Determining Optimal Discount Policies in B2B Relationships

Viktoryia Buhayenko

and Erik van Eikenhorst

Department of Economics and Business, School of Business and Social Sciences, Aarhus University, Fuglesangs Alle 4,

8210 Aarhus V, Denmark

Faculty of Economics, Informatics and Social Sciences, Molde University College, Britvegen 2, 6410 Molde, Norway

1 STAGE OF THE RESEARCH

The research is preliminarily divided into three stages.

The current work is close to the end of the ﬁrst stage.

At this stage two heuristic algorithms have been de-

veloped for a non-restricted one-item case. They

have been tested on a simple example problem. It is

planned to test models on more instances.

2 OUTLINE OF OBJECTIVES

Distinctive objectives can be outlined for each stage:

1. development of heuristic algorithms for an unre-

stricted one-item case (stage 1, current stage);

2. extension of the problem to a multiple item case

with a capacity restriction, development of an al-

gorithm based on metaheuristics (stage 2);

3. aplication of an alternative approach — possibly

game theory — reformulation of the problem if

needed (stage 3).

3 RESEARCH PROBLEM

This research deals with the question of which dis-

counts (when and how much) a supplier needs to offer

to a set of heterogeneous customers to maximize his

proﬁt.

The supplier has a possibility to regulate the de-

mand using discounts — to increase the demand in

periods when the product is produced and to lower the

demand in periods with no production of the product.

Savings for the supplier arise from reduction of set up

and inventory cost, but the buyer gets extra inventory

cost. The discount offered by the supplier should be

large enough to make the buyer order anyway at the

period wanted by the supplier.

The major assumptions of this study are the fol-

lowing:

• Monopolistic situation of the ﬁrm or very high

barriers for switching suppliers imply that price

levels of other ﬁrms need not be considered here.

• Perfect information about the demand and cost of

each buyer.

• Buyers and the supplier have no capacity or ware-

house restrictions (valid only for the ﬁrst stage).

• Buyers are considered to be rational in their reac-

tion to the discount by always choosing the lowest

cost option available.

• Buyers have full information in advance about fu-

ture discounts.

• Simple discount in the form of a single price re-

duction.

The topic is very different from standard yield man-

agement where total demand can be affected with

prices. In the researched case, the total demand re-

mains the same; there is only a question of when this

demand is ordered and produced. If the price is low

in one period, buyers will be induced to order in that

period and there will be less demand in the adjacent

periods. Thus, the problem is very distinctive in the

fact that price changes in one period will affect the

demand in other periods.

This approach can be applicable in situations

when the buyer represents a retailer with his own cus-

tomers which considers being more proﬁtable not to

introduce discounts to them. This can appear in a

number of cases:

• A physical cost of changing prices can be too high

— labor costs of changing shelf prices and rela-

belling, the costs of producing, printing, and dis-

tributing price books. These costs can take up to

40% of the reported proﬁts (Levy et al., 1997).

They are still vital especially for small retail shops

lacking contemporary digital devices.

• Price decisions can be time consuming for the cor-

porate management (Bolton et al., 2010).

• The retailer can lose its customers’ loyalty,

Buhayenko V. and van Eikenhorst E..

Determining Optimal Discount Policies in B2B Relationships.

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

who might consider him being too opportunis-

tic (Mauri, 2007).

• Prices of the retailer can be ﬁxed by the govern-

ment or a franchise giver.

• Customers are heavy equipment or airplane man-

ufacturers and have stable demand for spare parts

which are only a minor component of the ﬁnal

product (Lal and Staelin, 1984).

In these cases despite getting items for lower price

from his supplier, the buyer has no incentive to in-

troduce discounts to its own customers. Obviously

the demand of the ﬁnal customers doesn’t increase.

In this situation the buyer’s demand is insensitive to

price changes, in other words price elasticity of the

demand is low.

4 STATE OF THE ART

Many articles in the ﬁeld of operations management

have analysed ordering decisions while quantity dis-

counts are in place (Benton and Park, 1996). The

problem of when and how much discount to offer is a

problem that has received less attention, although it is

of equally great practical importance as how to act on

a given discount.

