NETWORKING OR TRANSHIPMENT? OPTIMISATIONALTERNATIVES FOR PLANT LOCATION DECISIONS: AN

EXAMPLE FROM AUSTRALIAN WOOL MARKETING LOGISTICS

H.I. Toft± and P.A. Cassidy

No.16 - February 1985

Ian Toft, Senior Lecturer, Department of Econometrics, University ofNew England, Armidale, Australia.

Peter Cassidy, Head, Department of Marketing and Applied Economics,School of Business, Brisbane College of Advanced Education, Brisbane,Australia.

This work was supported by a grant from the Wool Research Trust Fundon the recommendation of the Australian Wool Corporation.

ISSN 0157-0188

ISBN 0 85834 582 X

NETWORKING OR TRANSHIPMENT?

OPTIMISATION ALTERNATIVES FOR PLANT LOCATION DECISIONS:

AN EXAMPLE FROM AUSTRALIAN WOOL MARKETING LOGISTICS

ABSTRACT

A search for rigor, relevance, and efficiency in formulating plantlocation problems has resulted in longstanding experimentation withalternative formats and solution techniques. Most notably, these havespanned the full range of programming methods fromtransportation/transhipment models through concave and quadraticprogramming to mixed-lnteger formulations. This quest has been regardedas a staple of agricultural economics and discussed extensively inthe operations research literature. While researchers have reportedmajor drawbacks with mixed-integer programming the most popular currentapproach, by way of contrast, newly emergent proponents of minimum costflow network analysis seem to have overemphasized the virtues of anotherwise promising innovation. To illustrate that not all ,traditional’methods need be ’limiting’ and ’unaccommodating’ by comparison withnetwork analysis, elements of a refined transhipment formulation asapplied in a logistics study of Australian wool assembly, packaging,storage and shipment, is described and contrasted with networkingmethods.

2

Analysis of regional competition and the spatial location of

production has a heritage extending back at least as far as the nineteeth

century, to the early elaboration of the doctrine of comparative

advantage accredited to Ricardo & Millo Early location theorists,

notably Von Thunen, Weber, Englaeder, Lunhardt, Ohlin, Palander, Hoover

Losch, and Fetter, followed and extended the classical general

equilibrium theory of Walras, Pareto & Wicksell, to embrace spatial

considerations. International trade theorists pioneered in their turn,

and contributions to further understanding of spatial factors were made

by Yntema, Mosak, and Graham among others. Such has been the

cross-fertilisation from these different strands, that the

interrelationships between classical, neo-classical, and modern trade

theory on the one hand, and location theory on the other, have been

described by Seaver as best illustrated by regarding them as the direct

parental stock of a hybrid offspring, spatial research.

Despite this long standing continuum of advance in theoretical

aspects, it was not until the formulation of the activity analysis model

of production and allocation by Koopmans, and Dantzig, in 1951, that any

empirical content could be given to the theoretical prospects offered to

date. Agricultural applications of this powerful method led the field,

as the pioneering 1952 contribution by Baumol attests to. Equally this

interest has been sustained over the intervening thirty year period, to

the extent that in 1983, this area could rightfully be described as ’a

staple of agricultural economics’ (Kilmer, Spreen and Tilley, P.731).

Fuller, Randolph and Klingman, (hereafter FRK), in their 1976

contribution, outlined the extensive treatment in the American

Journal of Agricultural Economics of spatial equilibrium analysis, and in

particular, questions of optimal plant location. Other studies have

continued this trend since in this same publishing vehicle, but FRK’s

article constitutes the starting point for the present one, as it ably

contrasts the development and application of competing methodologies

designed to formulate and solve plant location problems. Moreover, FRK

break new ground by detailing a further innovatory approach of

considerable promise, network analysis. In the present authors view,

however, their claims made regarding the supremacy of network

for~nulations over the totality of existing plant location approaches is

misconstrued. This article sets out a counter view from a stand point of

the alternative of investing in refinement of the more ’traditional’

models.

EVOLUTION AND RANGE OF MODELS : THE COMPETING METHODS

Tracing through methodological development in agriculturally

oriented spatial studies a progression is seen, where the basic

transportation model of linear programming as applied by Fox, Judge,

Judge & Wallace, and Stemberger, to derive equilibrium spatial material

flows and prices, gives away to an emphasis on the more ambitious problem

of optimal (cost-mlnlmlsing) plant location er~, as epitomised by the

studies of Stollsteimer and Polopolus. This latter group of models set

out to specify the least cost number, size and location of processing

plants, but was restrictive in the way it could handle the full interplay

between assembly of raw product, processing, and final production

distribution. As point trading models however, these formulations proved

superior to the less operational continuous space and uniform density

constructs, as applied to related questions by Olson and Williamson

previously.

