Integer Programming

1
Integer Programming
Optimisation Methods
2
Lecture Outline

• 1 Introduction
• 2 Integer Programming
• 3 Modeling with 0-1 (Binary) Variables
• 4 Goal Programming
• 5 Nonlinear Programming

3
Introduction

• Integer programming is the extension of LP that
  solves problems requiring integer solutions.
• Goal programming is the extension of LP that
  permits more than one objective to be stated.
• Nonlinear programming is the case in which
  objectives or constraints are nonlinear.
• All three above mathematical programming models
  are used when some of the basic assumptions of LP
  are made more or less restrictive.

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Summary Linear Programming Extensions

• Integer Programming
• Linear, integer solutions
• Goal Programming
• Linear, multiple objectives
• Nonlinear Programming
• Nonlinear objective and/or constraints

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Integer Programming

• Solution values must be whole numbers in integer
  programming .
• There are three types of integer programs
• pure integer programming
• mixed-integer programming and
• 01 integer programming.

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Integer Programming

• The Pure Integer Programming problems are cases
  in which all variables are required to have
  integer values.
• The Mixed-Integer Programming problems are cases
  in which some, but not all, of the decision
  variables are required to have integer values.
• The ZeroOne Integer Programming problems are
  special cases in which all the decision variables
  must have integer solution values of 0 or 1.

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9.3 The Branch-and-Bound Method for Solving Pure
Integer Programming Problems

• In practice, most IPs are solved by some versions
  of the branch-and-bound procedure.
  Branch-and-bound methods implicitly enumerate all
  possible solutions to an IP.
• By solving a single subproblem, many possible
  solutions may be eliminated from consideration.  
• Subproblems are generated by branching on an
  appropriately chosen fractional-valued variable.

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• Suppose that in a given subproblem (call it old
  subproblem), assumes a fractional value between
  the integers i and i1. Then the two newly
  generated subproblems are
• New Subproblem 1 Old subproblem Constraint
• New Subproblem 2 Old subproblem Constraint
• Key aspects of the branch-and-bound method for
  solving pure IPs
• If it is unnecessary to branch on a subproblem,
  we say that it is fathomed.

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• These three situations (for a max problem) result
  in a subproblem being fathomed
• The subproblem is infeasible, thus it cannot
  yield the optimal solution to the IP.
• The subproblem yield an optimal solution in which
  all variables have integer values. If this
  optimal solution has a better z-value than any
  previously obtained solution that is feasible in
  the IP, than it becomes a candidate solution, and
  its z-value becomes the current lower bound (LB)
  on the optimal z-value for the IP.
• The optimal z-value for the subproblem does not
  exceed (in a max problem) the current LB, so it
  may be eliminated from consideration.
• A subproblem may be eliminated from consideration
  in these situations
• The subproblem is infeasible.
• The LB is at least as large as the z-value for
  the subproblem

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• Two general approaches are used to determine
  which subproblem should be solved next.
• The most widely used is LIFO.
• LIFO leads us down one side of the
  branch-and-bound tree and quickly find a
  candidate solution and then we backtrack our way
  up to the top of the other side
• The LIFO approach is often called backtracking.
• The second commonly used approach is
  jumptracking.
• When branching on a node, the jumptracking method
  solves all the problems created by branching.

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• When solving IP problems using Solver you can
  adjust a Solver tolerance setting.
• The setting is found under the Options.
• For example a tolerance value of .20 causes the
  Solver to stop when a feasible solution is found
  that has an objective function value within 20
  of the optimal solution.

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Harrison Electric Company

• The Company produces two products popular with
  home renovators old-fashioned chandeliers and
  ceiling fans.
• Both the chandeliers and fans require a two-step
  production process involving wiring and assembly.
• It takes about 2 hours to wire each chandelier
  and 3 hours to wire a ceiling fan. Final assembly
  of the chandeliers and fans requires 6 and 5
  hours, respectively.
• The production capability is such that only 12
  hours of wiring time and 30 hours of assembly
  time are available.

