Joint Pricing and Inventory Decisions for Perishable
Products
Onur Kaya (okaya@ku.edu.tr)
College of Engineering, Koc University
Aylin Lelizar Polat
College of Engineering, Koc University
Abstract
We consider perishable products and focus on the optimal pricing and inventory
decisions for these products in a deterministic setting. In this system, the demand rate is
affected with not only the prices of the products but also with the freshness and age of the
products. In addition, the products decay at a certain rate while they are in stock. We
analyze both a single product and a multi product model and considering the interactions
between the products, we aim to determine when to change the prices of the products, the
optimal values of these prices and when to replenish the inventory with new products.
We analyze the effect of different parameters on the optimal solution through numerical
experiments and obtain managerial insights.
Keywords: Pricing, Inventory, Perishable Products, Revenue Management
Introduction
In this study, we consider the inventory management and pricing of perishable products
such as milk, yogurt, eggs, bread, fresh fruits and vegetables, considering a price and
time dependent demand function. Effective management of perishable products is an
important issue for many companies and management of these products is especially
difficult for the managers because of the perishability risks of these products in a very
short time. Different than durable products, as the perishable products age, the demand
for these products start to decline and in a short time these products can become
completely obsolete. Thus, not only the amount of these products but also their status or
age effect the inventory and pricing decisions of these products making the problem
much harder to solve.
There are many studies in literature considering the inventory and pricing
decisions for perishable products, however most of these studies consider the inventory
and pricing decisions separate from each other and coordinated inventory and pricing
decisions are not studied as much. In addition, in most of the studies in literature, single
product systems are considered and demand is assumed to be time-independent. However,
in reality, due to customer choices, the actual demand depends on the age or status of the
products at hand in addition to their price. When the products are new or fresh, the
demand rate is high, however as the products age, they will be demanded less and the
customers can move to other types of products. Because of the change in demand over
time, it might be better to charge different prices at different times and employing a
1
dynamic pricing strategy might improve the profits substantially. In the literature most of
the inventory models assume that inventories can be held in stock indefinitely to satisfy
future demands. On the other hand most of the commodities either deteriorate or become
outdated, hence they are unstable. Most commonly used goods are exposed to
deterioration; like fruit, vegetable, meat, etc. Therefore if the rate of deterioration is not
sufficiently low, its impact on modeling of the inventory cannot be ignored.
In recent years, we have observed an increased adoption of dynamic pricing for
perishable products in manufacturing and retail industries. For instance, in food industry
decaying products like bread and vegetables have very short shelf life. They are generally
priced at the retail price when they come fresh. However they are sold at discounted
values when they come to their expiry date. So retailers can attract customers who are
more price sensitive with the aim of generating more revenue through their sales. It is a
complex task to determine a right price for the product which requires a wealth of
information about the status of the products at hand and customer preferences for
products with different freshness and price values. In the past, companies faced high
costs in changing prices and had insufficient information about customer demand. As a
result, companies always fixed the product's prices over a long period resulting in static
prices. However, today, with the improvements in technology, companies have much
more data about the customers and can change their prices at relatively low costs
suggesting a dynamic pricing policy to be more profitable.
In literature, deterministic joint inventory and dynamic pricing problem is generally
analyzed considering a single product. Raafat (1991) presents a survey for the inventory
management of deteriorating items and Goyal and Giri (2001) present a review of
advances of deteriorating inventory literature since the early 1990s. Abad (1996) assumes
that prices can be changed continuously at no cost and determines the optimal price p(t)
as a continuous function of time. Transchel and Minner (2009) consider discrete pricing
of non-perishable products assuming a fixed cost for changing prices. Mukhopadhyay et
al. (2004) consider perishable products, but they assume a single price to be charged over
the whole cycle and do not consider price changes. In addition, they assume that the
demand function is independent of time. Sana (2010) considers an EOQ model for
perishable items where demand is price dependent and partial backorder is allowed. They
determine the optimal prices and the order size but they assume that the timings of the
price changes are fixed. Rajan and Rakesh (1992) consider the relationship between
pricing and ordering decisions for perishable products by allowing a continuous price
change over the cycle. Broekmeulen and Donselaar et al. (2009) analyze a perishable
product system considering the age of products and they propose a replenishment policy
and compare their results with the optimal policy that does not take into account the age
of inventories.
