Hurt, Verner G.

Stennis, Earl A.

English

Research Papers in Economics

SOUTHERN  JOURNAL  OF  AGRICULTURAL  ECONOMICS  JULY,  1975
A NEGATIVE-COST  APPROACH TO THE
FORMULATION  OF  A  TRANSHIPMENT  PROBLEM*
Earl A.  Stennis  and  Verner G.  Hurt
The  transhipment  problem  formulation  has  gressed  with  a  negative sign.  Costs  in  submatrix
been  and  is  still  being  used  extensively  by  re-  D are those reported by King and  Logan for trans-
searchers  to  solve  spatial  equilibrium  and  plant  porting processed  meat from the region  of slaugh-
location  problems.  Hurt  and  Tramel  [2],  King  ter of the region  of demand.  Live animal supplies,
and  Logan  [4],  Rhody  [7],  and  Judge  et.al.  processing capacities  and requirements  are indicat-
[3]  have  all treated  the subject  of  alternative  for-  ed in the border totals.'
mulations  of  transhipment  problems.  This  paper  Regular  and negative-cost  formulations  of the
(1)  proposes  an  alternative  formulation  of  these  transhipment-plant  location  model  can  be  ex-
problems  using a  negative-cost technique  and,  (2)  pressed  in  mathematical  terms  as  follows:
suggests  a matrix reduction  scheme  which  will  re-
duce computational  time for  some problems.  Regular Formulation:
Minimize:  X  X  Tij Xij  +-  Hi Si  +  X  Y  tij  Li
THE  KING-LOGAN  TRANSHIPMENT  i  i  i  i 
MODEL  Negative-Cost  Formulation:
King and Logan used a three region, two-stage  Minimize:  X  X  Ti  Xi  - Hi U' +  E  tij Li
formulation  of  the  transhipment  model  [4, p.98] 
which was reformulated  by Hurt and Tramel[2, p.  Both  models  are  subject  to  the  following  con-
764].  As  a  point of  departure,  the  same  sample  straints:
problem  will  be  used to present  the negative-cost  X  Lij =  Si
formulation.  This  should  enable  the  reader  to
more  readily  determine  differences  between  for-  (live animal shipments from region i equals  supply
mulations  and decide  which formulation  best serv-  in region i)  X  Xij  S i
es his particular needs. Table  1 presents the matrix  i
format for the basic problem.  Table 2 is the matrix  (meat  shipments  from  region  i  equals  animals
of costs,  supplies,  and requirements.  Costs  in the  slaughtered in region i)
submatrix  A  are  those  reported  by  King  and  Si  Si - (Li  - Lij)
Logan for live animal  shipments.  Submatrix  B has  i
no  relevance  to  the  problem  in  this  formulation  (animals  slaughtered  in region  i  equals  supply  in
and  sufficiently  high  costs  have  been  inserted  to  region i adjusted for live animal shipments)
prevent  entries.  In  submatrix  C,  all  elements  ex-  Y  Xij =  D
cept the main diagonal have been given  sufficient-  i (total meat shipments to region j equals demand in ly high costs to prevent their entry in the final sol-
ution.  The main  diagonal  of submatrix  C is com-  region  )
posed of the  cost of processing for that respective  S  - Dj
plant with the  variance  that processing  cost  is in-  (supplies must equal demands)
Earl A.  Stennis is  associate  economist  and Verner G.  Hurt  is Agricultural  Economics  Department  Head  at  Mississippi  Agricultural and  Forestry Experiment  Station.
* Mississippi  Agricultural  and  Forestry  Experiment  Station  Journal  Series  No.  2864.
1 For  a discussion  of  the relevant  theory  see  [1],  [4],  and  [5].
165Table  1.  A  NEGATIVE-COST  FORMULATION  OF  A  THREE  REGION  SLAUGHTER  PLANT
LOCATION  MODEL:  MATRIX  FORMAT a
Processing  plant  Consuming region
(region)  S
__l  1  2  3  1  2  3
Producing
region
1  A  B  Raw product
supply
2  Raw product
shipments
3
Processing
plant (region)
1  C  D  Processing
capacity
2  Excess capacity  Final product
shipment from
3  region of pro-
cessing to
demand  region
Rj  Processing  Final product
J
capacity  demand
a Submatrix  A  provides  for  shipment  of  live  animals  from  each  producing  region  to  each  slaughter
plant,  submatrix  B has  no  relevance  to  the  problem,  submatrix  C  permits  excess  slaughtering  capacity,
