LEONARD M. LODISH*
A mathematical programming model and heuristic solution procedure are developed
to realign sales territories. Unique model aspects are: (1) the objective function
is the anticipated profit generated by the sales force; (2) the interrelated problem
of account specific call frequency determination is simultaneously considered; (3)
travel time is considered, including combining calls on accounts into trips.
Sales Territory Alignment to Maximize Profit
INTRODUCTION
Many sales managers spend much time trying to
determine more profitable territory alignments of their
sales force. Given a limited sales force, how should
the boundaries of territories be determined? This
article reports on a mathematical programming model
and heuristic solution procedure for the alignment of
territories to maximize profit. The model attempts
to put the important factors of the decision into a
logical, consistent structure, A key aspect of the model
is that the interrelated problem of determining optimal
account sales call frequencies is simultaneously considered and shown to be necessary for a maximal
profit solution. Solving the mathematical program is
very difficult for real problems. Heuristics to obtain
good but not necessarily optimal solutions which
management will implement are discussed.
Currently, many sales managers are using various
ad hoc procedures to help in territory alignment. When
examined rigorously, these rules of thumb usually lead
to alignments which are not compatible with maximizing profit [6], Most of the current procedures are
primarily based on available, measurable, "hard" sales
or potential data. However, the implicit models underlying the procedures are incomplete. The underlying models need other, less measurable, "softer" data
and judgments to become compatible with maximizing
profit. In particular, the mathematical programming
model shows that the sales response functions of
individual accounts to changes in sales call effort are
an important factor affecting the profitability of alternative territory alignments. These response functions are judgmentally determined in most cases.
PREVIOUS APPROACHES TO THE
ALIGNMENT PROBLEM
Analytical literature on territory alignment is very
sparse. Most marketing writings discuss the problems,
but as Easingwood [3] commented, "exhortations
rather than solutions are offered." See [1], for example.
One analytical approach to the sales districting
problem is that of Hess and Samuels [4]. They report
practical implementation of a model which grew out
of the legislative districting problem. The model attempts to assign geographic units (e.g., countries, zip
codes) to salesmen such that each salesman has equal
"activity" and a compactness measure of the resulting
territories is minimized. The compactness measure
is a weighted sum of squared distances from the
geographic unit to the salesman's home base. The
weight which multiples the squared distances is the
activity of the geographical unit.
Hess and Samuels discuss and have used alternative
activity measures such as number of calls required
by the accounts in the unit, sales potential of the
unit, the unit's number of customers, or present sales.
They believe that the most appropriate activity measure should be related to the salesman's time required
by the unit. However, their reported applications
showed only one out of seven using salesman's time
as an activity measure.
Even if time is used as an activity measure in their
procedure, the optimal amount of time required by
each geographic unit is determined outside the model.
The optimal time to spend in each geographic unit
depends on the profitability of alternative uses of that
* Leonard M, Lodish is an Associate Professor of Marketing
at the Wharton School, University of Pennsylvania, This research
was supported by Management Decision Systems, Inc, The author
wishes to acknowledge editorial help from Ronald Frank, Paul
Green, John D, C, Little, and Sharon Casselman,
30
Journal of Marketing Research
Vol, Xlt (February 1975), 30-6
SALES TERRITORY ALIGNMENT TO MAXIMIZE PROFIT
time in other geographic units. As the following model
shows, as long as sales are responsive to changes
in time applied in each unit, the determination of the
optimal time to be spent in each unit is interrelated
with the districting problem.
The use of squared distance as a function to be
minimized is another limitation of Hess and Samuel's
procedure. As Cloonan [2] pointed out, the use of
squared distances is really not as proportional to travel
time as are unsquared distances. Also, if salesmen
combine trips to more than one unit before returning
home, then the objective function (using either squared
or unsquared distances) will not be proportional to
travel time.
Easingwood [3] develops an analytical approach
to determining regions and territories with equal workload. However, his workload measure does not include
travel time, implicitly assuming it to be equal for all
accounts. He also does not consider the profitability
implications of alternative sales call frequencies other
than those assumed in the workload definitions.