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Harrison Electric Company
If each chandelier produced nets the firm 7 and
each fan 6, Harrisons production mix decision
can be formulated using LP as follows
maximize profit 7X1 6X2
subject to 2X1 3X2 12 (wiring hours) 6X1
5X2 30 (assembly hours) X1, X2 0
(nonnegative) X1 number of chandeliers
produced X2 number of ceiling fans produced
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Harrison Electric Company
With only two variables and two constraints, the
graphical LP approach to generate the optimal
solution is given below
6X1 5X2 30
Possible Integer Solution
Optimal LP Solution (X1 33/4, X2 11/2,
Profit 35.25
2X1 3X2 12
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Integer Solution to Harrison Electric Co.
Optimal solution
Solution if rounding off
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Integer Programming

• Rounding off is one way to reach integer solution
  values, but it often does not yield the best
  solution.
• An important concept to understand is that an
  integer programming solution can never be better
  than the solution to the same LP problem.
• The integer problem is usually worse in terms of
  higher cost or lower profit.

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Branch and Bound Method

• Branch and Bound breaks the feasible solution
  region into sub-problems until an optimal
  solution is found.
• There are Six Steps in Solving Integer
  Programming Maximization Problems by Branch and
  Bound.
• The steps are given over the next several slides.

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Branch and Bound Method The Six Steps

• Solve the original problem using LP.
• If the answer satisfies the integer constraints,
  it is done.
• If not, this value provides an initial upper
  bound.
• Find any feasible solution that meets the integer
  constraints for use as a lower bound.
• Usually, rounding down each variable will
  accomplish this.

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Branch and Bound Method Steps

• Branch on one variable from Step 1 that does not
  have an integer value.
• Split the problem into two sub-problems based on
  integer values that are immediately above and
  below the non-integer value.
• For example, if X2 3.75 was in the final LP
  solution, introduce the constraint X2 4 in the
  first sub-problem and X2 3 in the second
  sub-problem.
• Create nodes at the top of these new branches by
  solving the new problems.

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Branch and Bound Method Steps

• 5.
• If a branch yields a solution to the LP problem
  that is not feasible, terminate the branch.
• If a branch yields a solution to the LP problem
  that is feasible, but not an integer solution, go
  to step 6.

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Branch and Bound Method Steps

• 5. (continued)
• If the branch yields a feasible integer solution,
  examine the value of the objective function.
• If this value equals the upper bound, an optimal
  solution has been reached.
• If it is not equal to the upper bound, but
  exceeds the lower bound, set it as the new lower
  bound and go to step 6.
• Finally, if it is less than the lower bound,
  terminate this branch.

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Branch and Bound Method Steps

• Examine both branches again and set the upper
  bound equal to the maximum value of the objective
  function at all final nodes.
• If the upper bound equals the lower bound, stop.
• If not, go back to step 3.

Minimization problems involve reversing the roles
of the upper and lower bounds.
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Harrison Electric Co
Figure 11.1 shows graphically that the optimal,
non-integer solution is X1 3.75
chandeliers X2 1.5 ceiling fans
profit 35.25

• Since X1 and X2 are not integers, this solution
  is not valid.
• The profit value of 35.25 will serve as an
  initial upper bound.
• Note that rounding down gives X1 3, X2 1,
  profit 27, which is feasible and can be used
  as a lower bound.

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Integer Solution Creating Sub-problems

• The problem is now divided into two sub-problems
  A and B.
• Consider branching on either variable that does
  not have an integer solution pick X1 this time.

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Optimal Solution for Sub-problems
Optimal solutions are Sub-problem A X1 4 X2
1.2, profit35.20 Sub-problem B X13, X22,
profit33.00 (see figure on next slide)

• Stop searching on the Subproblem B branch because
  it has an all-integer feasible solution.
• The 33 profit becomes the lower bound.
• Subproblem As branch is searched further since
  it has a non-integer solution.
• The second upper bound becomes 35.20, replacing
  35.25 from the first node.