King and Logan, building on the insights of Orden, revitalised the

usefulness of linear programming transportation codes in application to

more realistically specified plant location problems. This new

framework, termed a transhipment model, came to predominate for a period

in the U.S. literature, and in other countries still has a continuing

vogue.I Specifically, such formulations derive a flow and location

pattern at least cost by allowing the added possibility for commodity

shipments to be forwarded to their destinations via a series of

intermediate (supply or demand) points, rather than simply being shipped

from m surplus regions to n deficit ones. Not only the optimal flow

patterns of raw product assembly and final distribution could be derived,

but plant location transhipment models of the King and Logan genre detail

the numbers, sizes, and locations of processing plants as well.

Economies of scale in processing were incorporated by means of heuristic

techniques in deciding the integral question of plant sizes.

Furthermore, questions of the inclusion of capacity limitations in

processing, time staging, multiple products, bounding, and inequality

constraints, were demonstrated for this type of model by Hurt & Tramel,

and Leath & Martin, in realistic extensions.

At this stage of developing methodology, the compounding growth of

computer power, coupled with an on-going interest in extending realism

and r.iqour spurred attempts to apply other promising activity analysis

approaches to solve plant !ocation problems. As FRK describe, Kloth &

Blakely, and Candler, Snyder & Faught, explored separable programming and

concave programming respectively, in attempts to improve on King and

Logan’s heuristics, where non-llnear long run total processing costs were

evident. With this continuing advance of computer power and the wider

I See the 1983 application incorporated in IAC, R~ort on the MeatProcessing Industry, Australian Government Publishing Service, Canberra,1983.

availability of varying solution algorithms, researchers turned their

attention to linear programming models solved by mixed integer

techniques. Pioneering studies here range from the 1977 grain facilities

location research of Hilger, McCarl, and Uhrig, to recent work (1983) on

optimal size and siting of U.S. beef slaughter plants by Faminow and

Garhan. Without exception, these and related mixed integer applications

reported serious difficulties. In these explorations either theproblem

dimensions realistically specified exceeded the capability of any known

mixed integer code (FRK and Tcha & Lee)~ or the solutions obtained to a

simplified model lost guarantees of optimality and still proved costly in

running time; or in addition, solutions terminated after substantial

computer time, when revised, proved highly unstable. In reruns, ’what

was considered a minor change in the problem resulted in a solution quite

different from the initial advanced integer solutions’ (Faminow & Garhan

p.429).

Concurrent with this less promising experimentation with

sophisticated versions of the activity analysis programming format, FRK

were advancing an alternative innovatory format utilizing minimum-cost

network flow modelling, solved by a special purpose primal simplex

algorithm. In application, this mode which can be shown to be

mathematically equivalent to a mixed integer zero-one programming format,

proved attractive in locating the spatial configuration of cotton-ginning

plants in the U.S. south-west. A further application in 1983 by Garcia

Diaz, Fuller, and Phillips, related to export wheat marketing facilities,

while solved by the more usual out-of-kilter algorithm, repeated similar

claims to FRK’s of demonstrating network analysis supremacy over mixed

integer formulations. Indeed, by inference, these claims extended to

dominance over the whole range of alternative ’traditional’ approaches to

the plant location problem generally.

Typically network models were seen to ~e much faster and less costly

to solve than mixed integer formats° Similarly, they were said to allow

of more realistic problem specifications, due to their ease of

accommodating large scale specifications~ As ~n indication, Garcia Diaz,

Fuller and Phillips ’show that an agricultural system model involving in

excess of 3,500 nodes (constraints) and 73,000 arcs (activities) can be

solved by a network flow algorithm in less than 2 ~inutes’ (po124)o

Results such as these are certainly impressive with respect to

efficiency of central processing unit time taken, and possibly problem

realism allowed. However, before we ,throw-out the baby with the

bath-water’, by assuming that ’traditional’ methods are now superceded in

plant location studies, the authors believe it prudent to reappraise the

powers of at least one ,traditional’ approach. Specifically this article

illustrates the alternative continuing appeal and robustness of the

transhipment formulation of the linear programming transportation model,

in applications concerning the same areas of research into agricultural

marketing systems as earmarked by the proponents of network flow models.

As a vehicle designed to this end, a refined transhipment formulation is

applied to the problem of providing low cost options for the assembly,

2packaging, transport, and shipment of the total Australian woo! clip.

