ANDRIS A. ZOLTNERS"
A recent article described a mathematical programming model and heuristic
solution procedure to realign sales territories. This report presents two linear
integer programming models for sales territory alignment to maximize profit.
Emphosis is placed on the development of models which are easy to implement.
Integer Programming Models for Sales
Territory Alignment to Maximize Profit
Lodish [2] describes a user-oriented computer system for sales territory alignment. The system determines which salesman should cover which account
subareas, how many trips each salesman should make
to each of his assigned subareas, and how much time
each salesman should spend calling on each account
within each assigned subarea. These decisions are
made in an attempt to maximize total expected profits
over all the sales territories under consideration.
The system is based on two mathematical programs,
MPl and MP2, which are used interactively to assign
Ssalesmen, indexed by s(s = 1, 2, ..., S), to accounts
in / subareas, indexed by / (j = 1, 2, ..., / ) . For
salesman .s assigned to subarea j , the programs determine (1) the number of trips («,j) salesman s should
make to subarea /; (2) the number of calls (JC^,) salesman
s should make on account / (ieAj) in subarea /, where
Aj is the index set of accounts in subarea j ; and hence
(3) the total time (t,j) during the planning horizon
that salesman s spends in area j not including travel
timfc to the subarea.
The total time salesman 5 spends in sale-related
activities during the planning period is
H,j = Estimated round-trip time from the salesman's home base to subarea /.
W^j = A zero-one decision variable which is equal
to one if salesman s covers subarea / and
zero otherwise.
The models constrain total sales time to be less
than or equal to T^, the total sales time available to
salesman s. The total time constraints for each salesman, and several others which require that each
account subarea be assigned to a salesman, are the
major constraints in the models.
The alignment is made in an attempt to maximize
profit. The expected profit from salesman 5 making
an average of .vj, calls on account / is a, r^,, (A ,,,), where
r,i (.v,;) is the expected long-term sales rate to account
/ if an average of -x^,,. calls are made by salesman
sper planning period and a, is the average contribution
to profit per sales dollar to account i. The total expected
profit is the sum of these expressions for each account
in each subarea and for each salesman,
profit is the sum of these expressions for each account
in each subarea and for each salesman.
Mathematical programs MPl and MP2 are reproduced in Appendix A with the notation Lodish developed. His notation differs slightly from the notation
introduced herein; in this notation the sales response
function (r^, (x^,)), the number of calls on account
( (Xjj), and the average time each salesman spends
with account i (h^^) are all salesman specific. It should
be noted, however, that the sales response function
and the average time with account i can be the same
regardless of the salesman.
The following integer programming model, IPl, is
equivalent to models MPl and MP2. That is, solving
IPl optimally will produce the same solution as would
be obtained by solving models MPl and MP2 optimally.
where:
hj, = Estimated average time salesman s spends
with account /.
*Andris A. Zoltners is Assistant Professor of Marketing, Northwestern University.
426
Journal of Marketing Research
Vol. XIII (November 1976), 426-30
INTEGER PROGRAMMING MODELS FOR SALES TERRITORY ALIGNMENT
(IPl)
(1)
Maximize
subject to:
.V,-V..
+ «.,«.,•) ^ r, for s = 1,2, ..., S
for s= 1,2,
and all it•A'
(3)
for s = 1,2, ..., s
/ = 1,2, ..., j
(4)
s
(5)
for i = 1,2,
for s= 1,2, ..., s
i = 1,2,
n ,j- integer
(7)
for s = 1,2, ..., s
J = 1,2,
(8) 0 < A,, < Maximum,, integer for 5 = 1,2, ..., S
and all i
(6)
Wjj binary
The objective function (1) maximizes total profitability. Constraints (2) require that the salesman's total
time available for calling on accounts and travel to
and from account subareas is not exceeded. Constraints (3) insure that salesman s does not call on
account ieAj more than the number of times that he
travels to subarea j . It should be noted that the optimal
solution will have n,,j = rn^x .x,,. If it were not, then
some of the salesman's time would be consumed
(constraints 2) without some compensating profit.
Because the model maximizes profitability and because the sales response functions are monotonically
nondecreasing with respect to calling effort, n^j
= niax A,, in the optimal solution. Constraints (4)
insure that between one and N,j trips will be made
by salesman s to subarea j if subarea j is assigned
to salesman s. Moreover, these constraints insure that
no trips will be made to subarea j if salesman s is
not assigned to subarea j . This, in turn, affects the
average number of calls salesman .s can make on each
account ieAj via constraints (3). If there is no reasonable upper limit on the number of trips a salesman
should make to a subarea, then N,j can be set to
an artificially large number. Constraints (5) assure that
each subarea is assigned to a salesman. If, in addition,
the model is used to determine which subareas should
be assigned to a salesman and which should not, the
equalities in these constraints should be replaced by
inequalities. Constraints (6)-(8) describe the values
that the decision variables are allowed to assume.
Eor example, constraints (8) limit the number of sales
calls for salesman s to account i to be at most
Maximum,:.