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Optimal Solution for Sub-problem
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Sub-problems C and D
Subproblem As branching yields Subproblems C and
D.
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Sub-problems C and D (continued)

• Subproblem C has no feasible solution at all
  because the first two constraints are violated if
  the X1 4 and X2 2 constraints are observed.
• Terminate this branch and do not consider its
  solution.
• Subproblem Ds optimal solution is
• X1 4 , X2 1, profit 35.16.
• This non-integer solution yields a new upper
  bound of 35.16, replacing the original 35.20.
• Subproblems C and D, as well as the final
  branches for the problem, are shown in the figure
  on the next slide.

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Harrison Electrics FullBranch and Bound
Solution
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Subproblems E and F

• Finally, create subproblems E and F and solve for
  X1 and X2 with the added constraints X1 4 and
  X1 5. The subproblems and their solutions are

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Subproblems E and F (continued)
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Goal Programming

• Firms usually have more than one goal. For
  example,
• maximizing total profit,
• maximizing market share,
• maintaining full employment,
• providing quality ecological management,
• minimizing noise level in the neighborhood, and
• meeting numerous other non-economic goals.
• It is not possible for LP to have multiple goals
  unless they are all measured in the same units
  (such as dollars),
• a highly unusual situation.
• An important technique that has been developed to
  supplement LP is called goal programming.

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Goal Programming (continued)

• Goal programming satisfices,
• as opposed to LP, which tries to optimize.
• Satisfice means coming as close as possible to
  reaching goals.
• The objective function is the main difference
  between goal programming and LP.
• In goal programming, the purpose is to minimize
  deviational variables,
• which are the only terms in the objective
  function.

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Example of Goal Programming
Harrison Electric Revisited

• Goals Harrisons management wants to
• achieve, each equal in priority
• Goal 1 to produce as much profit above 30 as
  possible during the production period.
• Goal 2 to fully utilize the available wiring
  department hours.
• Goal 3 to avoid overtime in the assembly
  department.
• Goal 4 to meet a contract requirement to produce
  at least seven ceiling fans.

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Example of Goal Programming
Harrison Electric Revisited
Need a clear definition of deviational variables,
such as
d1 underachievement of the profit target d1
overachievement of the profit target d2 idle
time in the wiring dept. (underused) d2
overtime in the wiring dept. (overused) d3
idle time in the assembly dept. (underused) d3
overtime in the wiring dept. (overused) d4
underachievement of the ceiling fan goal d4
overachievement of the ceiling fan goal
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Ranking Goals with Priority Levels

• A key idea in goal programming is that one
  goal is more important than another. Priorities
  are assigned to each deviational variable.

Priority 1 is infinitely more important than
Priority 2, which is infinitely more important
than the next goal, and so on.
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Analysis of First Goal
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Analysis of First and Second Goals
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Analysis of All Four Priority Goals
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Goal Programming Versus Linear Programming

• Multiple goals (instead of one goal)
• Deviational variables minimized (instead of
  maximizing profit or minimizing cost of LP)
• Satisficing (instead of optimizing)
• Deviational variables are real (and replace slack
  variables)

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9.2 Formulating Integer Programming Problems

• Practical solutions can be formulated as IPs.
• The basics of formulating an IP model

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Example 1 Capital Budgeting IP

• Stockco is considering four investments
• Each investment
• Yields a determined NPV
• Requires a certain cash flow at the present time
• Currently Stockco has 14,000 available for
  investment.
• Formulate an IP whose solution will tell Stockco
  how to maximize the NPV obtained from the four
  investments.

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Example 1 Solution

• Begin by defining a variable for each decision
  that Stockco must make.
• The NPV obtained by Stockco is Total NPV
  obtained by Stocko 16x1 22x2 12x3 8x4
• Stockcos objective function is max z 16x1
  22x2 12x3 8x4
• Stockco faces the constraint that at most 14,000
  can be invested.
• Stockcos 0-1 IP is

max z 16x1 22x2 12x3 8x4 s.t. 5x1
7x2 4x3 3x4 14 xj 0 or 1 (j 1,2,3,4)
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Set Covering as an IP

• In a set-covering problem, each member of a given
  set must be covered by an acceptable member of
  some set.
• The objective of a set-covering problem is to
  minimize the number of elements in set 2 that are
  required to cover all the elements in set 1.