HOW THE MODEL WAS DEVELOPED
Before formally specifying the model, an informal
discussion of the part played by serendipity in the
model development should provide insight into the
model's basic concepts. The model was developed
as an outgrowth of experiences using CALLPLAN
[5], a model-based system for helping each salesman
to determine norms for the number of calls to make
on each account and prospect in order to maximize
the profit contribution of his territory. In a typical
CALLPLAN implementation 10 salesmen begin use
of the system at a 2-day introductory seminar.
During the first day the system input is explained
and the required judgments about each territory are
made by the salesman with the help of his manager.
The system input is basically the sales time and travel
time implications of alternative sales call frequency
policies and response functions relating sales volume
to alternative call frequencies. The data are entered
interactively by the salesmen at time-sharing terminals.
Optimal call frequency norms for each account are
then calculated and printed. The second day of the
seminar usually involves fine tuning of sales response
estimates, travel time, and sales time required by calls
on accounts.
During the second day typical questions that salesmen ask are: "How do I analyze whether to go on
a 'fire-fighting' call?", "Can I decide on whether to
pursue a new account without having to go back to
the computer?", "How much more sales would I get
if I could spend more time in the territory?" The
answer to these questions is related to the marginal
sales or profit if one more (or less) hour was optimally
spent with the territory's accounts. Because the
CALLPLAN solution algorithm uses incremental
31
analysis, printing out the required marginal profit of
an additional hour is easy.
Once this marginal value is printed, a natural managerial activity is comparing marginals for all the
territories. In the majority of cases the variation of
the marginals is quite large. In a typical 10-territory
district the highest marginal value is greater than 10
times the smallest. Taking sales time from territories
with small marginal value and adding it to those of
high marginal value will increase total profitability.
Marginally low productivity time is substituted for
marginally high productivity time on different accounts. Transferring time from one territory to another
is done by switching accounts, i.e,, changing territory
boundaries. The following mathematical programming
model attempts to put these marginal concepts into
a complete conceptual framework.
THE MATHEMATICAL PROGRAMMING MODEL
The mathematical program is structured with a profit
objective function and constraints to reflect time
limitations of individual salesmen. The model assumes
that salesmen repetitively call on accounts and prospects and that there is some response relationship
between time spent with the account and the account's
sales volume. The territories to be realigned are divided
into mutually exclusive, collectively exhaustive subareas. The decision variables are: which salesmen
should cover each subarea, how many trips he should
make to the subarea, and how much time he should
be spending in calling on accounts once he is in the
area. This section first describes the definition of
subareas, the decision variables, constraints on the
decision variables, the profit response functions for
each subarea, and the objective function for the
mathematical program. Next the subproblems are
described which are needed for determining the subarea response functions.
The / subareas (indexed by ; for j = 1, ,,,,;) are
defined so that accounts which are usually called on
during the same trip will be included in the same
subarea. The subareas are defined by a natural grouping of accounts that are in them rather than by more
traditional zip codes or census tracts. Each salesman
s (for s = 1, ,,,, S) is assumed to be based in one
place and have a specified round trip travel time to
and from each area. Let u^j denote the average time
per round trip for salesman s to area j . Changing
the location of a salesman would involve changing
his travel time to each subarea. The model assumes
that the time per trip to a subarea is independent
of number of trips made to other subareas. This
assumption is satisfied when the salesman returns
home before making his next trip.
The decision variables to be output by the model
are of three types. The first is a designation of which
salesman is to be assigned to each subarea. The
32
JOURNAL OF MARKETING RESEARCH, FEBRUARY 1975
collection of all the subareas assigned to the salesman
is his territory.
Let
Wjj, = (1 if salesman s is assigned to subarea /
( 0 if salesman s is not assigned to subarea
J.
