NZOR Volume 10 number 1 January 1982

35

GRAPH THEORY - A SURVEY OF ITS USE IN OPERATIONS RESEARCH

L.R. FOULDS

UNIVERSITY OF CANTERBURY CHRISTCHURCH, N.Z.

SUMMARY

One of the common themes in O.R. is the modelling approach. Unfortunately many accurate models of O.R. problems turn out to be intractable when subjected to standard techniques. This paper shows how graph theory and networks may be profitably used to model certain discrete O.R. problems from a different view-point. Effective algorithms and heuristics for the models and real world applications are referenced.

1. IntroductionOne of the common themes in O.R. is the modelling

approach. Unfortunately many accurate models of O.R. problems turn out to be intractable when subjected to standard techniques. However certain discrete problems can be profitably analysed using graph theoretic models. This paper introduces useful concepts from graph theory and shows how they may be used to look at certain O.R. problems from the viewpoint of the graph theorist.

We begin in the next section with a discussion of the difficulty of problems as a justification for using the alternative graph theoretic models. Sections 3, 4, and 5 discuss models of certain O.R. problems based on graphs, directed graphs and networks, respectively. A relatively small number of references are cited. They develop in some depth what is discussed here and should be taken as signposts by the reader interested in undertaking a more exhaustive literature search.

2. PROBLEM COMPLEXITY

Many models of common processes that are analysed by O.R. techniques are discrete in the sense that their decision variables must assume values from a discrete rather than a continuous set. Examples include: vehicleManuscript submitted August 1981, revised October 1981.

36

and machine scheduling, facility location, plant layout, and certain assignment and allocation problems. The determininistic versions of such models are usually concerned with optimizing some measure of system performance subject to constraints on the values the variables can assume. Thus many are expressed in terms of integer programming. These models form a branch of applied mathematics which has extensive overlap with O.R. and is known as combinatorial optimization (CO).

Unfortunately there are just too many solutions to any nontrivial problem for complete enumeration to be feasible. For example, consider the problem of finding the shortest path which visits each of a given set of n cities. Even if each of the n! possible paths could be evaluated in a billionth of a second it would still take over 16,000 years to find the best for n=21. It is therefore of interest to attempt to design algorithms which are more effective than complete enumeration.

We turn now to the question of evaluating the effectiveness of an algorithm. The concept of effective­ness was placed on a firm scientific foundation by Edmonds [30] whose work caused the following convention to be generally adopted by those concerned with algorithm efficiency:

An algorithm is considered to be effective if it can guarantee to solve any instance of the problem for which it was designed by performing a number of elementary computational steps and the number can be expressed as a polynomial function of the size of the problem.

It is assumed that computation time is linearly proportional to the number of elementary computational steps required to implement the algorithm. The size of a specific instance of a problem is defined to be the number of symbols required to describe it.

It is a valid question to ask whether an effective, or polynomial-time, algorithm can be devised for a given CO problem. There exist CO problems for which it has been shown that no effective algorithm exists, and others for which polynomial-time algorithms have been devised. This second class of problems is denoted by P (polynomial).

There exists a third class of problems whose status is unknown. It is possible to devise algorithms for each problem but no effective algorithm is known for any of them. However neither has there yet appeared a proof showing that any are intractable. Our problem of finding the shortest path through a given set of cities lies in this last class which is denoted by NP (nondeterministic polynomia I).

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Within NP there is a subset of problems which are called NP-Complete. A problem is termed NP-Complete if it: (1) belongs to NP and (2) has the property that if an effective algorithm is found for it then an effective algorithm can be found for every problem in NP. In this sense the NP-Complete problems are the hardest in NP.To establish the status of a CO problem for which no effective algorithm is known it is usual to employ the concept of reducibility.

A problem pi is said to be reducible to problem p2 (written pi « P2) if the existence of an effective algorithm for p2 implies the existence of an effective algorithm for Pi . The following result is often used to establish that a problem p2 a NP is NP-Complete.Theorem 2.1: If pi is NP-Complete and pi^pa then p2 is also NP-Complete.Proof: Garey and Johnson [40, p.38].

Many of the problems in NP have defied the attempts to find effective algorithms of some of the best mathe­maticians over the past 30 years. There is also more objective circumstantial evidence that P ^ NP. Thus it seems unlikely that an effective algorithm exists for any of the NP-Complete problems.

Unfortunately, many of the O.R. models of the common processes given earlier as examples are NP-Complete. Of course a manager will find no comfort at all in his O.R. consultant telling him that the literature does not contain an efficient method which guarantees an optimal solution for his problem in reasonable computational time. Come what may, he somehow has to schedule his flight crews, route his delivery vans, or whatever. At this point the O.R. analyst has a number of options: he can attempt to (1) develop the methodology that will provide optimal solutions efficiently, (2) find algorithms that will solve certain special cases of the problem, (3) look for efficient algorithms that solve a relaxed version of the problem,(4) come up with algorithms that seem likely to run quickly most, but not all, of the time, (5) give up the quest for optimality and provide approximate methods that run quickly but have no guarantee of optimality.

Because of the theory just discussed, aim (1) is often unrealistic. Aims (2) and (3) are occasionally appropriate. However there is a very real danger.Churchman [21] and others have warned of the pitfalls of substituting the real problem with an artificial one which corresponds to standard models and known techniques.Nothing can give O.R. a worse name than the consultant who "bends" his client's problem into a form which is

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amenable to solution by his pet method, producing solutions which are of little practical use. However aim (4) is obviously worthy. It is usually satisfactory to employ an algorithm which almost always runs in reasonable time. The simplex algorithm for linear programming is a good example. However if such a method cannot be found we are left with aim (5).