and submatrix  D  provides  for shipment  of  meat  from each  slaughter  location  to  each  region  of  final
demand.  Total  slaughtered  in each  area  is  the  diference between processing  capacity and excess  capacity.
0  <  Si, Si, Dj, Lij,  Xij  tij = animal  transfer  cost  from  region  i  to  re-
gion j
(non-negativity  constraints:  there  can be  no nega-  g
tive  supplies,  slaughters,  demands,  or  shipments)  Hi = slaughter cost per head in region i
Where:  ac  = dressing percentage
Xi; = meat shipment from region i to region jS  = supply of  slaughter  cattle  in  region  i,  ad-
Lij = live shipment from region i to region j  justed for dressing percent,  a
Si = slaughter of cattle in region i  D  demand for meat in region
Ti  = meat  transfer  cost  from  region  i  to  re-
gion j  Ui = unused slaughter  capacity in region  i
166Table  2.  A  NEGATIVE-COST  FORMULATION  OF A  THREE  REGION  SLAUGHTER  PLANT
LOCATION  MODEL:  MATRIX  OF COSTS,  SUPPLIES,  AND  REQUIREMENTS
Processing  plant  Gonsuming  region
(region)
. S. 1  2  3  1  2  3  Si
Producing
region  A  B
1  0  2  3  *  *  *  15
22  2  0  4  *  *  8
3  3  4  0  *  *  *  5
Processing
plant  (region)  C  D
1  -5  *  0  1  2  30
2  -7*  1  0  3  30
3  *  -8  2  3  0  30
R.  30  30  30  7  14  7
J
*Denotes  a  cost sufficiently  high  to preclude  entry in  the minimum  cost solution.
A constant can be added to or subtracted from  to  solve the problem.  The solution  is  presented  in
all elements  in  a column  or row without changing  Table  3 and  is the same  as the  solution previously
the  optimum  least-cost  distribution  patterns.  It  is  reported  [2],[4].
with this  license that the negative-cost formulation  There  are  three  justifications  for  using  this
There  are  three  justifications  for  using  this is created.  Technically,  this  is essentially  the same  i  i  i 
proposed  alternative  formulation:  (1)  It  is  in- as the  Hurt-Tramel  formulation  except  that  proc-  herently  simpler  to  formulate  the  cost  matrix,  as
essing  cost  for  a  given  plant  has  been  subtracted  resi  c  o  otae  toe  incorporate
from  all  elements  oftecoumprocessing  costs  do  not  have  to  be  incorporated from  all  elements  of  the  column  or  row  where  it  with  transfer  costs  in  either  submatrix  A  or  D. with  transfer  costs  in  either  submatrix  A  or  D. had  originally  been  incorporated.  This  results  in
* m  x p d in T e 2  T  , t  Processing  costs  are  simply  inserted  as  negative the  matrix  presented  in  Table  2.  Thus,  the  nega-  values  on the  main  diagonal  of  submatrix  C.  (2) values  on the  main  diagonal  of  submatrix  C.  (2) tive-cost  formulation  simply  says  we  are  maximi-  The  adjustment  of  the  cost  matrix  after  each
zing  savings  for  non-use  of  processing  capacity  iteration  is  greatly  simplified  when  an  iterative
rather  than  minimizing  processing  costs--which rather  than  minimizing  processing  costs-which  procedure  such  as  the  one  utilized  by  King  and
is equivalent.  Logan  is  employed.  It is  not necessary  to  recon-
The  usual transportation  procedure  was  used  struct the  entire transfer  cost submatrix - which-
167Table  3.  A  NEGATIVE-COST  FORMULATION  OF  A  THREE  REGION  SLAUGHTER  PLANT
LOCATION  MODEL:  MINIMUM  COST  SOLUTION
Processing  plant  Consuming  region
(region)
S.
1  2  3  1  2  3
Producing
region  A  B
1  15  0  0  0  0  0  15
2  0  8  0  0  0  0  8
3  0  0  5  0  0  5
Processing
plant  (region)  C  D
1  15  0  0  7  6  2  30
2  0  22  0  0  8  0  30
3  0  0  25  0  0  5  30
R.  30  30  30  7  14  7
J
ever  is used, A or D. It is necessary  only that the  matrix  generator  needed  for  a  specific  problem).
new processing  costs be inserted on the main  diag-  Despite  the  attributes  of  matrix  generators,  ex-
onal of  submatrix C.  (3)  A matrix reduction tech-  perience  has  shown  that  it  is  still  desirable  to
nique that will be presented later  in this paper pro-  reduce  a problem's  complexity  to.  a minimum.