Because each salesman has a limited amount of
time available, the number of subareas he can cover
depends on how much time he is to spend getting
to and from each subarea and how much time he
will be spending on accounts once he is in the subarea,
A second decision variable output is n^j, the number
of trips salesmen s should make to area j. The third
output is t^j, the amount of time during the planning
period salesman 5 should spend in area j , not including
travel time to the subarea. To retain consistency with
our previous work, time is expressed as an integer
number of time units, such as hours or half hours.
The constraints on the decision variables generate
five equation types. The model assumes that each
subarea will not be split up among salesmen, since
all accounts in the area would normally be called on
during the same trip. This constraint generates two
equations types:
(1)
and
(2)
J. is a binary integer variable, for / = 1, ,,,,
/and 5= 1, ...,S.
Each salesman has an amount of time r, available
for selling and traveling over the planning period. The
time he is spending in the areas he is covering should
be less than T^. Specifically, the number of trips made
to each area times the travel time per trip plus the
time in the area need to be summed over all areas
covered by the salesman, generating the next constraint:
(3)
Two definitional constraints are needed to specify
the model's objective function. Let qjequal the amount
of time being spent in subarea jand Ij equal the number
of trips made to subarea j . Then:
(4)
and
(5)
Because the decision variables and constraints all
relate to utilizing the time and trips of each salesman.
the objective function should relate the time and trips
to sales and profits. If all external influences are
constant, then the sales and profits obtained in a
subarea are assumed to be related to the time spent
making calls on the area's accounts and the number
of trips made to the area. Sales and profits also depend
on whether the time spent in the subarea is used in
the most profitable manner. Let R^ (q,, L) denote the
profit response in subarea jduring the planning period:
(1) if Ij trips are made to the subarea; (2) if q^ time
units, exclusive of trip time, are spent in the area;
and (3) if the time and trips are allocated to calls
on individual accounts in the most profitable way.
If desired, a salesman subscript can be added to the
response function to reflect differences in expected
salesmsin performance. The objective of the mathematical program for the alignment problem is to find
values of the decision variables within constraints (1)
through (5), to maximize the total profitability from
all areas or:
(6)
Maximize z = V i?, (q^, /,),
Call this mathematical program MPl,
THE SUBAREA RESPONSE EUNCTIONS
If a given amount of time t^ and number of trips
Ij are to be made to a subarea j , what is the optimal
number of calls to make on accounts in the area?
The anticipated profit for this optimal call policy is
the needed subarea response function value Rj (q^,,
Ij). Because many values of q^, and Ij are feasible
within constraints (1) through (5), many subproblems
need to be solved. These subproblems are reduced
forms of the individual salesman's optimal account
call frequency problem which the CALLPLAN system
attempts to solve. We next outline relevant aspects
of the CALLPLAN problem for solving the subproblems.
The decision variable of the CALLPLAN problem
is Xj, the average number of calls to make on account
i during the planning period. Each account i is located
in a subarea g.—equivalent to the subareas in (1)
through (5), Given that the salesman is in the account's
subarea, each call on account i is assumed to take
an average of h,time units, including travel time within
the subarea to get to the account. The number of
trips Ij made to the subarea j is assumed to be the
maximum number of times any account in the area
is to be called on. Let A^ denote the set of all accounts
j such that gj = J. Then Ij = Max [x, for all i e
Aj].
The number of calls made on an account is assumed
to affect its sales. The salesman on the account, along
with his manager, estimate the account sales response
to call frequency functions. Let r, (Xj) denote the
expected long-term sales rate to account i if an average
33
SALES TERRITORY ALIGNMENT TO MAXIMIZE PROFIT
of Xj calls are made per planning period (Xj = 0, ,.,,
Maximum,,). The sales response function is multiplied
by an account specific adjustment factor a^, the
average contribution per sales dollar at account i, to
obtain anticipated profit contribution. To parameterize
each account response function, estimates are made
of sales rates at five call levels: if the present call
frequency policy is continued, increased or decreased
by 50%, and if zero or a saturation level of calls
are made. Other points on the curve are obtained
by fitting smooth curves through the five points.