The purpose of this paper is to explore a new way in which the structure of the model of a difficult problem can be exploited in order to attain at least one of the five aims. This way involves building a new, graph-theoretic model. The interested reader is referred to the classic text on pure graph theory (GT) by Harary [48] and to the more applied texts by Chachra Ghare and Moore [18], Christofides [19], Deo [26], Minieka [67], Robinson and Foulds [81], and Swamy and Thulasivaman [86] all of which contain applications to O.R. Graph theoretic notation and terminology used in this paper is fully defined in [48] and [81] . We denote a graph with point set V and line set E by (V,E).

3. THE APPLICATION OF GRAPH THEORY IN O.R.

The purpose of this paper is to explain, now that GT has become a systematic tool, how it can be used to yield new insights into O.R. models. One simple use of graphs in problem-solving in any field including O.R. is the following. It is often convenient to depict the relationship between elements of a system by means of a graph. Thus one often sees a transportation, assignment, or PERT problem represented by a graph (in these cases a directed g r a p h ) . Such representations are often an aid in describing a problem and are useful as such. However, this paper is concerned not just with ways of representing an O.R. problem in terms of GT but more with using the results of GT to actually solve the problem. Not every conceivable GT application is documented. Rather, some of the major applications are presented to give an overall flavour to this area, with the topics of facilities layout and traffic network design treated in some depth.

3.1 GRAPH THEORETIC ALGORITHMS

In this paper we adopt the following specialized definition of an algorithm. An a l g o r i t h m for a problem is a scientific procedure which is guaranteed to converge to an optimal solution of the problem. This subsection discusses effective GT algorithms for certain O.R. problems.

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The C h i n e s e P o s t m a n ' s P r o b l e m

Consider a postman who begins delivering mail in a city suburb having picked it up from the post office. He wishes to cover each street in his area at least once and finally return to the post office, travelling the least possible distance. This problem is known as the Chinese Postman's Problem (CPP) as it was first considered by the Chinese mathematician Kwan Mei-ko [66]. Further applications abound, including: the collection of household refuse, milk delivery, the inspection of power, telephone, or railway lines, the spraying of salt-grit on roads in winter, office block cleaning, security guard routing, scheduling of snow ploughs and even museum touring.

The problem can be formulated as a GT model in which a weighted graph G is devised with its points, lines and weights representing intersections, streets, and distances respectively. An E u l e r i a n tour in a graph is a closed circuit of the lines of the graph which includes each line exactly once. Clearly CPP is equivalent to finding the least weight Eulerian tour in the graph. If G is Eulerian (contains an Eulerian tour), any Eulerian tour is optimal. There exists an efficient algorithm for devising such a tour which is due to Fleury (see Lucas [63]). If G is not Eulerian some lines will have to be covered more than once. An efficient algorithm for this case has been presented by Edmonds and Johnson [31]. Basically the algorithm adds duplicated lines of minimal total weight so as to make G Eulerian and then uses Fleury's rule.

Good [45] provided the following interesting appli­cation of a directed Euler tour in a digraph. The position of a rotating drum in a computer can be recognized by means of binary signals produced at a number of electrical contacts on the surface of the drum. The surface is divided into 2m sections, each comprising either conducting or insulating material. If an electrical contact is adjacent to a conducting (insulating) section the signal 1(0) is sent.

The problem is to minimize the number say k of consecutively placed contacts so that each position of the drum gives a unique reading. It can be solved as follows. Any position corresponds to a k-digit binary number. It can be shown that m contacts are sufficient by reducing the problem to finding a directed Euler Tour in a diagraph.

The Tim e t a b l i n g P r o b l e m

Consider a school with m teacher s: T i , T 2 , ... 1 Tm ;and n classes: Ci, C 2 , ... , C . Given that teacher T. isn 1

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required to teach class Cj for p^j periods schedule a com­plete timetable in the minimum possible number of periods. The problem can be formulated as a GT model in which a bipartite graph G = (V,E) is devised with V = Vi U V 2 where Vi = {T i ,T2, ••• ,Tjp} and V 2 = (Ci,C2, ... , C n }, and points T^ and Cj are joined by p^j lines.

The timetabling problem is equivalent to colouring the points in V, so that no two adjacent points have the same colour, with as few colours as possible - each colour representing a distinct period. (See Haken and Appel [46] for a famous result on graph colouring.) Bondy and Murty [9] have presented an efficient algorithm for bipartite graph colouring. They also solve the more realistic problem of assuming that there are only q classrooms available. The complication of preassignments (i.e. conditions specifying the periods during which certain teachers and classes must meet) has been studied by Dempster [27] and de Werra [28].

The C o n n e c t o r P r o b l e m

A railway network connecting a number of cities is to be set up, the objective is to make it possible to travel by some path between every pair of cities. Given the cost of construction wij, of linking cities vi and v j , design a network with minimum possible construction cost. Other applications include: electrical network design, natural gas reticulation, communication network construction, and city utilities cable layout. A less direct application appears in the form of a subroutine in some solution methods for the travelling salesman problem which is discussed later. The problem can be formulated as a GT model in which a weighted graph G = (V,E) is devised with points, lines, and weights representing cities, feasible connections, and construction costs respectively. The problem is equivalent to finding a spanning subgraph of G of minimum weight. There are two well-known efficient algorithms for this problem due to Kruskal [56] (which is more suitable when there are relatively few connection possibilities) and Prim [79] (which is more suitable when most connection possibilities are ava ilable). Efficient implementations for the latter algorithm are given by Dijkstra [29] and by Kevin and Whitney [53].