vides the third  justification.  No  problem  is  incurred  when  the  negative-
It might  be  argued  that  simplification  is  not  cost formulation  is  used  in  a  limited  plant  capa-
necessary  with a matrix generator;  but while  mat-  cities  case  as presented  by Hurt  and Tramel  [2].
rices  can  be  constructed  with  matrix  generators,
as  a  practical  matter,  researchers  working  with  A  MULTI-REGION,  MULTI-PLANT,
smaller  problems  often  manually  construct  their  MULTI-PROCESSING,  MULTI-PRODUCT
problems.  This  avoids  computing  center  turn-  TRANSHIPMENT  FORMULATION
around  time  and  in  some  cases  eliminates  the
need  for  assistance  from  computer  programmers  A negative-cost  model  for multi-product  com-
(not  all  economic  researchers  are  competent  pro-  modity space and multi-product  plants (processing
grammers  and, while canned transportation  algori-  both  final  and  intermediate  products)  has  been
thms  are  available,  such  is  not  the  case  for  the  formulated  [8]  and  is  equivalent  to the  problem
168of this type  formulated  by  Hurt and  Tramel  [2].  inal capacity,  the  corresponding  row  and  column
rhe authors  feel that negative-cost  formulation  of  can be removed  without changing future solutions.
transhipment  problems  is  useful  for many  appli.  Usually,  this  significantly  reduces  computational
cations.  However,  depending  upon  the  number  time. However, it would not be necessary to delete
of  plants  producing  more  than  one  product,  the  all inactive  processing.  If,  in  the learned  opinion
Hurt-Tramel  formulation  may  be  superior  for  of  the  researcher,  a  plant  should  be  left  in  for
transhipment  problems  involving  multi-product  further  consideration,  this  could  be  done.  The
plants.  This  view is based  on the fact that  matrix  economic  feasibility  of  utilizing  matrix  reduction
row  space  will  be  increased  by  the  number  of  is  dependent  upon  matrix  size.  Experience  has
multi-product  plants  (the  rank  of  the matrix  will  shown  that  the  feasibility  of  matrix  reduction  is
not necessarily  be  increased  by this  number).  Due  questionable  for matrices  less  than  100 x 100  and
to  space  requirements,  a  sample  multi-region,  where  less  than  three  processing  plants  are  de-
multi-plant,  multi-processing,  multi-product  tran-  leted on an iteration.
shipment problem  and comparison is omitted here.  A  technique  utilized  to  increase  efficiency
when  using  an  iterative  transhipment  procedure MATRIX  REDUCTION  involves  using  the  solution  for  each  iteration  as
Despite  the rapidity with which  modern digia  stal  a  tating basis for the  next. This  often-used  tech-
computers  handle  computations,  large  regional  nique  will  work  even  if  matrix  reduction  is  em-
spatial equilibrium  problems  can  run  into  a  sub-  ployed because: (1) for a feasible solution (whether
stantial amount of computer time and money. One  optimum  or  not),  m +  n - 1  active  routes  are
such  problem processed  by  an  IBM 360/40  com-  required,  (2) there are two active  routes associated
puter in the Computing Center at Mississippi State  with  an  unused  processing  plant  even  if  one  of
University required  slightly over eight hours for  a  these is effectively  zero, and (3) when  a processing
single  solution  using  the  usual  transportation  al-  plant is deleted,  the corresponding row and column
gorithm  [10].  is deleted  as  well as  the two active  routes reflected
When  approximating  optimum plant  location,  for this plant,  thus  still  leaving  m  n - active
using  an  iterative  technique  such  that  used  by  routes  and afeasible  solution.
King and Logan  [4],  there are  at least three tech-
niques which  can be utilized to save computer time  SUMMARY
and expense.  With  today's  modern  digital  computers  it  is
First,  by  using  the negative-cost  formulation,  possible to formulate  and solve  increasingly  larger