427
There are several advantages to viewing the sales
territory alignment problem as an integer program.
As the problem generally is simplified by approximating the sales response functions by piecewise linear
concave functions over the call frequency range of
interest [see 1,8], the problem can be formulated
in a straightforward linear framework. Consequently,
after only a slight modification, IPl can be solved
by use of any all-purpose linear integer programming
code. One might be tempted to disregard IPl as
computationally infeasible because of its potentially
large size. However, the integer programming literature has grown considerably in the last several years
and provides good techniques for determining good
solutions. For instance, many heuristic techniques
provide good, but not necessarily optimal, solutions.
Any of these can be modified to solve IPl. Further,
the problem structure becomes more apparent when
formulated as an integer program. IPl is a specially
structured integer program. Constraints (2) and (5)
are the only substantive constraints. They require that
each salesman's available sales time not be exceeded
and that each account subarea be assigned to exactly
one salesman. Constraints (5) are multiple choice
constraints which limit considerably the potential
solutions. Constraints (2) are mutually exclusive constraints. That is, each of the constraints involves
different variables. Taken together, these constraints
provide a structure which can be exploited readily
when solving. Hence, an integer programming algorithm can be developed that is specifically designed
for this problem. Generally, the most efficient integer
programming algorithms and computer codes are the
result of a desire to solve specially structured problems.
An easy-to-implement model, IP2, for the sales
territory alignment problem can be obtained by modifying IPl so that it contains only constraints of types
(2) and (5). This is accomplished by establishing a
priori a set of possible values for the total time, T^^
= S,^^. hjXi -\- H,j«,j, that salesman .j allocates to
account subarea / in the event that he is assigned
to this subarea. Recall that IPl solves for n^j, the
number of trips salesman s makes to subarea j ; x^i,
the number of sales calls to account ieA^; and consequently for Tjj. Model IP2, however, determines which
salesman should call on which subarea at what level
of sales effort, where the potential total sales time
allocated to subarea j is one of a predetermined set
of K values, say
The sets Tis,j) can be determined in several ways.
Typically, they would reflect each salesman's and/or
sales managers's best estimates of reasonable subarea
call strategies. Potential values for T^^^ can be deter- ,
mined on an individual account basis. For example,
salesman s would determine K call strategies for
428
subarea j in the event that he is assigned to it. These
call strategies can be specified in terms of x^^, the
number of calls made to account i. Hence, define
Xji as the number of calls made to account jeAj
according to strategy k (k = 1,2, ..., K). For each
potential call strategy the number of trips to subarea j is n^j = rn^x x% and the total sales time allocated to subarea j is
JOURNAL OF MARKETING RESEARCH, NOVEMBER 1976
subject to:
(10)
for each 5 = 1,2, ..., S
S
K
f o r e a c h i = I , 2 , ...,J
(11)
(12)
W^^,. = O o r l .
The objective function maximizes total profitability.
Constraints (10) insure that each salesman's total sales
Alternatively, potential values for T^^^. can be deter- time, Tj, is not exceeded, and constraints (11) insure
that each account subarea is assigned to a salesman.
mined at the subarea level. They can be specified
Constraints (12) require that the decision variable W^j^
explicitly or stated in terms of an interval. If stated
assumes
values 0 or 1. The decision variable can be
in terms of a min-max sales time interval, the set
interpreted
as follows.
T{s,j) can be defined as the set of values uniformly
spaced between the minimum and maximum values,
11 if a salesman s is assigned to subarea
inclusively. For each T^j^, the salesman will decide
\
for himself how to allocate sales time to each account
requiring total sales time T^j^
ieAj. On the other hand, he can use the following
FO otherwise
integer program, IP3, to determine a profitable allocaModel IP2 assumes that each salesman-subarea
tion of his time in subarea j .
combination has exactly K potential call strategies.
(IP3)
This assumption was made primarily to make the
exposition simpler. In fact, the number of potential
Maximize ^ a.r^.
call strategies will differ with each salesman-subarea
combination. Generally, there will be many potential
subject to:
call strategies for those subareas a salesman is likely
to call on and fewer call strategies for the other
subareas. Impossible salesman-subarea assignments
are removed from the model. If salesman s is to be
0 <
assigned to subarea / then the model determines the
Hjj
for each i
requisite calling effort after variables W^^j^ have been
,i,n^j integer.
removed for h^ s.
It should be noted that once a salesman's territory
CONCLUSION
has been established, IP3 can be extended to solve
for the salesman's optimal CALLPLAN. This model,
Several linear integer programming models for the
IP4, is formulated in Appendix B. Both 1P3 and IP4
sales territory alignment problem have been presented.
are specially structured integer programs.
Model IPl is equivalent to Lodish's MPl and MP2.