S h o r t e s t Path P roblems

Consider once again the connector problem just discussed. Before actually constructing the railway network it is conceivable that a planner may wish to cal­culate certain shortest paths in the graph of potential connections. There are a multitude of applications of

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the problem of finding shortest paths in a weighted graph. Where all construction costs are nonnegative, an efficient algorithm for finding the shortest path between two specified points has been given by Dijkstra [29], In some cases, some construction "costs" may represent profits and hence would be negative. Algorithms for a general cost matrix were given by Ford [37], Moore [70] and Bellman [5], When shortest paths between all pairs of points are required, Dijkstra's algorithm can be used repeatedly. However Floyd's algorithm [33]], as elaborated upon by Murchland [71], is in general about 50% faster.

The O p timal A s s i g n m e n t P r o b l e m

Consider a factory in which n workers: Wi, W 2 , ... ,W are to be assigned to n machines Mi, M 2 , ... , M n , in a n one-to-one fashion. Each worker has been tested on each machine and a table of standardized times are available providing information about relative worker abilities.The problem is to assign the workers to machines so as to minimize the total of the standardized time of the assign­ment. The problem can be formulated as a GT model in which a weighted, bipartite graph G = (V,E) is devised with V = Vi U V 2 where Vi = {Wi , W 2 , ••• fW n ) an<^V 2 = {Mi,M 2 , ••• ,M n }. The line joining Wi and Mj is weighted with tij, the standardized time for worker Wi on machine M j . The problem is equivalent to finding a minimum weight perfect matching on G. An efficient algorithm for this problem, called the Hungarian method, has been given by Kuhn [55].

The Loca t i o n o f Centres

Consider a network of roads whose points represent communities. There are a number of emergency centres, such as hospitals, police or fire stations, to be located on the network, not necessarily in the communities.The optimality criterion is often taken to be the mini­mization of the distance of the furthest community to a centre. There is a second, related problem: for a given critical distance locate the smallest number of centres so that all communities lie within this critical distance from at least one centre. The problem can be formulated as a GT model in which a weighted graph is (G = V,E) is devised with points, lines, and weights representing communities, roads and distances respectively. The problem is to identify a set of points, possibly inserted into the lines of G, which satisfy the above criteria. Christofides [20] has called these new points abso l u t e p-ce n t r e s and has provided an efficient algorithm which can solve either the minimax or the number of centres minimization problems.

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With regard to the location problem just discussed, it is often the case that nonemergency facilities such as switching centres for telephone networks, substations in electric power networks, post offices, libraries, or goods depots are to be located. With such scenarios it is usual to modify the optimality criterion from one of "minimax" to one of "minimum", i.e. to minimize the total cost of travel from each community to the nearest centre. (These travel costs are commonly some function of the importance of the community, given for example by the demand for the service of the facility. Problems of this type come under the general heading of f acility location on a network. Unfortunately there are no efficient GT algorithms for this problem. The reader is referred to Hanan and Kurtzberg [47] for further details on what is available.

C o m m u n i c a t i o n s N e t w o r k R e l i a b i l i t y

Consider a number of centres which are to be connected by communications links. A measure of the reliability of the system is the smallest number of links whose breakdown makes communication between every pair of centres impossible. Each potential link joining centres i and j has a construc­tion cost c i j . The system is to be designed so that (for a given number k) at least k links must break down before complete pairwise communication becomes impossible. The optimality criterion is total construction cost which is to be minimized. The problem can be formulated as a GT model in which a weighted graph G = (V,E) is devised with points, lines and weights representing centres, potential links, and construction costs respectively. The problem is to determine a minimum weight k-connected spanning subgraph of G. For k=l this problem reduces to the connector problem already discussed. Unfortunately there are no known efficient algorithms for problems for which k > 1. However the special case in which every possible link is available and their construction costs are all equal can be solved by the efficient GT algorithm of Bondy and Murty [9].

P r oject S e l e c t i o n

Suppose an organization has n projects: P i , P 2 , ••• >p n which must be carried out. Project P^ requires some subset R-l c: {T i , T 2, ••• , Tq } of the total set of q resources available. Each project can be completed in a single unit of time but two projects requiring the same resource cannot be executed similtaneously. What is the maximum number of projects that could be executed at the same time? The problem may have to be solved repeatedly as more projects become available in later periods. It can be

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formulated as a GT model in which a graph G = (V,E) is devised with points and lines representing projects and resource requirement nonoverlap respectively. That is Pi and Pj are connected whenever Ri D Rj = <J>. The problem reduces to finding a maximal independent set of G.

A very efficient GT algorithm for this problem has been presented by Bron and Kerbosh [14].

3.2 GRAPH THEORETIC HEURISTICS

Analysts in business and industry are often faced with problems of such complexity that the standard algorithms of O.R. are inappropriate. There are a number of reasons why this might be so: (1) The dimensions of the problem may be so large that the application of the fastest-known algorithm on the fastest computer may take a prohibitive amount of computational time. This is certainly true for certain vehicle routing problems. (2) The problem may be virtually impossible to formulate in explicit terms. The aims of different managers involved in operating a system may be conflicting or ill-defined. In fact it may be difficult to express many features of a problem in quantitative terms. (3) Data collection may be beset with problems of accuracy and magnitude. For example in large- scale location problems the analyst may be faced with calculating an enormous number of location-to-location distances. In order to provide this information in reasonable time it may be necessary to make approximations. Sometimes the use of approximate data makes the concept of an optimal solution meaningless.

The idea of approximate methods, which are easy to use but which give up the guarantee of optimality is not new. Indeed as early as 300 AD Pappas, writing on Euclid, suggested this approach. Descartes and Leibnitz both attempted to formalize the subject. It became known as the study of heuris t i c s and h e u r i s t i c was the name of an area of academic study whose aim was to investigate the methods of discovery and invention. It was allied with logic, philosophy and psychology. The name itself was derived from the Greek word h e u r i s k e i n - to discover.In O.R. today the term h e uristic is used to describe a method which, on the basis o f e x p e r i e n c e or j u d g e m e n t , seems likely to y i e l d a g o o d s o l u t i o n to a p r o b l e m , but w hich cannot be g u a r a n t e e d to p r o d u c e an o p t i m u m .