PROBLEM SETTING

Australian wool production is significant on a world scale

comprising around 25% of global output, and amounting to nearly 35% of

total apparel types. Of this production, around 97% (the majority in raw

form) is exported, predominantly to Northern Hemisphere customers. These

2 A full account of this modelling and research is contained in

CASSIDY, McCARTHY and TOFT.

exports flow largely, but not exclusively through three main ports:

Melbourne, Sydney, and Fremantle. Australian wool growing is widely

dispersed geographically with some 67,000 participating growers spread

throughout all states. Obviously this large scale transfer of wool from

dispersed Australian grazier producers to the end-user (in the main

Japanese, European, and Comecon spinning mills), involves provision of

extensive marketing and logistics services.

An accelerating pace of innovatory activity has been a feature of

Australian wool marketing and handling over the last decade. In

particular, the marketing/exchange sub-system has experienced major

advance in selling and display techniques arising from the innovation of

objective measurement of wool characteristics. This advance enables

buyers to purchase under guaranteed sale by certificate and sample.

Essentially these changes allow wool to be sold without the necessity of

buyers physical inspection of sale lots and brings in the possibility of

centralised selling irrespective of the locale of wool storage.

Progressively, as selling arrangements intrude less and less on the

warehousing/handling/distrlbution sequence this latter sub-system is

allowed more leeway to focus on the direct economic forces acting on it:

viz, the need to adapt to lower wool physical distribution ’pipeline’

costs.

Wool in traditional farm bales is suitable and robust for the first

stage land transportation phase. At the shipping port interface however,

the farm bales must be dumped (compressed) to ocean shipping densities.

As a result of the plethora of differing dump presses presently in

existence in wool shipping centres, wool for ocean carriage now gets

converted into a multitude of dumped bale sizes and shapes. In an era

when conventional break/bulk shipping was in use, this wide variation of

~ckage sizes was not material. Given the changeover to container

shipping in the 70’s, and most recently the negotiation of freight rates

based on more efficient use of shipping volu~e (’box~ rates in effect),

this underlying diversity is now a drawback. It is imperative for

cost~efficiency purposes, that export wool packaging conforms to two

major requirements. First, that it occupies minimum spaee ioeo is

packaged to super high densities. Secondly, these basic packages are

container~compatible to allow full use of the ’box’. Commercially

oriented research and development on wool pressing technologies to

accommodate these dual requirements, presently focuses on two main

generic super dense baling/packaging contenders~ the three bale

compactor, and the Jumbo bale.

Technological advance in super dense dumping has further spillover

effects, setting the stage for innovation at other Junctions in the wool

distribution pipeline. Both of the super dense dumping technologies

noted offer potentially large associated savings in storage space, both

prior to, and after the ocean shipment lego Logically, the further back

in the logistics chain that the bulky farm bale is transformed into a

smaller package, the more likely cost savings in space and warehousing

generally (a prime delivery cost element) might be made. To this effect,

commercial trials of super dense packaging have been instigated in some

selling centres on a prewauction sale basis, and storage costs seen to be

reduced in this fashion° Alternative centres continue the traditional

process of converting farm bales to dumped packages post~sale, Just prior

to loading for ocean shipment. With the advent of sale by sample and

centralised selling possibilities, the actual locale of storage (not only

the question of storage in differing packaging forms) must be addressed

as well. Questions of fostering production area oriented regional

storage and handling systems, versus centralised selling centre (state

9

capital cities), or portside storage, are brought into focus when

considering total system cost efficiencies.

Essentially then, optimisation of the Australia-wide assembly,

packaging, and distribution network, means modelling a system that is

compatable with the innovatory marketing sub-systems envisaged, but which

in turn, allows a full scale interplay between the competing sets of

packing, storage, and distribution innovations available, in defining a

solution. An optimal solution would specify the wool flows in which the

costs of the physical processes of assembly, storage, packaging,

transportation, and shipment were minimized. Equally, such a solution

would designate where, and by what methods the series of value-added

packaging and marketing transformations that constitute the optimal

system should take place. Additionally, in the non-capacitated case, the

model should not only aim to pinpoint the locale, but also to denote the

size of whatever packaglnE/pressing/storage investment was seen as

necessary.

To accomplish this analysis, a linear programming transhipment

formulation, solved by the transportation MODI code was employed. This

model appealed as the most efficient and robust approach compared with

the alternatives, as the following sections describe.