Nevertheless, there is a number of articles ad-

dressing the problem of offering an optimal discount

schedule from the supplier’s side (Crowther, 1964),

(Monahan, 1984), (Lal and Staelin, 1984), (Rosen-

blatt and Lee, 1985), (Lee and Rosenblatt, 1986),

(Banerjee, 1986). Later papers in this ﬁeld base their

research on EOQ assumptions a well (Busher and

Lindner, 2004), (Chen and Robinson, 2012). These

papers share one more unifying feature — an assump-

tion that customers’ demand is independent of dis-

counts. That is an assumption valid for the current

paper as well.

The literature researched above indicates that a sit-

uation with more than two players is not considered

until recently. If a number of customers is consid-

ered, they are assumed to be homogeneous, or het-

erogeneous only in their demand. Discount sched-

ules offered by the supplier are constant and involve

limited number of break points (very often only one).

The same discount schedules are introduced for all

the customers. The models presented in the articles

are based on EOQ assumptions. To the best of our

knowledge capacitated problems as well as cases with

multiple items have not been considered yet.

More recent papers incorporate the price elastic-

ity of demand, which makes their research closer to

revenue management.

This research differs from the approaches stated

above in the following way:

• some of the EOQ assumptions are not applied;

• dynamic demand and ﬁnite time horizon are sup-

posed;

• a number of heterogeneous customers are consid-

ered, who are different not only in their demand

but in their holding and order costs;

• discounts are different for every single customer;

• discounts can vary from period to period.

5 METHODOLOGY

The methodology described in this paper concerns

stage 1 of the research. Currently two heuristic al-

gorithms have been developed. Later it is planned to

apply a metaheuristic (stage 2) and a game theoretical

(stage 3) approaches.

5.1 Initial Situation

In the current situation the customers decide their or-

ders based on the Wagner-Whitin algorithm (Whitin

and Wagner, 1958), the supplier receives the orders

and applies the Wagner-Whitin algorithm to schedule

his production based on these orders.

5.2 Cost Compensation Heuristic

Exact solutions to the problem are very hard to obtain

and would require an exponential amount of binary

variables, representing each possible order schedule,

for each customer.

Therefore, a heuristic solution approach has been

developed which involves a separation between the

problem when production and orders should take

place, and the amount of discount that has to be of-

fered to each costumer in each period to make them

order at the periods indicated.

The following parameters are used for deﬁning the

algorithm:

demand for every customer i(i = 1, . . ., n or i ∈ N)

in every period t (t = 1, . . . , m or t ∈ M). Demand

of the supplier is the summation of his customers’

orders in that period;

ﬁxed order processing/set up costs for every cus-

tomer i and the supplier i = 0;

inventory holding interest rate for each customer i

and the supplier to carry a monetory unit of inven-

tory from period t to period t + 1, assumed to be

constant;

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unit price for every customer i and the supplier in

every period t. Initially, unit price is set equal to

the original price without discount. The unit price

for the supplier is given for symmetry reasons in

order to calculate inventory holding costs;

costs for every customer i and the supplier before

introduction of discounts.

The model operates with the following decision vari-

ables:

inventory for every customer i and supplier i = 0

in every period t;

binary variable for every customer i and supplier

i = 0 in every period t, 1, when the order/set up is

planned, 0 otherwise;

total amount of compensation for every customer

order quantity for the customer i and the supplier

i = 0 in every period t.

The solution procedure includes the following

three steps:

STEP 0.

The problem for each customer and the supplier is

solved using the Wagner-Whitin algorithm. Order and

production patterns before introduction of discounts

are obtained at this stage. They are not used in further

calculations and are a benchmark for results received

at Step 1.

Furthermore, a new parameter ω

itk

for every cus-

tomer i for every period t and k (t ≤ k) is introduced

and calculated. The parameter consists of the costs of

ordering the demand from period t to period k in pe-

riod t and the minimum cost incurred up to period t.

It is calculated according to the Wagner-Whitin algo-

rithm.

Costs c

for customer i before introduction of dis-

counts are received at this step:

= min

t∈M

itm

, for ∀i > 0 (1)

remains unchanged at this and the next step and

is the same for every customer i and every period t.