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Piece-wise linear functions as IP

• 0-1 variables can be used to model optimization
  problems involving piecewise linear functions.
• A piecewise linear function consists of several
  straight line segments.
• The graph of the piecewise linear function is
  made of straight-line segments.
• The points where the slope of the piecewise
  linear function changes are called the break
  points of the function.
• A piecewise linear function is not a linear
  function so linear programming cannot be used to
  solve the optimization problem.

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Piece-wise linear functions as IP

• By using 0-1 variables, however, a piecewise
  linear function can be represented in a linear
  form. Suppose the piecewise linear function f (x)
  has break points .

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Piece-wise linear functions as IP

• Suppose the piecewise linear function f (x) has
  break points .
• Step 1 Wherever f (x) occurs in the optimization
  problem, replace f (x) by
  .
• Step 2 Add the following constraints to the
  problem

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Piece-wise linear functions as IP

• If a piecewise linear function f(x) involved in a
  formulation has the property that the slope of
  the f(x) becomes less favorable to the decision
  maker as x increases, then the tedious IP
  formulation is unnecessary.
• LINDO can be used to solve pure and mixed IPs.
• In addition to the optimal solution, the LINDO
  output also includes shadow prices and reduced
  costs.
• LINGO and the Excel Solver can also be used to
  solve IPs.

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9.4 The Branch-and-Bound Method for Solving Mixed
Integer Programming Problems

• In mixed IP, some variables are required to be
  integers and others are allowed to be either
  integer or non-integers.
• To solve a mixed IP by the branch-and-bound
  method, modify the method by branching only on
  variables that are required to be integers.
• For a solution to a subproblem to be a candidate
  solution, it need only assign integer values to
  those variables that are required to be integers

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9.5 Solving Knapsack Problems by the
Branch-and-Bound Method

• A knapsack problem is an IP with a single
  constraint.
• A knapsack problem in which each variable must be
  equal to 0 or 1 may be written as
• When knapsack problems are solved by the
  branch-and-bound method, two aspects of the
  method greatly simplify.
• Due to each variable equaling 0 or 1, branching
  on xi will yield in xi 0 and an xi 1 branch.
• The LP relaxation may be solved by inspection.

max z c1x1 c2x2 cnxn s.t. a1x1
a2x2 anxn b x1 0 or 1 (i
1, 2, , n)
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9.6 Solving Combinatorial Optimization Problems
by the Branch-and-Bound Method

• A combinatorial optimization problem is any
  optimization problem that has a finite number of
  feasible solutions.
• A branch-and-bound approach is often the most
  efficient way to solve them.
• Examples of combinatorial optimization problems
• Ten jobs must be processed on a single machine.
  It is known how long it takes to complete each
  job and the time at which each job must be
  completed. What ordering of the jobs minimizes
  the total delay of the 10 jobs?

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• A salesperson must visit each of the 10 cities
  before returning to his home. What ordering of
  the cities minimizes the total distance the
  salesperson must travel before returning home?
  This problem is called the traveling sales person
  problem (TSP).
• In each of these problems, many possible
  solutions must be considered.

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• When using branch-and-bound methods to solve TSPs
  with many cities, large amounts of computer time
  is needed.
• Heuristic methods, or heuristics, can be used to
  quickly lead to a good solution.
• Heuristics is a method used to solve a problem by
  trial and error when an algorithm approach is
  impractical.
• Two types of heuristic methods can be used to
  solve TSP nearest neighbor method and
  cheapest-insertion method.

