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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox Chapter IntroductionH — Operations Research

## Scope of the Chapter

This chapter provides functions to solve certain integer programming, transportation and shortest path problems. Additionally ‘best subset’ functions are included.

## Background to the Problems

General linear programming (LP) problems (see Dantzig (1963)) are of the form:
• find x = (x1,x2,,xn)T$x={\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)}^{\mathrm{T}}$ to maximize F(x) = j = 1ncjxj$F\left(x\right)=\sum _{j=1}^{n}{c}_{j}{x}_{j}$
• subject to linear constraints which may have the forms:
 n ∑ aijxj = bi, j = 1
i = 1,2,,m1 (equality)
 n ∑ aijxj ≤ bi, j = 1
i = m1 + 1,,m2 (inequality)
 n ∑ aijxj ≥ bi, j = 1
i = m2 + 1,,m (inequality)
xjlj, j = 1,2,,n (simple bound)
xjuj, j = 1,2,,n (simple bound)
$∑j=1naijxj=bi, i=1,2,…,m1 (equality) ∑j=1naijxj≤bi, i=m1+1,…,m2 (inequality) ∑j=1naijxj≥bi, i=m2+1,…,m (inequality) xj≥lj, j=1,2,…,n (simple bound) xj≤uj, j=1,2,…,n (simple bound)$
This chapter deals with integer programming (IP) problems in which some or all the elements of the solution vector x$x$ are further constrained to be integers. For general LP problems where x$x$ takes only real (i.e., noninteger) values, refer to Chapter E04.
IP problems may or may not have a solution, which may or may not be unique.
Consider for example the following problem:
 minimize 3x1 + 2x2 subject to 4x1 + 2x2 ≥ 5 2x2 ≤ 5 0x1 − 0x2 ≤ 2 and 0x1 ≥ 0,x2 ≥ 0.
$minimize 3x1 + 2x2 subject to 4x1 + 2x2≥5 2x2≤5 0x1 - 0x2≤2 and 0x1 ≥ 0,x2≥0.$
The hatched area in Figure 1 is the feasible region, the region where all the constraints are satisfied, and the points within it which have integer coordinates are circled. The lines of hatching are in fact contours of decreasing values of the objective function 3x1 + 2x2$3{x}_{1}+2{x}_{2}$, and it is clear from Figure 1 that the optimum IP solution is at the point (1,1)$\left(1,1\right)$. For this problem the solution is unique.
However, there are other possible situations.
(a) There may be more than one solution; e.g., if the objective function in the above problem were changed to x1 + x2${x}_{1}+{x}_{2}$, both (1,1)$\left(1,1\right)$ and (2,0)$\left(2,0\right)$ would be IP solutions.
(b) The feasible region may contain no points with integer coordinates, e.g., if an additional constraint
 3x1 ≤ 2 $3x1≤2$
were added to the above problem.
(c) There may be no feasible region, e.g., if an additional constraint
 x1 + x2 ≤ 1 $x1+x2≤ 1$
were added to the above problem.
(d) The objective function may have no finite minimum within the feasible region; this means that the feasible region is unbounded in the direction of decreasing values of the objective function, e.g., if the constraints
 4x1 + 2x2 ≥ 5,  x1 ≥ 0,  x2 ≥ 0, $4x1+2x2≥5, x1≥0, x2≥0,$
were deleted from the above problem.
Figure 1
Algorithms for IP problems are usually based on algorithms for general LP problems, together with some procedure for constructing additional constraints which exclude noninteger solutions (see Beale (1977)).
The Branch and Bound (B&B) method is a well-known and widely used technique for solving IP problems (see Beale (1977) or Mitra (1973)). It involves subdividing the optimum solution to the original LP problem into two mutually exclusive sub-problems by branching an integer variable that currently has a fractional optimal value. Each sub-problem can now be solved as an LP problem, using the objective function of the original problem. The process of branching continues until a solution for one of the sub-problems is feasible with respect to the integer problem. In order to prove the optimality of this solution, the rest of the sub-problems in the B&B tree must also be solved. Naturally, if a better integer feasible solution is found for any sub-problem, it should replace the one at hand.