TRANSHIPMENT FORMAT AND REFINEMENTS

As a survey of earlier writers makes clear, the linear programming

transportation model has considerable flexibility. Orden showed that by

introducing arbltlary constants into a linear programming transportation

model, transhipment could be allowed in transportation movements. King

and Logan incorporated Orden’s insights in a linear programming

transportation model format which introduced processing as a cost, with

I0

p~ocessing costs related to per unit costs on a long run cost curve and

solved for optimal plant locations and sizes° Hurt and Tramel produced

an alternative linear programming transportation model which was applied

in King and Logan’s problem setting, using what the present authors’ term

a ~pivotal submatrix’. This particular example allowed for capacity

constraints to be introduced using this technique° Submatrices of this

type were utilized as well by Leath and Martin~ who extended the generic

model to analyze a multi~stage, multi~product problem. These latter

authors, in addition, illustrated how inequalities could be included as

constraints, leading to greater real world relevancy for this construct.

It is the experience of the current authors, that the ’pivotal

submatrix~ allows of a building block approach which confers considerable

flexibility in model formulation, and the resultant framework can be

conveniently manipulated and solved using readily available linear

programming transportation packages, as distinct from specialized

algorithms. An illustration of the use of ’pivotal submatrices~ is given

in Figure I, which depicts the application to a commodity flow of two

processes in sequence. Here submatrices D and H are ,pivotal

11

Figure I: ’Pivotal Submatrices’ DefininE A

Sequence of T~o Processing Activities

I 2 3 ~ 5 6 7 8 9

I

2

4

5

6

7

8

9

A

F

bI b2 b3 b4 b5 b6 b7 b8 b9

aI

a2

a3

a4

a5

a6

a7

a8

a9

Here aI = supply of raw material, Location I

a2 = supply of raw material, Location 2

a3 = supply of raw material, Location 3

b7 : demand for finished product, Location 7

b8 : demand for finished product, Location 8

b9 = demand for finished product, Location 9

bI (: a4) : capacity, Process I, Location 4

b2 (: a5) =capacity, Process I, Location 5

b3 (: a6) :capacity, Process I, Location 6

b4 (: a7) = capacity, Process 2, Location 10

b5 (: a8) = capacity, Process 2, Location 11

b6 (= a9) = capacity, Process 2, Location 12

In submatrix A, cij is the cost per unit of transporting one unit of

raw material from i to J and applying Process I at J.

Submatrices B and C have ’prohibitively’ high costs.

Submatrix D (a ’pivotal submatrix’) has zero per unit costs on the

main diagonal, and ’prohibitively’ high costs elsewhere o

In submatrix E, cIiJ is the cost per unit of transporting from i to

j one equivalent unit of material to which Process I has been applied, and

applying Process 2.

Submatrlces F and G have ’prohibitively’ high costs.

Submatrix H (a ’pivotal submatrix’) has zero per unit costs on the

main diagonal and ’prohibitively’ high costs elsewhere°

11In submatrix I, cij is the cost per unit of transporting from i to

j one equivalent unit of material to which Processes I and 2 have been

applied°

In the Australian wool study described, a modification was developed

of the ’pivotal submatrix’ approach, which extended the work of both Hurt

and Tramel, and Loath and Martin. This refinement made use of two or

more ’pivotal submatrices’ in parallel. This rearrangement allowed two

or more competing processes, which turnout the same end product, to be

considered with economy in the overall matrix size° Additionally, this

path also allows inequality constraints to be introduced relatively

simply, in contrast to Leath and Martin’s alternative methods.

Figure 2 illustrates the use of two ’pivotal submatrices’ applied in

parallel. In this illustrative example there are six submatrlces, each

of dimension 3 x 3.

Figure 2: Two ’Pivotal Submatrices’ Applied in Parallel

I 2 3 4 5 6 7 8 9

1

2

4

5

6

A

C

C

aI

a2

F

a4

a5

a6

bI b2 b3 b4 b5 b6 b7 b8 b9

Here number of units available in Location I

number of units available in Location 2

number of units available in Location 3

bI = capacity available to Process X, Location 4

b2 = capacity available to Process X, Location 5

b3 = capacity available to Process X, Location 6

b4 : capacity available to Process Y, Location 4

b5 = capacity available to Process Y, Location 5

b6 = capacity available to Process Y, Location 6

a4 = bI + b4 = capacity available to X or Y, Location 4

a5 = b2 + b5 = capacity available to X or Y, Location 5

a6 = b3 + b6 = capacity available to X or Y, Location 6

b7 : number of units demanded for the output of Process X or Y,

Location 7

b8 = number of units demanded for the output of Process X or Y,

Location 8

b9 = number of units demanded for the output of Process X or Y,

Location 9

In submatrix A, cij is the cost per unit of transport from i to J

and applying Process X.