The salesman on the accounts, along with his manager, have been the main source for the sales response
estimates. Other information on competitive activity
and account sales potential can be helpful to the
salesman and manager in making the estimates. For
example, many companies have relatively firm sales
potential numbers for their accounts. Salesmen will
use that number as a base to estimate the sales level
at saturation call effort. The potential number will
be adjusted to reflect account policies of multiple
suppliers and the maximum share any one supplier
will be allowed.
The first stage of CALLPLAN determines the
needed subarea aggregate response functions for the
alignment problem MPl, Specifically, let MP2 be the
following mathematical program. Find Xj for all i e
A, so that:
(7)
= Max
(10)
(8)
UAj
and:
(. < /,for all ie
MP2 solves the problem of how many calls to make
on each account in a subarea, given a constraint on
time within area and trips to the area. The value of
the optimal solution to MP2 is the objective function
value for the alignment problem, MPl, for a particular
area /, selling time q., and trips Ij. As shown in [5]
the solution to MP2 is a straightforward incremental
analysis. Thus, in order to solve the alignment problem,
the optimal call frequency problem for accounts in
each subarea must be solved simultaneously. As long
as the number of calls made on an individual account
is assumed to be related to the account's sales, then
the optimal alignment will be dependent on the optimal
call frequency policy.
We now can pull MPl and MP2 together to state
the alignment problem as a main program and many
subprograms:
MPl (Main Program) find W^j, n^j, and t^j for
Maximize z =
Subject to:
(11)
(12)
0 < W^j < 1, W^j integer for s = 1, ,,,, S
and j = 1, ..., J
(13)
W^. (n,.«,,. + t^j) < T.for s = 1, ,,„ S
(14)
(15)
MP2 (Subprogram) for all feasible values of q^ and
Ij for J = 1, ,.,, /, Find Xj for all ieAj to:
(16)
Maximize
"i''i
I
i ^j- I J
Subject to:
(17)
(18)
Subject to:
(9)
s = I, ,,., Sand j = 1, ,.,, Jto:
and 0 < X j < Ij for all i such that gj = /,
This formulation assumes that the home bases of
the salesmen remain constant. A decision variable for
alternative territory centers which affect the time per
trip to the subareas could be added at the price of
more complexity. Also, as shown in [5], a travel cost
per trip to each area could be subtracted from the
objective function of MP2. However, in practice the
travel time considerations are much more important,
THE MODEL IN PERSPECTIVE
The model should not be solved exactly and implemented without any adjustments. Large scale restructuring of the sales force typically involves many
emotional consequences. Behavioral variables, such
as salesmen's personal relationships with certain accounts, their habit patterns, their prestige, and their
remuneration may be affected by territory realignment
and are not included explicitly in the model.
On the other hand, if the behavioral variables can
be simultaneously considered, the model of MPl
provides a reasonable structure for guiding improvement of the sales force alignment. The way the model
is solved in practice considers the behavioral variables
by involving the sales manager directly with the
34
JOURNAL OF MARKETING RESEARCH, FEBRUARY 1975
computer in the solution procedure. The computer
essentially provides the manager with problem territories and directions needed to improve the alignment.
How far the manager moves to remedy the problems
is left within his control because of the other phenomena not explicitly considered by the computer,
THE HEURISTIC SOLUTION OE THE
MATHEMATICAL PROGRAM IN PRACTICE
The key concept of the heuristic is that if the
territories are optimally aligned in terms of MPl, then
taking time from one territory and putting it in another
would not increase overall profits, Equivalently, the
marginal profit of time for each salesman should be
equal, assuming that each salesman is utilizing his
time to maximize the profit potential of the accounts
in his territory. If the marginal profit of each territory
is not equal, more time is needed with the accounts
in territories of high marginal values and less time
with those territories of low marginal values. Accounts
need to be switched from high marginal value territories to low marginal value territories. Because the
total sales force size is assumed constant, the scarce
resource of the model is the total time for selling
and travel available for all salesmen. Thus the relative
differences of costs of each salesmen will not affect
the model's solution. If the model were to be used
to investigate alternative sales force sizes, then the
differing costs of additional sales force time would
be relevant.