In this study we consider a deterministic model and focus on the coordinated
pricing and inventory decisions. Different from the studies stated above, we consider both
single and multiple product systems with perishable products that decay at a certain rate
and the demand is both time and price dependent. We assume that the prices can be
changed at any time at a certain cost and we aim to find the best times to change the
prices as well as the optimal values of these prices. Every time an order is given for fresh
products, a fixed order cost is incurred in addition to the inventory holding costs for the
products kept at hand. The objective of the model is to maximize the profit per unit time
2
by choosing the optimal lot size and the pricing strategy where the retailer can change
price over time. We analyze this system for both single and multi product cases
considering the demand interactions between the products. We provide explicit results for
the optimal solution in the single product model and present an efficient heuristic to
determine the best solution for the multi product model. We observe that dynamic pricing
provides substantial benefits compared to the static pricing, allowing the companies to
substantially increase their profits.
Single Product Model
In this section, we consider a single perishable product and focus on the pricing and
inventory decisions for this product. We denote the time between two successive orders
for new products as an inventory cycle and during an inventory cycle, the price of the
product is changed several times in order to obtain the maximum possible profit. We let h
denote the inventory cost per unit per unit time, N denote the number of different prices
used in an inventory cycle and f denote the cost of changing the price such that Nf
denotes the total cost of changing prices in an inventory cycle. In our model, ti, i=1,2..N-1,
denotes the time of the ith price change with t0=0 and tN denotes the end of an inventory
cycle at which time new and fresh products are ordered. Figure 1 presents an illustration
of an inventory cycle.
Figure 1- Illustration of the Inventory Process
We let pi denote the price during the time interval [ti-1, ti). In this model, due to
perishability of the product, the inventory is depleted partly to meet the demand and
partly for deterioration. We assume that each product at hand decays at a constant rate θ
independent of the other products such that the decay rate of all inventory is given by
w(t)=θI(t) at any time t, as a function of the instantaneous inventory level at that time,
denoted as I(t). Note that Q= I(0) denotes the batch size ordered at the beginning of each
cycle. Every time an inventory replenishment order is given, a fixed order cost A and per
unit cost c is incurred such that the total cost at the time of order will be A+cQ. We
3
describe the inventory function I(t) by the differential equation (1) which consists of the
decay rate and the price and time dependent demand rate, D(p,t), such that:
𝐼 ′ (𝑡) = -D(p,t)- θI(t)
(1)
For any t in the interval [ti-1, ti], the inventory equation below satisfies equation (1)
and the boundary condition Ii(ti) = I(ti).
𝐼𝑖 (𝑡) = 𝑒
−𝜃𝑡
𝑡𝑖
∫ 𝐷(𝑝, 𝑠) 𝑒 𝜃𝑠 𝑑𝑠 + 𝐼(𝑡𝑖 )𝑒 𝜃(𝑡𝑖 −𝑡)
𝑡
For the purpose of obtaining explicit results, we consider a linear demand function
that is decreasing in price and time, such that D(p,t)=a-bp-kt. However, we note that the
same analysis as in this paper can be done and similar results can be obtained for
different types of demand functions. Solving the above differential equation for the linear
demand function results in the following inventory function at any time t where ti-1 ≤ t ≤ ti
for all i=1,2,…N.