In submatrix B, cij is the cost per unit of transport from i to J

and applying Process Y.

In submatrix C, ’prohibitively’ large per unit costs are used.

In submatrices D and E, zero per unit costs are used in the main

diagonal and ’prohibitively’ large per unit costs elsewhere.

11In submatrix F, cij is the per unit cost of transport from i to J

after processing through Process X or Y~

The number of units allocated to row one of submatrix F equals the

sum of the number of units allocated to column one of A, and column one

of B, and so on. The procedure can be extended with more than two

submatrices in parallel where there are more than two processes competing

to produce a homogeneous output.

As stated, ’pivotal submatrices’ linked in parallel, apart from

reducing matrix size, can facilitate the introduction of various

15

inequality constraints. Two illustrations of this are given. The first

is the introduction of a lower and upper limit on the flow through a

particular node. This is of direct interest in network analysis as

indicated by Fuller, Randolph and Klingman, and by Garcia-Diaz, Fuller

and Phillips. A transhipment model matrix formulation which will

constrain the quantity processed at location J to be in the interval

(kjL, kju) is given in Figure 3.

Figure 3: Parallel ’Pivotal Submatrices’ Introducing Bounded Flows:

viz, Quantity Processed at Location J in the Interval (kjL, kju)

I

2

I

2

C. o

A B

klL k2L k3L

F

I

D1 D2 D3

sI

s2

s3

kIu

k2u

k3u

Column totals associated with submatrices A and D

are kjL J = I, 2, 3

Column totals associated with submatrices B and E

are (kju - kjL) ; J = I, 2, 3

In submatrix A, cij is the per unit cost of transport from i to j

and of processing at J o

In submatrix B, cij,s are indentical with those in submatrlx A.

I ,In submatrix F, cij s are the per unit cost of transport of the

processed good from i to J.

This second illustration details the use of parallel submatrices to

impose constraints. Here the situation depicted is where the per unit

cost of a flow xij is cij for 0 < xij ~ kj and cljfor xij > kj, where

Icij> cij for corresponding oells i, J. This reproduces the cost

situation illustrated in Figure I, of FRK.

An illustrative formulation to achieve a constraint of this type is

given in Figure 4~

17

Figure 4: ,Pivotal submatrices’ introducing inequality constraints:

viz, Per Unit Cost cij up to Quantity kj,

Per Unit Cost cij beyond kj, eij > cij

I 2 3 I 2 3 I 2 3

I

2

I

2

cij

D

i

B

E

11cij

sI

s2

s3

k1

k2

k3

kI k2 k3 A A A DI D2 D3

Quantities of raw material available = si; i= I, 2, 3

Quantities of processed material demanded i: I, 2, 3

Column totals associated with submatrices A and D

are kj~ J= I, 2, 3

Column totals associated with submatrices B and E

are A, where3E si

i=l

3andA> ED

=i=l i

In submatrix A, cij is the per unit cost of transport from i to J

and processing at J (up to kj units).

18

Figure 5

COMPETING FLOWS IN THE WOOL DISTRIBUTION PiPELiNE

SUPPLY AREAS

LOCALPROCESSORSFARM BALES

ANDDENSE BALES

SUPER DENSE PACKING/COMi::~ESSING/STORING

SHIPPING PORTSOR PORTSIDE

REGIONAL CENTRES

A.W.C.STOCKPILE

LOCALPROCESSORSSUPER DENSE

DUMPED

ORDINARYDUMPING

(CO~PR~SS~N~

TOTALDEMAND

SHIP- SIDECONTA]NERTE. RMINAL

IIn submatrix B, cij is the per unit cost of transport from i to J

and processing at J (after first kj units).