The steps in the heuristic procedure are:
Step I
Solve MPl under some simplifying assumptions:
(1) that no territory boundaries are changed (i.e,, one
and only one salesman can cover each subarea); (2)
that there is no constraint on an individual salesman's
time; (3) that there is only a constraint on the total
time of all salesmen. These assumptions imply that
there is one salesman who is covering the territory
of all salesmen, but that his trips to geographic subareas
start at different points. Call this reduced problem
MP3, This problem is conceptually identical to that
solved by CALLPLAN [5], To be specific, the simplified problem MP3 is to find the optimal number of
calls to make on each account, x,, for all accounts
in all territories (i = 1, ,.,, I) in order to:
Maximize Va,r,
such that:
= max [Xjfor all ie Aj] for j = 1,
and
When MP3 is solved, the optimal amount of time
to spend in each subarea, tj, and the amount of trips
Ij to make, are a part of the solution, i.e., tj = S h^x^
for all ii.Aj and Ij = max [x. for all i such that g, = j ] ,
Step 2
Calculate the deficiency or surplus time in each
salesman's territory, call this T^.
AT^ = 2(/,,«. + (.) - r^for i = 1
S
where the sum is taken over all subareas j which
are in the territory of salesman s. If AT, is close
to zero for all s, stop; otherwise go to Step 3.
Step 3
Management decides which men should have areas
added and subtracted from their territories to lower
the deficiencies or surpluses in Step 2.
Step 4
Change the trip time Uj for each subarea which
has been reassigned and possibly change the response
curve r, (x) for those accounts which have been shifted
to reflect differences in salesmen's performance in
the new area. Go to Step 1,
We have solved MP3 in Step 1 by running sensitivity
analyses of the CALLPLAN solution for each territory. Since the CALLPLAN problem is solved incrementally for increasing time amounts in each territory,
it can efficiently calculate the marginal profit value
and optimal time and trips for each subarea for any
time level to be spent in a territory. By evaluating
the marginal values of alternative time levels in each
territory, it is straightforward to determine the optimal
amount of time to spend in each territory to equate
the marginals. These time levels would be optimal
for the whole district, except that the salesmen cannot
physically spend the new required amounts of time.
Some accounts or subareas need to be switched. When
they are switched, the travel time to get to their area,
as well as their sales response curve may change since
a new salesman will be calling on the account(s).
At this point the manager becomes an integral part
of the heuristic. The computer has given him broad
guidelines to which he must react—see Step 3, Once
he has made the switching decision and updated travel
times and sales response curves, the solution to MP3
is recomputed. If the new travel times and the sales
response changes show too much of an effect, the
manager might have "gone too far" and, hence, has
to modify some of his switches and rerun MP3. The
manager controls where the iterative process concludes and he controls the switches to be considered.
Pathological cases can be constructed in which the
heuristic will not converge to an optimal solution to
MPl. No claim is made that this heuristic will always
work. As shown below it has improved the profitability
35
SALES TERRITORY AUGNMENT TO MAXIMIZE PROFIT
of territory alignments (in terms of the model) in the
five situations in which it was implemented.
An Example
This heuristic procedure was used to realign the
territories of four men in a district of a large industrial
products firm. First, a CALLPLAN analysis was run
to determine the optimal call frequency for each
account in each of the men's territory. The following
values were obtained for the marginal profitability
of adding an hour to each territory:
Arnold
32.98
Bruce
72,22
Charlie
136,56
Donald
64.38
The high figure for Charlie indicated that more time
could be spent with the accounts he was covering
and the low figure for Arnold indicated that the
accounts he was covering were getting too much time
relative to the other men. Thus Arnold should be
covering more accounts and Charlie should be covering
fewer.
We next ran analyses of what would happen in
Charlie's territory if more time were added and in
Arnold's if less time were added. The runs showed
that if 219 hours were added to Charlie's accounts,
the territory's marginal value would go to $64.66, Also,
if 227 hours were taken away from Arnold's territory,
then the territory's marginal value would go to $65,62.
