A DISCRETE ARC-PAIR WITH FRACTIONAL FLOW
Basic
NB_at_0
Basic
A ROUND DOWN BRANCH CHOICE IS EXECUTED
Pre-Post-Opt Conversion Forcing fractional
flow, 0.15 to 0.
NB_at_UB
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A NEW DISCRETE NETFROM SOLUTION APPROACHEA
DYNAMIC BRANCH-AND-BOUND METHOD

• A DUAL-ARC CONSTRUCTION EXAMPLE

Basic
ROUNDDOWN BRANCH INFEASIBLE!
NB_at_UB
Basic
POST-OPT CONVERSION Free the Round Down Branch!
NB_at_0
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A NETFROM APPROACH FOR COMBINATORIAL OPTIMIZATION
PROBLEMS
COMPUTATIONAL RESULTS
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A NETFROM APPROACH FOR COMBINATORIAL OPTIMIZATION
PROBLEMS

• Discrete Generalized NETFORM algorithmic
  implementations were developed and compiled under
  Visual FORTRAN 6.1 within the MS Visual
  Development Environment.
• Codes compiled with Full Optimization option.
• Computational experimentations were completed on
• Dell Latitude, Pentium III, 1 GHZ Laptop, under
  MS Window 2000 operating system.
• Dell Latitude, Pentium IV, 1.2 GHZ Laptop, under
  MS Window XP operating system.

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COMPUTATIONAL RESULTSCONSTRAINT SATISFIABILITY
PROBLEMSBASIC BARNCH-AND-BOUND METHOD

• AIM TESTBED PROBLEMS Miyanos 3-Sat Problems
  with
• 
  at most one solution
• Each problem instance is converted to CSP
• Each CSP instance is converted to Netform
• No embedded network structure is present in the
  AIMs problems
• Problems solved by the basic branch-and-bound
  implementation GN2DV01

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COMPUTATIONAL RESULTSAIM SATIFIABILITY
PROBLEMSBASIC BARNCH-AND-BOUND METHOD
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COMPUTATIONAL RESULTSGENERALIZED ASSIGNMENT
PROBLEMSDYNAMIC BARNCH-AND-BOUND METHOD

• GAP TESTBED PROBLEMS Beasleys C Type
  Problems
• Each problem instance is converted to Netform
• Average degree of difficulty among problem types
• A through E.
• Embedded network structure is present with the
  side constraints
• Problems solved by the basic BB and dynamic BB
  implementations GN2DV01, GN2DV2.2, AND GN2DV2.3

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COMPUTATIONAL RESULTSGAP CLASS C PROBLEMS
BASIC BB IMPLEMENTATION GN2DV1.0
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COMPUTATIONAL RESULTSGAP CLASS C PROBLEMS
BASIC BB IMPLEMENTATION GN2DV1.0
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COMPUTATIONAL RESULTSGAP CLASS C PROBLEMS
COMPARING BASIC AND DYNAMIC BB IMPLEMENTATIONS
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COMPUTATIONAL RESULTSGAP CLASS C PROBLEMS
COMPARING BASIC AND DYNAMIC BB IMPLEMENTATIONS
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COMPUTATIONAL RESULTSGAP CLASS C PROBLEMS
COMPARING BASIC AND DYNAMIC BB IMPLEMENTATIONS
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COMPUTATIONAL RESULTSFIXED-CHARGE
TRANSPORTATION PROBLEMSDYNAMIC BARNCH-AND-BOUND
METHOD

• FCTP TESTBED PROBLEMS Minghes Testbed
  Problems
• Each problem instance is converted to Netform
• Embedded generalized network structure is present
• Problem solved by the basic and dynamic BB
  implementations GN2DV1.0, GN2DV2.3

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COMPUTATIONAL RESULTSFIXED-CHARGE
TRANSPORTATION PROBLEMSBASIC BARNCH-AND-BOUND
METHOD
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COMPUTATIONAL RESULTSFIXED-CHARGE
TRANSPORTATION PROBLEMSBASIC BARNCH-AND-BOUND
METHOD
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COMPUTATIONAL RESULTSFIXED-CHARGE
TRANSPORTATION PROBLEMSBASIC BARNCH-AND-BOUND
METHOD
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COMPUTATIONAL RESULTSFCTP PROBLEMS BASIC BB
IMPLEMENTATIONFCTP 10x10
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COMPUTATIONAL RESULTSFCTP PROBLEMS DYNAMIC BB
IMPLEMENTATIONMETHOD 2FCTP 10x10
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COMPUTATIONAL RESULTSFCTP PROBLEMS DYNAMIC BB
IMPLEMENTATIONMETHODS 1 2FCTP 10x10
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A NETFROM APPROACH FOR COMBINATORIAL OPTIMIZATION
PROBLEMSAN IMPORTANT OBSERVATION

• The main advantage in solving problems is through
• Speed in solving subproblems, and
• Cleverness of solution strategies.
• The advantage gained may be mainly valuable for
  finding a good solution in a decent amount of
  time.
• The Discrete Generalized NETFORM solver has a
  low relaxation strength!
• The low relaxation strength may mean that the
  solver take a long time to find and verify
  optimality.
• There are results suggesting that the new methods
  are doing well in providing optimality!

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A NETFROM APPROACH FOR COMBINATORIAL OPTIMIZATION
PROBLEMS
CONCLUSIONS AND FUTURE WORK
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A NETFROM APPROACH FOR COMBINATORIAL OPTIMIZATION
PROBLEMS

• A new basic BB algorithm for discrete
  generalized NETFORM is devised and implemented.
• A new dynamic BB approach is devised and
  implemented.
• The NETFORM modeling capability along with the
  new algorithms provide a unified modeling and
  solution approach to the discrete optimization
  problems.
• Computational experiments with three classes of
  combinatorial optimization problems are
  conducted.

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A NETFROM APPROACH FOR COMBINATORIAL OPTIMIZATION
PROBLEMS

• Computational results show that
• The new algorithms are viable alternative
  approaches to the current discrete optimization
  methods.
• The capabilities of the new algorithms are in
• Solving relaxed subproblems very efficiently, and
  
• Identifying high quality solutions within a
  fraction of CPU seconds
• highlight an important value of the current
  solver as a
• heuristic!

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A NETFROM APPROACH FOR COMBINATORIAL OPTIMIZATION
PROBLEMS

• Future work
• Focusing on the capability of the solver as an
  improvement method within
• Tabu Search metaheuristic and
• Scatter Search metaheuristic
• For solving discrete optimization problems.