𝐼𝑖 (𝑡) = (
𝑎−𝑏𝑝𝑖
𝜃
−
𝑘𝑡𝑖
𝜃
𝑘
+ 𝜃2 + 𝐼(𝑡𝑖 ))𝑒 𝜃(𝑡𝑖 −𝑡) −
𝑎−𝑏𝑝𝑖
𝜃
+
𝑘𝑡
𝜃
𝑘
− 𝜃2
(2)
We let Si denote the sales amount during time period [ti-1, ti) such that
𝑡𝑖
𝑆𝑖 = ∫ (𝑎 − 𝑏𝑝𝑖 − 𝑘𝑠)𝑑𝑠
𝑡𝑖−1
Then the profit per unit time can be written as:
1
𝑘𝑝𝑖
𝑁
2
2
𝜋 = 𝑡 {−𝐴 − 𝑐𝑄 − 𝑁𝑓 + ∑𝑁
𝑖=1(𝑎𝑝𝑖 − 𝑏𝑝𝑖 )(𝑡𝑖 − 𝑡𝑖−1 ) −
𝑘𝑡𝑖
𝜃2
𝑘
− 𝜃3 −
𝐼(𝑡𝑖 )
𝜃
𝑎−𝑏𝑝𝑖
) (1 − 𝑒 𝜃(𝑡𝑖 −𝑡𝑖−1 ) ) − (
𝜃
𝑘
2
(𝑡𝑖2 − 𝑡𝑖−1
) − ℎ[(−
𝑘
2
)]}
+ 𝜃2 ) (𝑡𝑖 − 𝑡𝑖−1 ) + 2𝜃 (𝑡𝑖2 − 𝑡𝑖−1
𝑎−𝑏𝑝𝑖
𝜃2
+
(3)
In this study, we need to determine the optimal values of N, pi and ti values for all
i=1,2,…N. We observe that for a given N, we can find the optimal pi and ti values using
the first order derivatives of the equation (3). Thus, we solve this problem using a twostage approach as follows, where 𝜋(𝑁) denotes the profit function for a given N.
Stage 1: 𝑀𝑎𝑥𝑁 𝜋(𝑁) − 𝐾(𝑁)
s.t. N ≥ 1
Stage 2: 𝑀𝑎𝑥𝑝,𝑡 𝜋(𝑁)
s.t. ti ≥ ti-1 ≥ 0
pi ≥ 0
D(pi,ti) ≥ 0
for all i=1,2,..N
for all i=1,2,..N
for all i=1,2,..N
We observe that for a given N, the second stage problem is jointly concave w.r.t p
and t and thus, we first solve the second stage problem for a given N and determine the
optimal price and time values as a function of N. Then, we solve the first stage problem
4
to determine the optimal value of N using the results of the second stage problem. For a
given N, the optimal price values can be found through equation (4) as functions of ti
values. Then, we solve the second stage problem by combining equation (4) with the first
order derivatives of the profit function (3) with respect to ti.
𝑎
𝑘
ℎ
𝑝𝑖 = 2𝑏 − 4𝑏 (𝑡𝑖 + 𝑡𝑖−1 ) − 2𝜃 +
𝑎−𝑏𝑝𝑖
ℎ [(
𝜃
𝑎−𝑏𝑝𝑖+1
𝜃
𝑘
ℎ(𝑒 𝜃(𝑡𝑖 −𝑡𝑖−1 ) −1)
2𝜃2 (𝑡𝑖 −𝑡𝑖−1 )
𝑎−𝑏𝑝𝑖+1
)
+ 𝜃2 + 𝐼(𝑡𝑖 )) (𝑒 𝜃(𝑡𝑖 −𝑡𝑖−1 ) + (−
𝜃
2
] = −𝑏𝑝𝑖2 − (𝑝𝑖 − 𝑝𝑖+1 )(𝑎 + 𝑘𝑡𝑖 ) + 𝑏𝑝𝑖+1
𝑓𝑜𝑟 ∀𝑖=1,2..N
+
𝑘𝑡𝑖+1
𝜃
(4)
𝑘
− 𝜃2 − 𝐼(𝑡𝑖+1 )) (𝑒 𝜃(𝑡𝑖+1 −𝑡𝑖 ) ) +
𝑓𝑜𝑟 ∀𝑖=1,2..N-1
(5)
Using a one-dimensional search over N, we find the optimal time and price values
for each N and then by comparing the profits in each case, we obtain the best N and the
corresponding optimal pi and ti values. Note that when N=1, this system corresponds to a
static pricing model in which the price is not allowed to change during a cycle. In the
numerical results section, we compare the results of this static pricing model with the
dynamic pricing case in which the prices are allowed to change during the cycle at a
certain cost.