Thus, if about 220 hours are subtracted from Arnold's
territory and added to Charlie's, the marginal values
of all 4 territories would be very close to equal.
The sales manager then looked at the required time
in each subarea, together with the output for Bruce's
territory (which is between Charlie's and Arnold's)
to see which subareas using approximately 220 hours
could be switched from Charlie to Bruce and then
which could be switched from Bruce to Arnold,
He decided to shift the Pennsylvania (84 hours)
and Wilmington (142 hours) areas from Charlie to
Bruce and the Burlington (61 hours) and Camden (174
hours) areas from Bruce to Arnold, The travel time
per trip changed only slightly as a result, since the
areas changed were at the boundaries of the present
territories. The sales manager also did not feel that
he needed to change any of the sales response curves.
The sales and profit effects of this redistricting involved a loss of $52,000 in projected sales and $21,000
in profit by spending about 220 hours less time with
the accounts Arnold was formerly covering. On the
other hand, by adding this 220 hours to Charlie's
territory, projected sales of $138,000 and profits of
$38,000 were added. The net projected increases for
the reallocation were $86,000 in sales and $17,000 in
profit. As discussed below, isolating sales increases
due to redistricting is difficult because there is no
control group. The four men involved and their man-
ager feel that they are more efficient with the new
districting.
LIMITATIONS AND DIRECTIONS EOR
EURTHER RESEARCH
Some of the problems of this approach are inherited
from the assumptions of the CALLPLAN system.
Travel during evenings is considered as having as much
time opportunity cost as day travel. In small territories
men may not stay overnight in geographical subareas.
The trip calculation in MP2 is incorrect in these
circumstances. The differential risk of achieving the
specified sales response of accounts are not considered
in the objective function. Many salesmen are risk
averters. However, the company looking at the aggregate of all accounts in territories is less sensitive to
differential risk at individual accounts.
Besides these inherited problems, other improvement opportunities involve aiding the manager in Step
3 of the heuristic by deciding which subareas should
be transferred. An integer linear program could be
solved to transfer areas to minimize travel time subject
to each area having a required number of trips and
each salesman having an equal work load. The output
of this program would then be adjusted by the manager
to reflect the other behavioral variables in the decision,
A program to suggest alternative territory centers
would also be useful at this stage. Obviously, the
whole problem MPl could possibly be solved as one
gigantic mathematical program. However, this would
be extremely expensive and would limit the sales
manager's control of the process.
CONCLUSION
Quantitatively assessing the benefits of this redistricting procedure is difficult in practice because a
control group is lacking. Five firms have realigned
territories utilizing the procedure and management has
credited many of the new changes to the computer
procedure. These are changes they otherwise would
not have made. Managers typically use the computer
output as a strategy guide, combining it with behavioral
factors to complete the decision.
Considered by itself, the mathematical programming
formulation of the redistricting problem should be a
useful guide to both researchers and practitioners.
The model shows the relationship between districting
and account call frequencies and delineates the necessary data and judgments which are needed to realign
territories to maximize profit.
REFERENCES
1, Allocating Eield Sales Resources. Experiences in Marketing Management No, 23, New York: National Industrial
Conference Board, 1970
36
2, Cloonan, James B, "A Note on the Compactness of
Sales Territories," Management Science, Part I 19 (December 1972), 469,
3, Easingwood, Chris, "A Heuristic Approach to Selecting
Sales Regions and Territories," Operational Research
Quarterly, 24 (December 1973), 527-34,
4, Hess, S, W, and S, A, Samuels, "Experiences with a
Sales Districting Model: Criteria and Implementation,"
JOURNAL OF MARKETING RESEARCH, FEBRUARY 1975
Management Science, Part II, 18 (December 1971), 41-54,
5, Lodish, L, M, "CALLPLAN, An Interactive Salesman's
Call Planning System," Management Science, Part II
18 (December 1971), 25-40,
6,
"Vaguely Right Sales Force Allocation Decisions," Harvard Business Review, 52 (January-February
1974), 119-24,