The Trave l l i n g S a l e s m a n Pro b l e m

The following scheduling problem often arises in the pharmaceutical industry. Batches of N drugs: D i , D 2 , . . . , Dj,

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are manufactured in a single reaction vessel, one at a time. If Dj is to follow Di the vessel has to be cleaned, at a cost Cij. The batches are to be manufactured in a continuous cyclic manner, so that once the last batch has been produced, the first batch is to be begun again. The problem is to find the production sequence with least total cleaning cost. The problem can be formulated as a GT model in which a weighted graph G = (V,E) is devised with points, lines and weights representing drugs, direct succession in the production sequence, and cleaning costs respectively. Then the problem is to determine a Ham i l t o n cycle of G of least cost. A Hamilton cycle in a graph G is a closed circuit of the lines of G which passes through each point of G exactly once. There are numerous other applications of this model including: mail box collection, school bus scheduling, electricity supply network design, and service vehicle routing. Unfortunately the problem is NP-Complete and hence no efficient algorithms are known. However there are a number of effective heuristics for the problem including those by Lin [60] and Christofides [20], the latter guaranteeing a solution value within 50% of the optimum.

A Sto r a g e P r o b l e m

Consider a factory which manufactures a number of chemicals which it then stores in a warehouse. Certain pairs of chemicals cannot be stored in the same compartment. What is the least number of compartments into which the warehouse must be partitioned for safe storage? This problem can be formulated as a GT model in which a graph G = (V,E) is devised with points and lines representing chemicals and incompatibilities respectively. The problem reduces to finding a least-colour colouring of G. The model also has application in bin packing, examination timetable construction and resource allocation. Unfortun­ately the problem is NP-Complete and thus a heuristic procedure is in order. Various available colouring heuristics have been compared by Matula et al. [65] and Williams [94].

Mine Ve n t i l a t o r Shaft L o c a t i o n

Consider an underground mining network of tunnels in which a ventilation system is to be located. The system comprises a pump and a number of units. The pump is usually sited at the main descent shaft. The units are usually located at the cutting faces (at the ends of the tunnels) although in an extensive network additional units may be sited at internal points as well. Each unit must be connected by some path of pipes, sometimes via intermediate units, to the pump. The cost of laying a pipe

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between any two points in the network is known. The problem is to determine how to connect all the units to the pump with minimal laying cost. This problem can be formulated as a GT model in which a weighted graph G = (V,E) is devised with points representing all possible sites and tunnel intersections and lines representing all feasible connections between these locations and weights representing laying costs. A subset U <= V is distingusihed, representing the given set of locations of the pump and the units (the pump is not specially distinguished). The problem reduces to finding a subgraph of G of least cost which spans U i.e. that contains a path between every pair of points in U.The problem is known as S t e i n e r ' s P r o b l e m in Graphs and is NP-Complete. Takahashi and Matsuyama [87] have devised an efficient heuristic for the problem.

The F a c i l i t i e s Layout Pro b l e m

The important industrial problem of plant layout has received significant attention in the last two decades.An important step in the plant layout process is the specification of which facilities are to be adjacent. It is assumed here that the facilities are to be located on a simply-connected plane region such as a factory floor or flat building site. This problem has been given different names by different authors e.g. "plant layout": Foulds and Robinson [35], [36], Apple [2], Moore [68], Hillier and Connors [49]; "facilities allocation", Buffa, Armour, and Vollman [12], and "layout planning", Huther[72]. In this paper we adopt the title f a c i l i t i e s layout p r o b l e m (FLP).

Many of the articles just mentioned are concerned with the layout of manufacturing plants. However FLP finds application in many other areas such as the design of service facilities such as fire stations, libraries, banks, universities, and hospitals; and other institutions such as office blocks and sports complexes. When the area to be laid out is a single floor of a building such as a factory or office block, the layout results in a partition of the floor into a number of areas or rooms.

We begin by presenting an FLP formulation due to Hillier and Connors [49]. If the areas required for the different facilities have significant variation, partition each facility into a number of subfacilities all with equal area. The planar site is then divided up in the form of a grid of elemental areas called locations each of the size of the subfacilities.

The problem then is to locate the subfacilities, guaranteeing that all the subfacilities of each facility are adjacent in a convenient configuration and materials handling costs are minimized.

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Let n = the number of subfacilities,= the number of available locations on the

planar site,

cij = the cost per unit time for assigning sub­facility i to location j ,

= the distance or cost of travel from location i to location j ,

f^j = the work flow from subfacility i to subfacility j .

= the set of locations to which subfacility i may be feasibly assigned.

a . = f..d. if i / k or j ^ r ijkr lk jr J

f ii<3. j j + c^j if i = k and j = r, and

x . . = 1 if subfacility i is assigned to location ^ j, 0 otherwise.

If there are more locations than subfacilities, a number of dummy facilities can be introduced with zero cij and fij values to make the numbers equal. fij values are set equal to relatively high levels for subfacilities i and j of the same facility to ensure their adjacency. Cij values are set to relatively high values when j J? S-̂ .

The problem then is to

nMinimize 7

subject to

n n

I 1 j=l k=l

n

Ii=l

a . ., x . . ijkr lj kr (1)

n

Ii=l

x . . = 1 j = 1,2,. . . ,n, (2)

n

Ij=l

X . .ID

= 1 i = 1,2,. . . ,n, (3)

X i j- 0 or 1, i =

j =1,2, . 1,2, .

. . ,n,

. . ,n.(4)

{1) — (4) is a quadratic assignment problem, Q A P . Unfortunately QAP is NP-Complete. So we develop a canonical FLP formulation which may lead to useful heuristic solution procedures.