Substitutable and Perishable Two Product Model
In this section, we consider two different products which are substitutable in nature such
that the demand for each product depends not only on that product’s price and freshness
but also it depends on the other product’s values. We employ a similar analysis here, as in
the previous section. We assume that both products are replenished at the same time such
that the length of the inventory cycle is the same for both products. Even though, this can
be seen as a restrictive assumption, this is a common procedure in reality since similar
products are obtained from the same supplier and they are ordered simultaneously.
Ordering these products separately also causes a high fixed order cost. However, during
the inventory cycle, their prices can be changed at different times.
We let ti1 and ti2 i=1,2..N-1, denote the time of the ith price change for products 1
and 2, respectively with t01= t02=0 and tN1= tN2 denote the end of an inventory cycle. Note
that at each time of a price change, only one of the products’ price is changed, however,
we don’t know which. This makes the analysis complicated since during an interval [ti-1,1,
ti1), the demand function of the first product might change if the price of the second
product changes at some time in this interval. Thus, we need to consider where each ti2
lies with respect to each ti1 value and because of this reason, we cannot write the profit
function of each product independently from the other in terms of these prices and times.
To overcome this difficulty, we concatenate the time arrays ti1 and ti2 to obtain a single
array ti which contains all times of the price changes in a strict ordering. We let pi1 and pi2
denote the price during the time interval [ti-1, ti). However, note that, in this case, at any ti
we do not know which price is changed, thus one of the pi values need to stay constant in
successive intervals while the other can change. Using the demand functions D1(p1, p2,
t)=a1-b1p1+e1p2-k1t and D2(p1, p2, t)=a2-b2p2+e2p1-k2t, we use the same analysis as in the
single product model and obtain the inventory and profit functions similar to the ones in
the previous section as below.
5
𝐼𝑖𝑗 (𝑡) = (
𝑎𝑗 − 𝑏𝑗 𝑝𝑖𝑗 + 𝑒𝑗 𝑝𝑖𝑘 𝑘𝑗 𝑡𝑖 𝑘𝑗
𝑎𝑗 − 𝑏𝑗 𝑝𝑖𝑗 + 𝑒𝑗 𝑝𝑖𝑘 𝑘𝑗 𝑡 𝑘𝑗
−
+ 2 + 𝐼𝑗 (𝑡𝑖 ))𝑒 (𝑡𝑖 −𝑡) −
+
−
𝜃𝑗
𝜃𝑗
𝜃𝑗
𝜃𝑗 𝜃𝑗2
𝜃𝑗
1
𝑘𝑗 𝑝𝑖𝑗
𝑁
2
2
𝜋𝑗 = 𝑡 {−𝐴𝑗 − 𝑐𝑗 𝑄𝑗 − 𝑁𝑗 𝑓𝑗 + ∑𝑁
𝑖=1(𝑎𝑗 𝑝𝑖𝑗 − 𝑏𝑗 𝑝𝑖𝑗 + 𝑒𝑗 𝑝𝑖𝑗 𝑝𝑖𝑘 )(𝑡𝑖 − 𝑡𝑖−1 ) −
2
) − ℎ𝑗 [(−
𝑡𝑖−1
2
𝑎𝑗 𝑝𝑖𝑗 −𝑏𝑗 𝑝𝑖𝑗
+𝑒𝑗 𝑝𝑖𝑗 𝑝𝑖𝑘
2
𝑎𝑗 𝑝𝑖𝑗 −𝑏𝑗 𝑝𝑖𝑗
+𝑒𝑗 𝑝𝑖𝑗 𝑝𝑖𝑘
(
𝜃𝑗
𝜃𝑗2
𝑘
+
𝑘𝑗 𝑡𝑖
𝜃𝑗2
𝑘
− 𝜃3𝑗 −
𝑗
𝐼𝑗 (𝑡𝑖 )
𝜃𝑗
(𝑡𝑖2 −
) (1 − 𝑒 𝜃𝑗 (𝑡𝑖 −𝑡𝑖−1 ) ) −
𝑘
2
)]}
+ 𝜃2𝑗 ) (𝑡𝑖 − 𝑡𝑖−1 ) + 2𝜃𝑗 (𝑡𝑖2 − 𝑡𝑖−1
𝑗
𝑗
where j={1,2} and k is the complement of j (i.e. k=2 when j=1 and vice-a-versa). Then
the problem can be written as
Max 𝜋 = 𝜋1 + 𝜋2
s.t. ti ≥ ti-1 ≥ 0
pij ≥ 0
Dj(pij, pik, ti) ≥ 0
pij = pi+1,j
for all i=1,2,..N
for all i=1,2,..N, j=1,2.
for all i=1,2,..N
if price of the other product is changed at time ti for j=1,2.
In this case, at any time ti since only one of the prices of the products is changed,