STEP 1.

In the previous step parameter c

which is going to

be used below and parameter ω

itk

which will be used

further in Step 3 have been calculated. The order and

production patterns can be determined as well which

are in place before introduction of discount. They can

be used as a reference while comparing the received

results.

Objective (2) is minimized for the supplier:

minimize

∑

t=1

∑

t=1

∑

i=1

(2)

subject to

∑

t=1

∑

t=1

∑

t=1

− P

≤ c

, for ∀i > 0 (3)

Constraint (3) ensures that there is no cost increase

for any customer i.

There are two additional constraints (4) and (5) for

the customers:

∑

u=t

− Q

≥ 0 , for ∀i > 0 , ∀t (4)

+ H

it−1

− H

= d

, for ∀i > 0 , ∀t (5)

For the supplier constraints (6) and (7) are in

place:

∑

i=1

∑

u=t

− Q

≥ 0 , for ∀t (6)

+ H

0t−1

− H

∑

i=1

, for ∀t (7)

For t = 1 constraints (5) and (7) are slightly modiﬁed

since H

it−1

equals to 0.

Constraints (5) and (7) ensure the continuity of the

ﬂow. They guarantee that the demand is satisﬁed from

production or inventory.

Constraints (4) and (6) link binary variables S

with continuous variables Q

forcing S

to take value

1, when Q

≥ 0, and 0, when Q

= 0.

It can be shown that the model offered above en-

sures both supplier and customers’ cost minimisa-

tion. Supplier’s costs and compensations offered by

the supplier to the customers are minimized in the ob-

jective of the model. The compensations are a lump

sum that would compensate the customer for order-

ing at the periods requested and represent the increase

in customers’ costs. Minimizing compensations the

model minimizes not only supplier’s but customers’

costs. This statement can be proved, since

∑

i=1

minimized in the objective function (2), constraint (3)

can be written as an equality and the objective func-

tion can be presented in the following way:

minimize

∑

i=0

∑

t=1

∑

i=0

∑

t=1

∑

i=1

∑

t=1

−

∑

i=1

(8)

DeterminingOptimalDiscountPoliciesinB2BRelationships

∑

i=1

and

∑

i=1

∑

t=1

are constants and can be

omitted in the objective function. The received for-

mula accounts for minimization of both supplier’s and

customers’ inventory and setup costs. A compara-

ble approach is employed by Lal and Staelin (Lal and

Staelin, 1984).

STEP 2.

After application of the model described in Step 1,

the new order and production patterns are determined.

All the variables are ﬁxed now. Only parameters ac-

counting for prices — v

— or discounts offered to

achieve these order and production patterns are calcu-

lated at this step.

The following new parameters are introduced at

this step:

itk

amounts (demand and inventory) which depend

on price in period t for an order from period t up

to period k (t ≤ k) for customer i;

period in which customers order according to the

order pattern received in Step 1;

period until which the order starting at

accord-

ing to the order pattern received in Step 1 lasts;

period in which customers order for period x

given the time horizon of y

(according to the or-

der pattern which is in place before discounts are

introduced);

period until which the order starting at x

lasts (ac-

cording to the order pattern which is in place be-

fore discounts are introduced).

Parameters enumarated above (except the ﬁrst one)

are assigned successively for every customer i. The

number of assignments for every customer i is equal

to the number of periods in which the order is placed

according to the order pattern received in Step 1.

Parameter ψ

itk

is calculated using formula (9) for

t = k and formula (10) for t < k for ∀i > 0:

itk

= d

(9)

itk

= ψ

itk−1

+ d

(k −t) + d

(10)

Parameter x

is successively assigned to periods when

customer i places an order according to the order pat-

tern determined at Step 1; parameter y

refers to the

period preceding the next order in the order pattern

determined at Step 1, or if there is no next order — to

the last period.

Parameter y

initially is presumed to be equal to

. This predetermines that y

is never larger than

. Even if according to the order pattern which is

in place if no discounts is introduced the next order

is placed later than it should happen according to the

order pattern determined at Step 1, the examined time

horizon is contracted to y

Parameter x

is asigned to period t ∈ M and t ≤ y

such that:

ity

= min

u∈M,u≤y

iuy

(11)

which refers to the minimum in column y

of the costs

matrix calculated applying the Wagner-Whitin algo-

rithm.