Icij > cij for corresponding cells (i~ J) in A and Bo

11In submatrix F, cij is the per unit cost of transport of the

processed goods from i to J o

THE AUSTRALIAN WOOL PHYSICAL DISTRIBUTION STUDY

An illustration of the Australian wool logistics flows to be

modelled are provided in Figure 5. A transhipment model using the MODI

algorithm was deemed an efficient approach to the task. In order to

apply this algorithm the data were set up in a matrix format as depicted

in Figure 6. This required sixty submatrices AI through L5, the overall

matrix size being 90 x 73. To keep the matrix size to a minimum, pivotal

submatrices were used in parallel where possible. Instances of this are

submatrices D3, E3 and F3 and submatrices B5 and GSo

The analysis seeks to minimize the cost of distributing wool from 59

supply areas to meet demand by three sectors: overseas demand at

shipping ports, demand by local wool processors at 15 locations, and

price stabilising stockpiling by the Australian Wool Corporation. Wool

stockpiled for this purpose is centralised at shipping ports. Three

types of flows can be identified. Firstly local processors can obtain

supplies direct from the 59 supply areas. This would be represented by

flows allocated to submatrix J1o Other than this first stage draw-off,

the remainder of the wool cllp moves through the assembly stage

(submatrix At), and wool flows from the 59 supply regions to 18 regional

centres and ports~

2O

Secondly super dense packaged wool is required by the three final

demand sectorso To this end, Wool which has been compressed into

super-dense bales by any of 3 separate competing processes situated at

regional sites or portside, will flow through one or more of Activities I

through 6. For example, wool flowing through Activity 1 would have

quantities in the optimal solution in submatrices AI, BI and any

combination of 15, K5 and L5.

Wool flowing through Activity 2 would feature in flows in

submatrices AI, C2, and any combination of D4, I4 and K4. Wool moving

through D4 would then flow through any combination of I3, K3 and L3.

Wool packaged in Activity 3 would flow through submatrix At, E2 and any

combination of I3, K3, and L3. Similarly wool moving through Activity 4

would flow through At, F2 and any combination of I3, K3 and L3. In

Activity 5 wool flows through At, G2, and any combination of I5, K5 and

L5. In Activity 6 wool flows through At, H2, and any combination of I3,

K3, and L3.

The third type of flow concerns wool in farm bales that does not get

compressed into super dense bales. In this case such wool will flow

through submatrices At, and any combination of 12, K2 and L2 as dense

(but not super dense) bales. Flows through Activities I through 6 are

constrained by the super dense baling capacities available at various

locations. Flows of wool as dense bales do not have this constraint. In

this way, the model allows a solution to be found in cases where the

total flow available is greater than total super dense compressing

capacity. The super dense baling capacities were varied in exploratory

runs on the basic model.

22

pattern of supply from different areas was not varied. Total supply

amounted to 4,002,470 farm bales (i.e. the 1981-82 wool clip).

Four basic settings of super dense compressing capacities were

used:-

(a) Zero super dense capacities were stipulated, where selected column

totals were set at zero to disallow wool flows through Activities 1

through 6. This case was seen as a ~Benchmark~ or calibrating

computer run, involving no innovative packaging technologies.

(b) A low capacity setting was specified: only presently installed,

super dense compressing faciltles were allowable°

(c) A high capacity setting was nominated: including presently

installed super dense plant as augmented by high density presses

capable of modification for compressing to super densities.

(d) Unlimited super dense compressing capacity was allowed.

Three of Activities I through 6 compete for the same super dense

compressing capacity. This problem was approached for both the low

capacity and high capacity eases by an initial allocation of trimpacking

capacity at shipping ports between Activities I, 4 and 5 in equal

proportions i.e. I/3 : I/3 : I/3. Analysis of initial computer runs

indicated that to minimize total cost this should be re-allocated to

Activity 4, that is, the allocation between Activities I, 4 and 5 should

be 0 : I : O,

In addition to variations in super dense compressing capacities, and

in the allocation of compressing capacities between Activities I, 4 and

5, various levels of costs as penalties were allowed in submatrix L2 for

wool available for export in high density rather than super dense bale

21

APPLICATION SOLUTIONS AND RESULTS

The model outlined in Figure 6 was solved by the MODI transportation

algorithm using a DEC20 computer. A number of different simulation runs

were completed based on varying policy-relevant parameters, with the

central processing unit time approximately 45 seconds per run.

In order to carry out a computer run, it was necessary to designate

per unit cost elements° These can be split into three types° Actual

assembly or processing costs~ which necessarily would be greater than or

equal to zero° A zero per unit cost could occur where the row and column

involved are at the same location and no processing is involved. Per

unit costs of zero also occur on main diagonals of ~pivotal submatrices’.

Finally there is a need to enter ,prohibitively large’ per unit costs to

rule out disallowed movements° Only the first two types of cost data had

to be entered into the computer, as the program initialized all per unit

costs at ’prohibitively large~ values as a starting point°

Apart from the per unit costs, it was necessary to enter the

designated row and column totals° Some of these~ in particular the

allowed levels of the super dense packaging capacities~ at different

sites, were varied between eomputer runs. Other variations included the

allocation of demand by local processors between wool purchased direct

off-farm by private treaty and Wool purchased through the auction system.