The efficiency in computations is enhanced by discarding inferior sub-problems. These are problems in the B&B search tree whose LP solutions are lower than (in the case of maximization) the best integer solution at hand.
The B&B method may also be applied to convex quadratic programming (QP) problems. Routines have been introduced into this chapter to formally apply the technique to dense general QP problems and to sparse LP or QP problems.
A special type of linear programming problem is the transportation problem in which there are p × q$p×q$ variables ykl${y}_{kl}$ which represent quantities of goods to be transported from each of p$p$ sources to each of q$q$ destinations.
The problem is to minimize
 p q ∑ ∑ cklykl k = 1 l = 1
$∑k=1p∑l=1qcklykl$
where ckl${c}_{kl}$ is the unit cost of transporting from source k$k$ to destination l$l$. The constraints are:
 q ∑ ykl = Ak l = 1
(availabilities)
 p ∑ ykl = Bl k = 1
(requirements)
ykl0.
$∑l=1qykl=Ak (availabilities) ∑k=1pykl=Bl (requirements) ykl≥0.$
Note that the availabilities must equal the requirements:
 p q p q ∑ Ak = ∑ Bl = ∑ ∑ ykl k = 1 l = 1 k = 1 l = 1
$∑k= 1pAk=∑l= 1qBl=∑k= 1p∑l= 1qykl$
and if all the Ak${A}_{k}$ and Bl${B}_{l}$ are integers, then so are the optimal ykl${y}_{kl}$.
The shortest path problem is that of finding a path of minimum length between two distinct vertices ns${n}_{s}$ and ne${n}_{e}$ through a network. Suppose the vertices in the network are labelled by the integers 1,2,,n$1,2,\dots ,n$. Let (i,j)$\left(i,j\right)$ denote an ordered pair of vertices in the network (where i$i$ is the origin vertex and j$j$ the destination vertex of the arc), xij${x}_{ij}$ the amount of flow in arc (i,j)$\left(i,j\right)$ and dij${d}_{ij}$ the length of the arc (i,j)$\left(i,j\right)$. The LP formulation of the problem is thus given as
 minimize   ∑ ∑ dijxijsubject to ​Ax = b,  0 ≤ x ≤ 1, $minimize ∑∑dijxijsubject to ​Ax=b, 0≤x≤1,$ (1)
where
aij =
 { + 1 if arc ​ j​ is directed away from vertex ​ i, − 1 if arc ​ j​ is directed towards vertex ​ i, 0 otherwise
$aij={ + 1 if arc ​ j​ is directed away from vertex ​ i, -1 if arc ​ j​ is directed towards vertex ​ i, 0 otherwise$
and
bi =
 { + 1 for ​i = ns, − 1 for ​i = ne, 0 otherwise.
$bi={ +1 for ​i=ns, -1 for ​i=ne, 0 otherwise.$
The above formulation only yields a meaningful solution if xij = 0​ or ​1${x}_{ij}=0\text{​ or ​}1$; that is, arc(i,j)$\mathrm{arc}\left(i,j\right)$ forms part of the shortest route only if xij = 1${x}_{ij}=1$. In fact since the optimal LP solution will (in theory) always yield xij = 0​ or ​1${x}_{ij}=0\text{​ or ​}1$, (1) can also be solved as an IP problem. Note that the problem may also be solved directly (and more efficiently) using a variant of Dijkstra's algorithm (see Ahuja et al. (1993)).
The travelling salesman problem is that of finding a minimum distance route round a given set of cities. The salesperson must visit each city only once before returning to his or her city of origin. It can be formulated as an IP problem in a number of ways. One such formulation is described in Williams (1993). There are currently no functions in the Library for solving such problems.
The best n$\mathbit{n}$ subsets problem assumes a scoring mechanism and a set of m$m$ features. The problem is one of choosing the best n$n$ subsets of size p$p$. It is addressed by two functions in this chapter. The first of these uses reverse communication; the second direct communication.