one of the pi values need to stay constant in successive intervals and this makes the
problem very complicated and we cannot obtain explicit results for the optimal pi and ti
values. However, we observe that if we know the ti1 and ti2 values, then we know which
price is changed at each ti and optimal prices in each time interval can be found explicitly
through the first order derivative of the profit function. Using this observation, we design
a heuristic algorithm to find the optimal solution.
To obtain the optimal results for this system, we employ a genetic algorithm. In this
algorithm, starting with a set of arrays of ti1 and ti2 values, we calculate the best pi1 and pi2
values using the derivative of the profit function and calculate the optimal objective
function value for the given time array. Then using mutation and crossover operators, we
obtain new time arrays for the timings of the price changes. The mutation operator in our
algorithm works as follows: At each iteration, we randomly select one of the values in
our time arrays and either increase or decrease it by a random amount leading to a new
solution. Then for this new time array we calculate the corresponding optimal price
values in each time interval using the derivatives of the profit function and then calculate
the objective function value. If the new solution leads to better solution than the previous
one, we continue doing the same operation until no further improvement is obtained. We
also use a crossover operator in our algorithm which works as follows: We take two time
arrays and we randomly pick one of the elements in one of these arrays. For example, let
[a1, a2, … ai, ai+1, … aN] and [b1, b2, … bj, bj+1, … bM] denote these two arrays and ai
denote the element that is chosen. Then we determine the closest time in the other time
array that is larger than the element that we picked, let it be bj+1. Then we exchange the
initial portions of these two arrays and obtain two new solutions as [b1, b2, … bj, ai+1, …
aN] and [a1, a2, … ai, bj+1, … bM]. Using these operators, we update our results and
continue in the same manner until no further improvements can be obtained in the
algorithm for a certain number of iterations or an upper bound is reached in the number
6
of iterations.
Computational Results
In this section, we numerically analyze the benefits of dynamic pricing for the single
product model and observe the impacts of the system parameters on the results. As a base
case, for the single product model, we use the parameters a=100, b=1, k=1, c=10, h=0.01,
θ=0.01, f=100 and A=1000. In Table 1, we present the results for the dynamic pricing
model as well as the results for the static pricing case, as a benchmark, in which the price
is not allowed to change during a cycle. In the first row of Table 1, we present the results
for this base case and in the following rows of this table, we do a sensitivity analysis by
changing one of these parameters as stated in the first column of Table 1.