Exploratory variations of the policy relevant demand spread of the

Australian Wool Corporation between stockpiling of high density or super

dense bales were made in addition° Stockpiling by the Australian Wool

Corporation was taken to be the residual resulting between supply from

the 59 supply areas in 1981~82 (the base year) and the sum of overseas

and local demand (by domestic first stage Australian processors). The

23

form. These variations tested the sensitivity of system solutions to

ocean shipments in packages costlier to ship than super dense dumped

wool. Equally these variations were seen as an aid to future freight

rate negotiations, and as a test of the investment in the new technology

recommended by the solutions. Some selected results from the wider

investigation are listed below.

Table I: Summary of Selected Computer Runs I through 5

Run Super Dense Shipping Activities

No. Capacity Penalty in theAssumptions Cost on Optimal

Non-Dense Bales SolutionsCents (Aus)/kg (Activities I

through 6)

I Zero capacity 4 Nil

2 Low capacity 4 2, ~3, 4, 6Allocation of , ,3 B C capacity£a~

ports solelyto Activity 4

High capacityAllocation of3 B C capacityat shippingports solelyto Activity 4

4 Unlimited 4Capacity

Unlimited 3Capacity

2, 3, 4, 5, 6

4

Change in costover Benchmark

Computer Run$ (Aus) million

Bench Mark

-$8.1

o$io.o

-$12.6

-$12.6

NOTES:

(a) 3 B C - Three Bale Compactor

(b) In relevant Computer Runs (2 to 5), Australian Wool Corporationstockpile demand is allocated on a 50/50 basis between super densebales and other forms, while local processor demand is allocated ona 25%/75% basis between farm treaty purchases and purchases throughthe auction system.

24

The results reported in Table I indicate the potential dominance of

Activity 4, if capital investment were to be channeled in this direction.

In a nutshell, packaging wool to super density via three bale compactor

techno!ogy (before sale at portside locations) proved dominant over other

competing packaging technologies, and over packaging at

productionmoriented regional centres.

As the model was highly efficient in terms of solution time taken,

computing input, and set up time, the authors were able to employ it as

reported, as a simulation type analytical tool to probe the sensitivity

of the wool distribution system design to potential change. In all 20

model runs were performed to answer policy relevant queries of the

following type:

I. What impact on the present wool distribution profile will result from

the interplay of the range of packaging innovations/investments now

mooted and/or in place?

2. Can investment in regional packaging compete with central port

locations?

~.3. How stable are these location specific investments to cost changes?

4. Is any particular central shipping port favoured or disfavoured?

In the light of the nodel’s optimal wool flows, is there adequate

storage capacity at active centres?

Is there enough super dense packaging capacity at optimal locations?3

3 An account of these model explorations is given in full inCassidy, McCarthy and Tofto

25

SUMMARY

The continuing quest by applied economists in search of rigor,

realism, and efficiency in their models of locatlonal analysis, has taken

researchers far in experimentation with alternative formats and solution

techniques. A continuum of applications from transportation models of

linear programming and their variants, through quadratic, concave, and

mixed integer programming, down to the recent adaption of network flow

analysis has been noted. While researchers have reported major

limitations with several of these formats eog o the inefficiency in

solution of mixed integer programming formulations, by ways of contrast,

the authors believe recent proponents of network methods have

overemphasized the virtues of their approach. FRK, and Garcia-Diaz,

Fuller, and Phillips, in their applications both claimed supremacy for

networking over ,traditional’ methods (including models of the

transhipment type). On the basis of the very criteria put forward by

these proponents, the claimed dominance of networking over transhipment

formats is disputed here on a number of fronts°

Firstly, the issue of realism in problem specification is said to

confer dominance on network methods° Essentially as FRK see it, this

question turns on the ability of networking to incorporate ’several

dimensions not conveniently incorporated into existing location models’

(p.435). Addressed here are the problems of pleeewlse linear costs,

including set-up charges, coupled with an increasing level of labour

costs (overtime working) reacting with the usual interplay of assembly

and storage/processing charges. It is eo~uterelaimed that all of these

dimensions can be contained conveniently and efficiently within the

refined transhipment alternative outlined. Specifically, the differing

costs of bounded flows in the transhlpment matrix (illustrated in Figure

26

4) efficiently encompasses the problem of this cost step-up flowing from

overtime plant utilization, and similarly a transhipment model can allow

for the stepwise linear variable cost function envisaged by FRK in their

Figure I. Equally the upper and lower bounding of quantity flows on

arcs, as seen in network analysis, is conveniently handled in

transhipment formulations, as the example in Figure 3 depicts. Realism

in any of the senses claimed by FRK for networking methods, can be

matched by the transhlpment formulatlon~ and likewise, a complete

interplay of assembly processlng/storage and distribution costs necessary

to reach a least cost system solution overall, is the epitome of this

approach.