If x

> x

, we assign y

= x

−1 and recalculate x

using formula (11). This step is repeated until x

≤ x

If x

= x

and y

= y

, then the correct price is

found, otherwise the new price which incorporates the

discount is found using formula (12) for x

< x

and

formula(13) otherwise, for ∀i > 0:

:= v

−

− min

t∈M,t≤y

ity

(12)

:= v

−

− min

t∈M,t≤y

ity

− ψ

(13)

The formula (12) differs for the case when x

< x

since the items ordered in period x

are not ordered

with the discounted price.

After calculation of the new price the costs matrix

which is calculated according to the Wagner-Whitin

algorithm is updated as well.

This is however a heuristic solution, because the

total discount calculated at Step 2 may be larger than

the increase in cost which is the compensation as-

sumed in Step 1.

5.3 Discount Interval Heuristic

While the previous heuristic procedure is based on

separation between determining the order pattern and

discounts, in the current procedure the order pattern is

determined based on the possible discount prices cal-

culated in advance. Moreover, the shortest path for-

mulation is used for the supplier in addition to the nor-

mal aggregated formulation (Eppen and Kipp, 1987).

Parameters d

, s

, h

, v

, x

, y

, ω

itk

, ψ

itk

and variables H

, S

, Q

are deﬁned in the same way

as in the previous model.

The following new parameters are introduced:

itk

reduction in price large enough for customer i to

order all demand from period t to period k in pe-

riod t given that no other discounts are offered;

λ original price of the item without a discount.

The following new variable is used in the model:

itk

binary variable, 1 if the total demand of every

customer i from period t to period k is ordered in

period t, 0 otherwise. This variable can be binary

due to the Wagner-Whitin property according to

which if an order is placed it will cover demand

for an integer number of periods.

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STEP 0.

The discount for every possible combination of

the starting period and the number of periods the or-

der is placed for for every customer i is computed in

advance at this step.

The algorithm described in Step 3 of the Cost

Compensation Heuristic is applicable here. But the

part where the prices are reset needs to be included

so that at any moment there is a discount in only one

period named x

. The amount of discount required is

saved:

= λ − v

(14)

The price itself is reset to be equal to λ and Wagner-

Whitin cost matrix is updated. This is necessary be-

cause it is not known which discounts will be selected

in the ﬁnal solution, keeping all the calculated dis-

counts will perturb the calculations. On the other

hand, this is a deﬁciency of the model since the dis-

counts applied in previous periods can inﬂuce future

discounts.

STEP 1.

In Step 0 paramaters p

itk

have been calculated and

are going to be used in the model below. The param-

eters account for the amount of discount offered. Pa-

rameters v

— prices for customer i in period t remain

unchanged after this step. Objective (15) minimizes

the income which is lost while introducing discounts

to customers as well as supplier’s inventory and set up

costs:

minimize

∑

i=1

∑

t=1

∑

k=t

itk

∑

u=t

∑

t=1

∑

t=1

(15)

Constraint (16) ensures that the amount produced by

the supplier together with the inventory left in period

t is enough to satisfy the demand which corresponds

to the summation of customer orders that satisfy their

demand from the current period t to a future period k.

It is equal to constraint (7) in the previous model.

subject to

+ H

0t−1

− H

∑

i=1

∑

k=t

itk

∑

u=t

, for ∀t (16)

For t = 1 constraint (16) is slightly modiﬁed since

0t−1

equals to 0.

Constraint (17) ensures that the possible maxi-

mum produced in period t is the summation of the de-

mand for all customers t for all periods t and is equal

to constraint (6):

∑

i=1

∑

u=t

− Q

≥ 0 , for ∀t (17)

The following constraints which are used while

determining the shortest path and called ﬂow balance

equations (Eppen and Kipp, 1987) are added to the

model.