The FLP will now be formulated in terms of graph theory. It is assumed that a r e l a t i o n s h i p chart is available which summarizes the desirability of siting each pair of facilities adjacently. Let

w. . = the c l o s e n e s s r a ting for siting facilities i 1 -* and j adjacently, assumed to be a real number,

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n = the number of facilities,

V = the set of facilities,

N = the set of pairs of facilities which must be adjacent in any feasible solution,

F = the set of pairs of facilities which cannot be adjacent in any feasible solution, and

E = (V&V)\(VUF) = the set of possible adjacenciesi.e. the set of all conceivable adjacencies (denoted by the unordered pair V&V) less the set NUF of adjacencies whose fate has been decided.

Consider the layout in Figure 1. The area occupied by each

r ~ — ------------------------------------- 1

Figure 1. A block layout and its associated planar g r a p h .

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facility has associated with it the point of a graph. If two facilities are adjacent (i.e. their boundaries meet at more than a single point) their points pi and pj are joined by the line {pi,pj} which intersects only their common boundary. This results in a drawing of a graph which depicts the adjacency structure of the layout. Note that the region exterior to the plane site is considered to be a facility - it may be desirable to locate some facilities adjacent to the boundary. It becomes apparent that we can pose the FLP as a graph-theoretic problem by defining a weighted graph (G,w) where G = (V,E). V and E have been defined earlier. The problem is to

As all closeness ratings are non-negative, as many adjacencies as possible should be made. This means that any optimal solution will correspond to a ma ximal planar graph.

One of the first papers using this approach was produced by Levin [59]. Then Krejcirik [54], developed what he called the RUGR algorithm, the first graph- theoretic heuristic. There followed a series of papers by Seppanen and Moore [83], Moore and Carrie [69], and Moore [68] which further developed the application of graph theory to FLP. Carrie, Moore, Roczniak and Seppanen [16] summarized the ideas of this series and presented a graph-theoretic heuristic.

The method begins by forming a weighted graph of the relationships between facilities whose points and weights represent facilities and closeness ratings respectively. An attempt is then made to identify a maximal planar subgraph of relatively high weight. (Four different strategies for line selection are explored by the a u t hors.)

{i , j } eE

subject to (V , E ' U N ) is a maximal planar graph

(5)

(6 )

{i,j}eN,

{i,j}eF,

(7)

(8 )

where 1 if facilities i and j are located adjacently

0 otherwiseand

E' = {{i,j } : xij = 1, {i,j}eE}.

This subgraph is then redrawn to reflect the relationship intensities in a form called a relationship diagram introduced by Muther [72]. Another graph called

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the dual is then constructed from this redrawn subgraph. This dual graph represents a layout apart from the fact that shapes and areas have not been taken into account.The last step of the method attempts to accommodate these in forming a layout block plan from the dual. In order to facilitate the use of graphs the authors use string processing grammars.

Another graph-theoretic approach has been developed by Foulds and Robinson [36]. Their heuristic circumvents the problem of having to test various promising high-weight sub-graphs for planarity. This is achieved by constructing a series of special graphs called deltahedra which are always maximal planar. The final deltahedron represents a layout that is hopefully of high quality. At this stage an improvement phase is used to identify potential improvements in the form of line replacement or point relabelling. The heuristic has been programmed in Pascal to run on a Burroughs B6700.* Two criteria have been used to judge its worth: (1) solution quality and (2) computa­tional speed. In order to be able to explore (1) it is important to be able to compare the values of the solutions produced with either optimal solution values of upper bounds thereof. Of course for large test problems optima are unavailable. However an upper bound on the optimal solution value can be calculated as follows:It is well-known that a planar graph with n points can have no more than (3n-6) lines. Thus an obvious upper bound on the value of any optimal solution is the sum of the weights of the (3n-6) heaviest lines.

Table 1 displays these quantities for test problems with randomly generated data where there has been no attempt to improve the final solution. The standard deviation of the weights for each problem is also given.The bound becomes progressively more blunt as the number of facilities increases. It can be seen that the heuristic seems to yield consistently good quality solutions to the model of equations (5)-(8).

Returning to the question of computational speed it has been shown by Griffin [41] that the heuristic is of polynomial order. Indeed the number of elementary computational steps taken to solve a problem is proportional to

9 n 2 + 9nlog3n + 6n - 40

*Footnote: The author is grateful to Dr P.B. Gibbons of the ComputerScience Department, University of Auckland, New Zealand, who wrotethe program, and to his own doctoral student, J. Giffin, who gathered the computational statistics.

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ProblemNo. n 0

UpperBound

DeltahedronSolution %

CPUTime(Secs)

1 10 5 2504 2483 99.16 0.272 10 10 2649 2593 97. 88 0.203 10 15 2670 2637 98.76 0.194 10 20 2725 2668 97.90 0.205 20 5 5709 5629 98.60 0.646 20 10 5997 5824 97.12 0.637 20 15 6245 6003 96 .12 0.658 20 20 6655 6324 95.03 0.649 20 25 7114 6699 94.17 0.66

10 30 5 9021 8858 98.19 1.0811 30 10 9581 9267 96.72 1.1312 30 15 10135 9718 95.89 1.1013 30 20 10962 10376 94.65 1.0914 30 25 11490 10703 93.15 1.0915 40 5 12349 12083 97.85 1.7516 40 10 13212 12658 95.81 1.7617 40 15 14057 13220 94 .05 1.7618 40 20 14957 13803 92.28 1.8019 40 25 15953 14704 92.17 1.82

Table 1. Computational Results for the Deltahedron Algorithm

where n is the number of facilities.