revision  of  processing  costs  to  reflect  ecohomies  and  more  complex  spatial  equilibrium  problems.
of scale  is  simplified.  It is not necessary  to  revise  Increases  in  computer  speed  and  efficiency,  for
an entire  submatrix  dealing  with  both  processing  the  most part,  more  than  offset  increases  in per-
and  transfer  cost-it  is  only  necessary  to  revise  hour  computer  costs-resulting  in  a  net  decrease
the main diagonal  of the submatrix  corresponding  in  computer  cost  for  most  operations.  Even  so,
to that processing  activity,  additional  gains  can  be  realized  from  devising,
Secondly,  since  solution  time  can  increase  evolving  and  utilizing  simpler  and/or  more  effi-
geometrically  rather than linearly as size  of a mat-  cient  software  for  the  research  problems  we  an-
rix increases,  it is  imperative  that  matrix  size  be  alyze.  This  paper  proposes  two  possible  ways  to
minimized.  As  noted  by  King  and  Logan  [4],  increase efficiency  when  the transportation  model
once  a  plant  location  has  been  excluded  from  a  is used to analyze plant location.2
solution,  diseconomies  of  scale  reflected  in  the  First,  a  simplified  method  of  formulating  a
processing  cost  associated  with  it thereafter  pre-  transhipment-plant  location  model  is  presented.
elude  its  being included  in  future  solutions.  This  This formulation  eliminates  the  necessity  of  com-
can  be  taken one  step  further.  The  row  and col-  bining processing  cost  with  either  the  charge  for
umn  associated  with  this  processing  plant  can  be  trans-locating  commodities  between  raw  material
deleted  entirely from  the matrix.  In the minimum  producers  and  processors  or  the  transfer  charge
cost solution,  all Xi  -= 0, where  i  minus the num-  for processor  to consumer  shipments.  This  charge
ber  of suppliers  equals  j  represent  unused proces-  is simply  inserted as  a negative value on  the main sing capacity.  In cases  where  this  is  equal to orig-  diagonal  of  the  processor  excess  capacity  sub-
2 The authors  do not claim that these  proposals  are all  entirely innovative.
169matrix.  (This  method,  while  feasible,  is  not  re-  while  decreasing computational  time,  will  not de-
commended  for  models  involving  multi-product  crease  accuracy  or  change  the  solution.  (In  the
plants.)  opinion  of  the  authors  such  a  procedure  would
The  second  suggestion  to  increase  efficiency  be economically beneficial  only with matrices with
-involves reduction  of matrix size. It is known that,  rank >  100.)
when  using  an  iterative  procedure  such  as  the
one used by King and Logan  [4], once a plant has  Given  the  limited  funds  any  research  agency
eliminated  in an optimum  solution,  it will not re-  has  available,  it  is imperative  that problem  formu-
turn  in  a  later  iteration.  It  follows  that  there  is  lations  and solution  algorithms  be  simplified  and
often  no  need  to  retain  those  rows  and  columns  made  more  efficient.  The  proposals  presented
associated  with  inactive  plants.  Their  removal,  above  adhere  to  this  philosophy.
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170

A Negative-Cost Approach to the Formulation of Transhipment Problems,&quot;

Alternative Formulations of the Transhipment Problem,&quot;

An Interregional Model: Its Formulation and Application to the Livestock Industry,&quot;

Interregional Competitive Position of the Hog-Pork Industry in the Southeast United States,&quot; unpublished Ph.D. Thesis,

Linear Algebra for Undergraduates,

Linear Programming Methods and Applications,

Optimum Location, Number, and Size of Processing Plants With Raw Product and Final Product Shipments,&quot;

Production, Processing and Consumption of Fluid Milk in the South,

A NEGATIVE-COST APPROACH TO THE FORMULATION OF A TRANSHIPMENT PROBLEM

A NEGATIVE-COST APPROACH TO THE FORMULATION OF A TRANSHIPMENT PROBLEM

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