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• Nearest Neighbor Method
• Begin at any city and then visit the nearest
  city.
• Then go to the unvisited city closest to the city
  we have most recently visited.
• Continue in this fashion until a tour is
  obtained. After applying this procedure beginning
  at each city, take the best tour found.
• Cheapest Insertion Method (CIM)
• Begin at any city and find its closest neighbor.
• Then create a subtour joining those two cities.
• Next, replace an arc in the subtour (say, arc (i,
  j) by the combinations of two arcs---(i, k) and
  (k, j), where k is not in the current
  subtour---that will increase the length of the
  subtour by the smallest (or cheapest) amount.
• Continue with this procedure until a tour is
  obtained. After applying this procedure beginning
  with each city, we take the best tour found.

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• Three methods to evaluate heuristics
• Performance guarantees
• Gives a worse-case bound on how far away from
  optimality a tour constructed by the heuristic
  can be
• Probabilistic analysis
• A heuristic is evaluated by assuming that the
  location of cities follows some known probability
  distribution
• Empirical analysis
• Heuristics are compared to the optimal solution
  for a number of problems for which the optimal
  tour is known

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• An IP formulation can be used to solve a TSP but
  can become unwieldy and inefficient for large
  TSPs.
• LINGO can be used to solve the IP of a TSP.

minimize s.t.

• cij distance from city i to city j
• xij 1 if tour visits i then j, and 0 otherwise
  (binary)
• ti arbitrary real numbers we need to solve for

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9.7 Implicit Enumeration

• The method of implicit enumeration is often used
  to solve 0-1 IPs. Many IP problems can be
  converted to 0-1 IP problems.
• Implicit enumeration uses the fact that each
  variable must be equal to 0 or 1 to simplify both
  the branching and bounding components of the
  branch-and-bound process and to determine
  efficiently when a node is infeasible.
• The tree used in the implicit enumeration method
  is similar to those used to solve 0-1 knapsack
  problems.
• Some nodes have variable that are specified
  called fixed variables.

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• All variables whose values are unspecified at a
  node are called free variables.
• For any node, a specification of the values of
  all the free variables is called a completion of
  the node.
• Three main ideas used in implicit enumeration
• Suppose we are at any node with fixed variables,
  is there an easy way to find a good completion of
  the node that is feasible in the original 0-1
  TSP?
• Even if the best completion of a node is not
  feasible, the best completion gives us a bound on
  the best objective function value that can be
  obtained via feasible completion of the node.
  This bound can be used to eliminate a node from
  consideration.

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• At any node, is there an easy way to determine if
  all completions of the node are infeasible?
• In general, check whether a node has a feasible
  completion by looking at each constraint and
  assigning each free variable the best value for
  satisfying the constraint.

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9.8 Cutting Plane Algorithm

• An alternative method to the branch-and-bound
  method is the cutting plane algorithm.
• Summary of the cutting plane algorithm
• Step 1 Find the optimal tableau for the IPs
  programming relaxation. If all variables in the
  optimal solution assume integer values, we have
  found an optimal solution to the IP otherwise,
  proceed to step2.
• Step 2 Pick a constraint in the LP relaxation
  optimal tableau whose right-hand side has the
  fractional part closest to 1/2. This constraint
  will be used to generate a cut.
• Step 2a For the constraint identified in step 2,
  write its right-hand side and each variables
  coefficient in the form x f, where 0 lt f lt 1.

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9.8 Cutting Plane Algorithm

• Step 2b Rewrite the constraint used to generate
  the cut as
• All terms with integer coefficients all terms
  with fractional coefficientsThen the cut is
• All terms with fractional coefficients lt 0
• Step 3 Use the simplex to find the optimal
  solution to the LP relaxation, with the cut as an
  additional constraint.
• If all variables assume integer values in the
  optimal solution, we have found an optimal
  solution to the IP.
• Otherwise, pick the constraint with the most
  fractional right-hand side and use it to generate
  another cut, which is added to the tableau.
• We continue this process until we obtain a
  solution in which all variables are integers.
  This will be an optimal solution to the IP.

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9.8 Cutting Plane Algorithm

• A cut generated by the method above has the
  following properties
• Any feasible point for the IP will satisfy the
  cut
• The current optimal solution to the LP relaxation
  will not satisfy the cut