Turning to the question of the efficiency in solution procedures of

these alternatives, the criterion offered here by network analysis

proponents is one of central processing unit time taken. In this

dimension, both network analysis and transhlpment models are vastly

superior to their competitors e.g. the inefficient mixed integer

approaches. On a strict comparison, however, between transhlpment and

network models~ there seems to be little material difference in raw

solution speed. Indeed Glover, Karney, and Klingman, in designing and

testing the speclalised primal-simplex algorithm used by FRK, note that

the primal transportation code is the faster solution algorithm (by some

10 percent) if the problem can be efficiently specified in the

transportation model framework. This conclusion is supported by the

results of our study where CPU time averages only 2.5 seconds. As a

further bias here, the present authors note that the transhipment model

is solved by widely available codes tailored to compatability on a huge

array of machine types. On the contrary~ FRK’s analysis made use of an

esoteric specially formulated primal-simplex network algorithm.

27

Taking the comparison one step wider, from simply model solution

time to include a consideration of overall model set-up time as well, may

produce a greater degree of variance. On the basis of FRK’s Figures 2

and 3, the authors consider it plausible to hypothesize that using the

’traditional’ transhipment framework could involve less of a researcher’s

time in problem specification and setmout, than would a network analysis

model.

Finally, FRK in their cotton-ginning application utilized a neat

implicit enumeration technique to multi~stage the optimal plant location

search process into sub-problems. Linking this enumeration stage with

network analysis proved efficient for the problem addressed. Such a

staging technique is not intrinsic to the network approach, and the self

same method could equally be applied with implicit enumeration and

transhlpment as the compatible solution modules employed°

Given the evidence assembled, and simply using the criteria offered

by network proponents, the authors conclude that at least one

’traditional’ approach is far from superceded in plant location analysis.

Rather, the two related methods of network analysis and transhipment

modelling should be looked on as likely alternatives. The actual

methodological selection for any plant location problem should be on the

simple basis of a ’horses for courses’ acco~odation, rather than

directed via any belief in some overall dominance by any one technique.

28

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Q

WORKING PAPERS IN ECONOMETRICS AND APPLIED STATISTICS

~ke Prior Likelihood and Best Linear Unbiased Prediction in StochasticCoefficient Linear Models. Lung-Fei Lee and William E. Griffiths,No. 1 - March 1979.

Stability Conditions in the Use of Fixed Requirement Approach to ManpowerPlanning Models. Howard E. Doran and Rozany R. Deen, No. 2 - March1979.

A Note on a Bayesian Estimator in an Autocorrelated Error Model.William Griffiths and Dan Dao, No. 3 - April 1979.

On R2-Statistics for the General Linear Model with Nonscalar CovarianceMatrix. G.E. Battese and W.E. Griffiths, No. 4 - April 1979.

Const~{ction of Cost-Of-Living Index Nmnbers - A Unified Approach.D.S. Prasada Rao, No. 5 - April 1979.

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Estimation of Household Expenditure Functions: An Application of a Classof Heteroscedastic Regression Models. George E. Battese andBruce P. Bonyhady, No. 7 - September 1979.

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A New System of Log-Change Index Numbers for Multilateral Comparisons,D.S. Prasada Rao, No. 9 - October 1980o

A Comparison of~{rchasing Power Parity Between the Pound Sterling andthe Australian Dollar - 1979. W.F. Shepherd and D.S. Prasada Rao,No. i0 - October 1980.

Using Time-Series and Cross-Section Data to Estimate a Production Functionwith Positive and Negative Marginal Risks. W.E. Griffiths andJ.R. Anderson, No. ii - December 1980.

A Lack-Of-Fit Test in the Presence of Heteroscedasticity. Howard E. Doran

and Jan Kmenta, No. 12 - April 1981.

the Relative Efficiency of Estimators ~G~ich Include the InitialObservations in the Estimation of Seemingly Unrelated Regressionswith First Order Autoregres~ive Disturbances. H.E. Doran andW.E. Griffiths, No. 13 - June 1981.

An Analysis of the Linkages Between the Consumer Price Index and theAverage Minimum Weekly Wage Rate. Pauline Beesley, No. 14 - July 1981.

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