Constraint (18) guarantees that, since it is as-

sumed that there is no initial inventory, the demand

for period 1 has to be ordered in period 1 and can be

ordered up to any future period k:

∑

k=1

i1k

= 1 , for ∀i > 0 (18)

Constraint (19) ensures that when the demand is or-

dered up to the previous period, then a new order must

be placed in this period up to some future period k.

This guarantees that the demand for all periods will

be satisﬁed:

t−1

∑

k=1

ikt−1

∑

k=t

itk

, for ∀i > 0 , ∀t > a (19)

STEP 2.

The order and production patterns are received as

a result of solving the model described above. They

can be displayed using the variable S

which shows

whether order is placed or not in period t. For the

supplier i = 0 it can be received directly. For every

customer i: S

= 1 when

∑

u=t

itu

= 1 for ∀t.

After determining order and production patterns

the same algorithm from Step 3 of the Cost Compen-

sation Heuristic is used to calculate actual discount.

This time it is applied with no changes. Actual prices

are calculated at this step.

This model is more difﬁcult to solve than the one

for the Cost Compensation Heuristic as a result of ex-

tra binary variables F

itk

which are used in the model.

At the same time the number of extra parameters cal-

culated in Step 0 is not as big as it would be for the

exact model where the discount given in period t de-

pends on the discounts offered in the previous periods.

In this heuristic it is assumed that only one discount is

given at a time. That is the reason why application of

the heuristic doesn’t guarantee the optimal solution.

6 EXPECTED OUTCOME

Currently the heuristic algorithms developed for the

simple unrestricted one-item case have been tested on

one problem. It is planned to do more tests on ran-

domly generated problems.

The data for the considered numerical example

was generated randomly for the problem size of 5 cus-

tomers and 20 periods. Realistically the production

costs of the supplier are higher than the order cost of

DeterminingOptimalDiscountPoliciesinB2BRelationships

the customers, while the inventory holding costs of

the supplier are lower than that of the customers. We

assume initial inventory to be equal to 0 in this exam-

ple.

Figure 1 shows order and production patterns be-

fore and after implementation of the heuristic proce-

dures described above. It displays the period in which

the order/set up is done. The ﬁrst row (customer 0)

represents set up schedules of the supplier.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

period

customer

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

period

customer

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

period

customer

Figure 1: Difference between the initial and ﬁnal or-

der/production patterns. a — initial order pattern. b —

order pattern. Cost Compensation Heuristic. c — order

pattern. Discount Interval Heuristic.

It can be noticed that in the initial (before the im-

plementation of the heuristics) pattern customers’ or-

ders rarely coincide with production periods. Cus-

tomer 3 orders almost in every period. Customers 2

and 4 order very often as well. Despite rather low fre-

quency of orders of customers 1 and 5 their order pat-

terns are not synchronized with production patterns.

In the ﬁnal patterns (after the heuristics were used)

orders of customers 1, 4 and 5 totally match produc-

tion periods. Customers 2 and 3 still order in-between

production periods but their orders became signiﬁ-

cantly less frequent.

Result for the supplier is summarized in tables 1

and 2 .

Table 1: Result for the supplier. Cost Compensation Heuris-

tic

Cost reduction due to coordination 45 120

Sales revenue lost because of discounts 34 805

Additional proﬁt 10 315

Table 2: Result for the supplier. Discount Interval Heuristic

Cost reduction due to coordination 30 740

Sales revenue lost because of discounts 14 462

Additional proﬁt 16 278

Due to the implemented discount pricing schedule

the supplier’s proﬁt was improved by 10 315 (Cost

Compensation Heuristic) and 16 278 (Discount Inter-

val Heuristic). It can be seen that Cost Compensa-

tion Heuristic results in more savings for the supplier

due to better coordination of the customers’ orders.

At the same time the supplier loses his proﬁt offering

more discounts to the customers. Discount Interval

Heuristic gives less cost reduction due to coordina-

tion but in the end the supplier gets more additional

proﬁt offering less discounts. In both cases the de-

mand doesn’t increase, the revenue of the supplier de-

creases because of the discounts, nevertheless, he gets

more proﬁt due to the coordination of orders.

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DeterminingOptimalDiscountPoliciesinB2BRelationships