The graph theoretic approach does not initially take shapes or areas of facilities into account. However there is some merit in allowing an analyst to input any non- quantifiable knowledge he has at this point. As the graph- theoretic methods terminate at this stage they do permit manual adjustment. The influence of facility shapes has been studied by Steadman [84],

There is another factor which makes the graph-theoretic approach of interest to layout engineers. It is straight­forward to calculate an upper bound on the value of any feasible solution (given an objective of maximization) to a specific FLP problem instance using graph-theoretic methods. A benchmark of this sort is often a valuable piece of information for the designer. Indeed Carrie et al. [16] predict that "the definition of the upper bounds of feasible solutions may be the most important contribution to facilities design made by graph theory."

4. DIGRAPH METHODS

A d i r e c t e d g raph or di graph for short is a graph in which each line has been given a direction. The lines have arrows attached to them to indicate their direction and are called arcs.

51

Many real-world systems which O.R. analysts study can be modelled as a binary relation between the elements of the system. Since this is what a digraph is, digraph theory (DT) can often be applied to analyse O.R. problems including those in Markov process (see Rosenblatt [82] and Howard [50]), game theory, job sequencing, assembly line balancing, transportation, and resource planning.In this section we discuss effective DT models and methods for certain O.R. problems.

Game Theory

Game theory has developed into an important O.R. tool over the last 40 years as it can be used to find the best way to perform a set of tasks in a competitive environment. The "games" involved can be classified according to: the number of "players", whether there is a stochastic element, whether complete information on the position of the game is available at every move, and whether there is a finite number of moves available at each position. The theory of digraphs is of limited use in the more general cases. However it can provide a useful approach, if not a complete analysis, for two-person, perfect information, determin­istic, finite games. Such a game can be modelled by a game d i graph with points and arcs representing positions and legal moves respectively. A game digraph has a unique s t a r t i n g point representing the initial position and at least one c l o s i n g p o i n t representing a position at which the game is terminated. The closing points are classified according to each outcome they represent such as "win for player A', "draw", and "win for player B " .Most game digraphs are acyclic, but when cycles are present there are rules to prevent endless cycle traversal. Each player tries to find a directed path from the starting point to a closing point which represents a win for him/her. What (s)he is concerned about is, given the game has reached a certain point, can (s)he force a win. If either player can claim this the point (position) is said to be won, as it is assumed that each player makes the best possible move at each point. The concept of the kernel of a digraph is useful in finding a winning strategy. A set of points K in a digraph is termed a kernel if (1) no two points in K are joined by an arc, (2) every point v not in K is joined by an arc (v,k) to some point k e K. The following theorem is of interest.

Theorem: Every acyclic digraph has a unique kernel.

Assume the player who makes the first move is labelled A, the other player, B. Assume further that the player who makes the last move is the winner. Then we have the important result:

52

T h e orem: If the starting point is not in the kernel of the game digraph, player A is assured of a win and A can win by always selecting points in K.

This theorem has the corollary that if the starting point is in the kernel B is assured of a win by always selecting points in K. For further reading on this topic see Berge [8], Busacker and Saaty [13], and Kaufmann [51].

P r o d u c t i o n P l a n n i n g a n d Control

One of the most common uses of digraphs in O.R. concerns depicting PERT and CPM relationships. The digraph depicting the precedence relationships is strictly speaking a network, but we include it in the present section as it is not clear how the theory of network flows aids the analysis. There are a number of advocates for each of the two approaches: activity-oriented digraphs with points representing activities and event-oriented digraphs with points representing events. The latter representation has the advantage that activity numbers indicate sequential relationships which makes it the more efficient for sol­ution by computer. However this often means that dummy activities have to be introduced.

Apart from the actual scheduling of activities it is often useful to construct the digraph (which is the pre­cedence diagram) for the purposes of assembly line balanc­ing. One common problem in this area requires the tasks to be arranged in a series of work stations so as to minimize the number of stations required without violating the precedence constraints. Chachra et al. [18] show how the theory of digraphs can be applied to solve this problem. Those authors also go on to consider various other problems in industrial engineering such as determining how many of what subcomponents must be fabricated to make a given number of final products, the determination of which process is best to make a given product, goods shipment scheduling (the transportation problem), and various organizational problems. They represent each of these problems in terms of digraphs and each model is a useful way to visualize the problem. However it is not clear how the theory of digraphs can be used to solve the problems.

5. NETWORKS

Study of O.R. network theory (NT) models has been given an important boost since George Dantzig's primal simplex algorithm [24] was specialized to the simplex method on a graph. Network optimization has become an

53

important practical branch of mathematical programming and network models are used as a fundamental planning tool in O.R. application areas (detailed late r ) . Specialized network algorithms are routinely solving models with thousands of constraints and millions of variables and are thus pushing back the frontiers of large-scale optimization. We discuss the representative sample of NT models ant methods from this vast field. A number of applicat-ons of these models are presented. Further examples of problems formulated in terms of networks are given in the survey of Kennington and Helgason [52].

The C a p a c i t a t e d T r a n s s h i p m e n t P r o b l e m (CTP)

Consider a digraph D = (V,A) in which it is desired to send a commodity between various points at minimal total cost. Each arc has a cost per unit of flow, and a capacity to accommodate flow. Each point i is assumed to obey a flow balance law, i.e. the amount of the commodity flowing into it equals the amount flowing out of it, except for a quantity bi. Let:

u^j = the capacity of arc (i,j),

xij = the amount of flow sent along arc (i,j),

bi = the supply of the commodity at point i. (Each point i for which bi < 0 is interpreted as a demand of -bi units.)