The expected profitability associated with sales time
As such it can be used in lieu of these models. IPl
commitment T,j^ is r,:^ = S^^^. a, r,,(x^) in those cases can be solved directly to yield an optimal solution.
where the number of calls, x^j, to each account was
Otherwise, good solutions can be obtained by means
part of the T^j^ determination. Subarea response estiof a heuristic algorithm. Lodish's marginal values
mates are needed in the case where each salesman
heuristic is an example of a heuristic algorithm for
is allowed to determine his own call policy. These
IPl. Model IP2 is a relaxation of IPl. It solves the
can be obtained in a manner analogous to that for
sales territory alignment problem in terms of sets of
account response functions. The expected profitabipotential call strategies. It is appealing from an implelity, Tjj.^, is then estimated from the subarea response
mentation viewpoint. Integer programming problems
functions.
similar to IP2 have been recognized in other applications and several good algorithms have been developed
The optimal sales territory alignment in relation to
to solve them. The interested reader should see [5],
the predetermined sales commitments, T^j^, and their
[6], and [7]. In fact, sales territories currently are
associated expected profitabilities, r^j^, can be deterbeing designed with the use of a model similar to
mined by solving the following integer program.
IP2. Computational experience to date has been excel(IP2)
lent. Seven territories have been designed from a total
of 197 geographic subareas in a fraction of a second
on a CDC Cyber 70 Model 74-18. This speed has
(9)
Maximize
allowed for the designing of numerous alternative sales
INTEGER PROGRAAAMING MODELS FOR SALES TERRITORY ALIGNMENT
territories based on varied assumptions.
A practical way to use models IPl and IP2 is to
have the solution procedure generate a set of nearly
optimal solutions in addition to the optimal solution.
Thus, the sales manager can decide the best territory
alignment from a set of potentially good ones. The
necessary methodology is outlined in [3] and [4].
The sales manager also can make final visual adjustments to the chosen alignment.
Finally, Models IP3 and IP4 solve subsets of the
total problem. Model IP4 is equivalent to Lodish's
CALLPLAN and can be incorporated into his marginal
values heuristic.
is a zero-one decision variable equal to one if
salesman s covers subarea j and zero if not.
is a decision variable equal to the number of
-calls the salesman makes to account i in subarea
j ; ieAj, A. is an index set of accounts in subarea
Jis the estimated round-trip time from the salesman's home base to subarea j .
is the total time available to salesman s.
is the amount of time being spent in subarea
jis the number of trips made to subarea j .
is the estimated average time the salesman
spends with account i.
is the expected long-term sales rate to account
I if an average of x- calls are made per planning
period.
is the average contribution per sales dollar at
account i.
is the subarea response function.
is the subarea in which account i is located.
h.
APPENDIX A
MATHEMATICAL
429
PROGRAMS MPl AND MP2
MPl
Maximize z = V RAq,,!,)
subject to:
APPENDIX B
AN INTEGER PROGRAMMING MODEL FOR
THE DETERMINATION OF A SALESMAN'S
OPTIMAL CALLPLAN
for j = 1,2,..., /
for .s = 1,2, ..., S
(IP4)
Maximize
for j = 1,2,
X^X^ a r ( x )
subject to:
f o r i = 1,2, ..., J
VV,j binary
'v, - 0 , ",jS: 0 integer
for 5
for /
=
=
1,2, ..., S
1,2, ..., J
X; < «j for each ieAj
for
1,2, ..., S
for i = 1,2, ..., J
«j > 0, integer for each / = 1,2, ..., J
0 < .X. < Maximum,., integer forfeachie A.
'MP2
Maximize
„..,.]
1.
subject to:
2.
3.
0 < J:, < \j for all i such that g,. = j
where:
is a decision variable equal to the number of
trips salesman s makes to account subarea /.
is a decision variable equal to the total time
salesman s spends in account subarea j not
including travel time to the subarea.
4.
5.
6.
REFERENCES
Lodish, L. M. "CALLPLAN, An Interactive Salesman's
Call Planning System," Management Science, Part U,
18 (December 1971), 25-40.
. "Sales Territory Alignment to Maximize Profit,"
Journal of Marketing Research, 12(February 1975), 30-6.
Piper, C. J. and A. A. Zoltners. "Some Easy Postoptimality Analysis for Zero-One Programming," Management
Science, 22 (March 1976), 759-65.
and
. "A Pragmatic Approach'to Zero-One
Decision Making," GSIA Working Paper WP-21-72-3,
Carnegie-Mellon University (August 1972).
Ross, G. T., P. Sinha, and A. A. Zoltners. "A Mathematical Programming System for Non-Selective Menu
Scheduling," Working Paper, University of Massachusetts (November 1975).
and R. M. Soland. "A Branch and Bound Al-
430
gorithm for the Generalized Assignment Problem,"
Mafhemarica/Programming, 8 (February 1975), 91-103.
7.
,
, and A. A. Zoltners. "A Note on the
Bounded Interval Generalized Assignment-Problem,"
Working Paper, University of Massachusetts (December
1975).
JOURNAL OF MARKETING RESEARCH, NOVEMBER 1976
8. Shanker, R. J., R. E. Turner, and A. A. Zoltners. "Sales
Territory Design: An Integrated Approach," Management Science, 22 (November 1975), 309-20.