## Recommendations on Choice and Use of Available Functions

 nag_mip_ilp_dense (h02bb) solves dense integer programming problems using a branch and bound method. nag_mip_ilp_print (h02bv) prints the solution to an integer or a linear programming problem using specified names for rows and columns. nag_mip_ilp_info (h02bz) supplies further information on the optimum solution obtained by nag_mip_ilp_dense (h02bb). nag_mip_iqp_dense (h02cb) solves dense integer general quadratic programming problems. nag_mip_iqp_dense_optstr (h02cd) supplies optional parameter values to nag_mip_iqp_dense (h02cb). nag_mip_iqp_sparse (h02ce) solves sparse integer linear programming or quadratic programming problems. nag_mip_iqp_sparse_optstr (h02cg) supplies optional parameter values to nag_mip_iqp_sparse (h02ce). nag_mip_transportation (h03ab) solves transportation problems. It uses integer arithmetic throughout and so produces exact results. On a few machines, however, there is a risk of integer overflow without warning, so the integer values in the data should be kept as small as possible by dividing out any common factors from the coefficients of the constraint or objective functions. nag_mip_shortestpath (h03ad) solves shortest path problems using Dijkstra's algorithm.
nag_mip_ilp_dense (h02bb) and nag_mip_transportation (h03ab) treat all matrices as dense and hence are not intended for large sparse problems. For solving large sparse LP problems, use nag_opt_qpconvex2_sparse_solve (e04nq) or nag_opt_nlp1_sparse_solve (e04ug).
nag_best_subset_given_size_revcomm (h05aa) selects the best n$n$ subsets of size p$p$ using a reverse communication branch and bound algorithm.
nag_best_subset_given_size (h05ab) selects the best n$n$ subsets of size p$p$ using a direct communication branch and bound algorithm.

## Functionality Index

 Feature selection,
 best subset,
 Given size,
 direct communication nag_best_subset_given_size (h05ab)
 reverse communication nag_best_subset_given_size_revcomm (h05aa)
 Integer programming problem (dense):
 print solution with specified names nag_mip_ilp_print (h02bv)
 solve LP problem using branch and bound method nag_mip_ilp_dense (h02bb)
 solve QP problem using branch and bound method nag_mip_iqp_dense (h02cb)
 supply further information on the solution obtained from nag_mip_ilp_dense (h02bb) nag_mip_ilp_info (h02bz)
 Integer programming problem (sparse):
 solve LP or QP problem using branch and bound method nag_mip_iqp_sparse (h02ce)
 Shortest path, through directed or undirected network nag_mip_shortestpath (h03ad)
 Supply optional parameter values to nag_mip_iqp_dense (h02cb) nag_mip_iqp_dense_optstr (h02cd)
 Supply optional parameter values to nag_mip_iqp_sparse (h02ce) nag_mip_iqp_sparse_optstr (h02cg)
 Transportation problem nag_mip_transportation (h03ab)

## References

Ahuja R K, Magnanti T L and Orlin J B (1993) Network Flows: Theory, Algorithms and Applications Prentice–Hall
Beale E M (1977) Integer programming The State of the Art in Numerical Analysis (ed D A H Jacobs) Academic Press
Dantzig G B (1963) Linear Programming and Extensions Princeton University Press
IBM (1971) MPSX – Mathematical programming system Program Number 5734 XM4 IBM Trade Corporation, New York
Mitra G (1973) Investigation of some branch and bound strategies for the solution of mixed integer linear programs Math. Programming 4 155–170
Williams H P (1993) Model Building in Mathematical Programming (3rd Edition) Wiley