The CTP problem can be formulated as

Minimize

subject to

I(i ,j)€A

Ij

(i ,j)€A

c . .x . . ID ID

x . . ID I

j(j ,i)€A

0 < x . ■ < u . . iD ID

x ..= b ., for all i£V,

for all (i,j )€ A ,

A number of other models, some we have already seen, turn out to be special cases of the above model. These include: (i) the transportation problem, in which every point is either a proper source or a proper sink and Uij=° for all (i,j)£A, (ii) the shortest path problem in which bi = 1 or -1 for each i£V and uij = 00 for all (i,j)£A, (iii) the maximum flow problem in which bi=l for exactly one i€V, bi=-l for exactly one i V and bi=0 otherwise. (iv) the minimum cost flow problem which is a combination of (i) and (iii) , and (v) the assignment problem which is a special case of (i). The CTP algorithm to be discussed here is usually so efficient

54

that there is often no need to use a special algorithm for these specialized cases.

The matrix form of the CTP is

Minimize CX

subject to AX = B,0 < X < U.

The matrix A is the incidence matrix for D. Thus the CTP is an upper-bounded LP with a very special structure. This structure gives rise to the following theorem.

Theorem (The Basis Tree Theorem): A set of columns of A comprise a basis <=> the corresponding set of arcs form a spanning tree of the undirected graph derived from D.

This theorem allows the upper bounded revised simplex method to be used to solve the CTP very efficiently. This has been done by Bradley, Brown, and Graves [11] who have available a primal network code called GNET. The use of graph theory in modifying the simplex method is fully detailed in Kennington and Helgason [52].

There is another method for the CTP, called the out- o f - k i l t e r method, which was developed by Fulkerson [39]. Unlike the method just mentioned it was devised solely for CTP and is not a specialization of a more general method. The method begins with a feasible set of flows and a set to dual variables. It comprises two phases: one in which changes are made to flows, the other in which changes are made to the dual variables. After an iteration of each phase an examination is made of each arc in the network. Conditions for optimality (based on the Kuhn-Tucker conditions) must be satisfied by each arc. Arcs which obey the conditions are said to be in kilter and those which are not, ou.t o f kilter. At each iteration an arc has its status changed permanently to in kilter.When all arcs are in this state the optimal solution is at hand.

Computational experience comparing the two algorithms has been reported by Barr, Glover, and Klingman [4],Glover and Klingman [42]. There is conflicting evidence as to which is superior.

The M u l t i c o m m o d i t y N etwork F low P r oblem

This problem is similar to the last except that several different commodities are sent between the points of the network. The problem can be modelled as

55

MinimizeK

I ckxk rk=l

AX k

mii , k = 1,.2, , K ,

KI DkXk < R,

k=l

0 < xk < uk , k = 1,r 2 , , K

A is once again the incidence matrix and the D are diagonal matrices. The jth component of R is the mutual arc capacity of the appropriate arc, to be shared among the K commod­ities. Kennington and Helgason [52] describe in detail two of the most common algorithms for the problem which take advantage of its special structure.

CTP w ith side c o n s t r a i n t s

The following model is a specialization of CTP in which there are further restrictions over and above arc capacity and flow balance:

Minimize CX + EZ,

subject to AX = B,SX + PZ = R,0 < X < U, 0 < Z < V.

A is once again the incidence matrix but S and P are arbitrary. The methods for the CTP can be modified to handle the size constraints: SX + PZ = R. This is achieved by partitioning the basis so that a portion of it corresponds to a directed spanning tree. All calculations involving this component of the basis are executed via labelling operations rather than by matrix multiplication. (See Kennington and Helgason [52] for full details.)

CTP w i t h c onvex costs

The CTP model can be generalized by replacing the objective function by g(X), a convex function of the arc flows X. This results in a nonlinear programming problem with special structure. It can be solved by a variety of adapted NLP techniques, each of which exploit the structure. These adaptions eliminate the need for matrix manipulation and all operations can be carried out on the network D. Kennington and Helgason [52] present adaptionsof piecewise linear approximation, the Franke- Wolfe method, Zoutendijk's method, and the convex simplex m e t h o d .

56

There are numerous presentations in the literature of successful applications of network models to solve real- world O.R. problems. These include the transportation of goods, analysis and synthesis of transportation and communication networks, equipment replacement, project planning, production and inventory control, machine loading, blending, and manpower training, and financial and energy planning. Golden and Magnanti [44] have a useful bibliography of the literature on deterministic network optimization through 1977, Glover and Klingman [43] discuss network applications in government and industry, and Assad [3] surveys multicommodity network flow techniq ues.

The L i n e a r M u l t i c o m m o d i t y Model

Rao and Zionts [80] discuss the general framework in which this model has been applied to a variety of trans- portation-allocation problems. Further applications involve the allocation of empty freight cars in railways (White and Bomberault [91]), the routine of fuel oil tankers (Bellmore et al. [6]), the racial desegregation of schools (Clarke and Surkis [22]), the analysis of waste management systems (Panagiotakopoulos [76]), and problems in operations management (Elmaghraby [32]).

C o m m u n i c a t i o n s Netw o r k s

In a communications system messages pass from one station to another via channels of given capacity. This system is often modelled by a network with flow, points, and arcs representing messages, stations, and channels respectively. Problems of realizability, analysis, and synthesis give rise to linear models and computer networks give rise to nonlinear models. Flows correspond to average message rates sent on different channels in store and forward computer networks (See Fratta et al. [38]). The objective is to minimize the total delay per message given certain transmission requirements.

Sometimes it is appropriate to include a concave objective function. As an example Yaged [95] has adopted a function to reflect the economics of scale in providing a given number of channels on a given link to satisfy flow requirements. Yaged [96] poses a dynamic model for the problem of minimizing the net value of installation costs. Other models with a concave objective function arise in computer routing problems where capacities are treated as decision variables and costs are assigned to installing capacity. (See Fratta et al. [38]) .

57

T r affic Networks

The traffic assignment model is concerned with an urban street network. It is assumed that each driver chooses the path he perceives to be of least cost from his origin to his destination and that Wardrop's first principle (Wardrop [90]) holds for each origin - destination pair p - q. This principle states:

The travel costs f rom p to q o f any pa t h s a c t u a l l y used are equal a n d no g r e a t e r than the cost o f any u nused path.

The objective is to predict the user-optimal flows (also called equilibrium flows) in the network which will occur assuming this assumption of driver behaviour. Of course if the arc capacities are sufficiently generous and travel costs are constant, then each driver will simply choose the shortest path from his origin to his destination. However in practice each arc often has a tight bound on its capacity to accommodate flow. This means that travel cost in the arc is nonlinearly dependent upon the level of its flow. This causes congestion and makes it harder to predict the equilibrium flows. This is because travel costs remain unknown until flow levels are estimated.It can be shown that the problem of finding the user optimal flows is the variational problem:

rX . .Minimize ] 1 A. . (t)dt

(i,j)CA Jo 13n n

Subject to D (j ,s) + I x®. = £ x® , j = 1,2, ..., n,i=l J k=l J s = l,s, ..., n.

ns

I Xi-j = x i , (i, j) eA,x=l J

X ® . > 0 , (i,j)eA,ID s =l,2, ... ,n

where we are dealing with a network D = (V,A) with a set of n points and

A. .(x. .) = the unit travel cost of arc (i,j) when it 1 -' 1-1 contains x^. units of flow,

x^. = the total flow in arc (i,j),

x?. = the flow in arc (i,j) with destination s, ID jJ and

D(k,j) = the number of drivers with origin i and destination j .

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Many of the early traffic assignment procedures were incremental in the sense that they start with no flow assigned and progressively allocate paths to increasing proportions of the 0 -D trip demands. When all flow has been allocated traversal costs are updated and the process is repeated. The aim is to converge to equilibrium flows after a number of iterations. This iterative approach represents an improvement over the early methods such as a l l - o r - n o t h i n g approach of Loubal [62] and the diversion approach of Overgaard [74].

One of the first incremental techniques was proposed by Martin and Manheim [64]. Many improvements appeared such as those of Van Vliet [89], Dial [25], and Trahan [88]. Later procedures no longer use the incremental approach but are often primal methods producing more realistic assignments. These include Bruynoogue et al. [51] and Dafermos [23]. More recent procedures using mathematical optimization techniques have been developed by Nguyen[73], Le Blanc et al. [58], Wigan and Luk [92], and Wigan [93].

Once all traffic travels according to equilibrium flow it may be found that improvements have to be made to the network to maintain adequate service as usage increases and road surfaces deteriorate. These improve­ments often take the form of the construction of additional streets or the upgrading of existing streets. The objective of the network design model is to invest funds subject to a budgeting constraint, so as to improve the network in an optimal fashion, i.e. to reduce congestion as much as possible. Many authors including Abdulaal and Le Blanc [1], and Foulds [34] have adopted total user cost as a measure of congestion.

The design problem has the objective

Minimize / A . . (x.^ )xt ./ • ■ \ 1 3 in 1 3 (1 ,3 )eA j j j

subject to T V c . . y . . < B ,i=l j=l 13 13

i * j

x* . < M y . .1 3 * 1 3

y . . = 0 or 1 , i=l,2, ... , n.1 1 3where c^. = the cost of improving arc (i,j) if it already

exists or of constructing it if it does not,

y - ■ = 1 if arc (i,j) is improved or constructed, 0 otherwise

B = the maximum amount that can be invested,

59

= the user optimal flow in arc (i,j)

(x?.)* = the user optimal flow in arc (i,j) with 1 -1 destination s,

M = a suitably large number.

Some of the literature on problems of this type will now be surveyed. There is an excellent text by Potts and Oliver [78] which has direct relevance. This book represents a clear statement of the basic principles underlying the theory and application of network flows in transportation problems.

A number of enumerative methods for network design have appeared. Boyce, Farki, and Weischedel [10], Foulds [34] and Los [61] have presented branch and bound methods which do not take congestion effects into account. The following methods do allow for congestion. Steenbrink [85] has an iterative approach based on finding system optimal flows.

Ochoa-Rosso [75] developed a basic multistaged mixed integer programming model which is solved using Bender's decomposition technique [7], taking advantage of the decomposable nature of the multistage problem. Bruynooghe [15] also developed an integer programming model. The objective function is a combination of construction costs and user travel times. A continuous function is used to derive lower bounds by relaxing the integrality require­ments. Le Blanc [57] proposed a branch and bound scheme in which lower bounds are calculated using the Frank-Wolfe algorithm. The result is an iterative, globally covergent algorithm for the production of lower bounds.

Chan [17] has motivated the use of branch and bound techniques for solving network design problems.He reports on a set of methodologies which combine the versatility of the enumerative type algorithm with some niceties of the algebraic formulations.

Abdulaal and Le Blanc [1] use continuous investment variables and develop an equivalent unconstrained non­linear programming problem. They solve this by pattern search with promising computational results. In a later paper, Le Blanc and Abdulaal [58] present another solution procedure which uses one-dimensional search techniques based on subgradients. This reduces the problem to a traffic assignment problem. Finally,Peterson [77] has presented a network optimization problem which can be modelled with cut-flow rather than conservation of flow constraints. This approach appears more efficient for many multicommodity formulations.

60

6. CONCLUSIONS AND SUMMARY

We have presented an introduction to the complexity of problems in combinatorial optimization, graph theoretic models and solution procedures to O.R. models ranging from the Chinese po s t m a n 1s problem to routing in computer networks.

It must now be clear to the reader that graph theory and its offshoots, the theory of digraphs and networks, have blossomed not only as a branch of mathematics but also as systematic tools in O.R. It has been shown that graph theoretic models often afford new insight into a wide variety of O.R. problems.

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