Table 1 - Single Product Model Results
Dynamic Pricing Model
Parameter Profit Order Cycle
Waste
Size
Length
Ratio
Base Case 1061
1112
42.8
0.09
a=200
4954
5721
91.2
0.24
a=50
209
182
10.8
0.03
b=2
392
832
37.9
0.08
b=0.5
2322
1293
46.8
0.11
k=2
1064
558
23.2
0.04
k=0.5
1008
2341
81.4
0.18
c=20
819
818
38.2
0.07
c=5
1154
1293
46.9
0.10
h=0.02
1059
1101
42.7
0.08
h=0.005
1065
1115
43.1
0.09
1023
1173
40.9
0.17
θ=0.02
0.05
θ=0.005 1072 1058 43.6
1039
1104
42.5
0.10
f=200
1084
1128
43.2
0.09
f=50
0.09
A=2000 1041 1135 43.1
1079
1087
42.6
0.08
A=500
Profit
949.6
4300
197.4
343.5
2189
948.4
915.6
716
1081.8
946
951
907
968.9
949.6
949.6
926
961
Static Pricing Model
Order Price Cycle
Size
Length
1106
56.4 43.6
5648
109.6 90.4
185
27.7 10.3
829
30.9 38.1
1282
106.6 46.7
549
54.9 22.5
2266
58.9 82.2
807
62.3 37.6
1290
53.2 46.8
1098
56.5 43.5
1111
56.3 43.7
1160
58.5 41.5
1069
55.5 44.5
1106
56.4 43.6
1106
56.4 43.6
1133
55.9 44.1
1093
56.6 43.4
Waste
Ratio
0.14
0.28
0.05
0.12
0.15
0.07
0.25
0.12
0.15
0.14
0.14
0.26
0.07
0.14
0.14
0.14
0.14
We observe that dynamic pricing results in a 12% increase in profits on average as
opposed to the single price model. In addition, the number of decayed products decrease
with dynamic pricing. However, the order size and the cycle length with dynamic pricing
can be higher or lower than the values with static pricing. When we look at the effects of
the parameters on the system, we observe that as the market size, a, increases, the order
size, cycle length and the profits increase substantially but a much higher wastage rate is
observed since more units are ordered in a cycle. In addition, the benefit of dynamic
pricing increases with a, since there is a larger room for improvement in a system with a
larger market size. When we consider the price sensitivity of demand, b, we observe that,
as b increases, less profit is obtained due to decrease in demand. The order size and the
cycle length also decrease and less amount of products are wasted as b increases. In
addition, as b increases, since the system will be more sensitive to pricing, the benefit of
dynamic pricing compared to the static pricing model increases. When b=0.5, the profit
7
with dynamic pricing is about 6% better than the profit with static pricing while this
value increases to 14% when b=2. When the time sensitivity of demand, k, increases, the
order size and the cycle length decrease and less amount of products is decayed. Next,
we consider the cost of the products, c, and observe that as c increases order size, cycle
length and profits decrease and less units are wasted. In addition, the benefit of dynamic
pricing also increases with c. The reason for this is that as the products become more
valuable, dynamic pricing and effective management of the products become more
important. We observe that as the inventory holding cost increases, profits decrease and
less units are ordered in each cycle leading to shorter cycle lengths. Similarly, when the
wastage rate increases, profit and the cycle length decrease, but we observe that more
units are ordered in each cycle in order to cover the high amount of decay and have
enough good products at hand for sale. Even though, the order size increases, since the
products decay at a faster rate, the cycle length decreases with θ. Next, we consider the
cost of changing the prices and observe that as f decreases, dynamic pricing will be more
useful and the prices are changed more often. As a results, profit increases and the
wastage rate decreases. Lastly, we consider the fixed order cost, A, and observe that as A
increases, more units are ordered and the cycle length increases as the wastage rate, while
the profits decrease.
When we look at the prices in the dynamic pricing model, we observe that, at the
beginning of the cycle a higher price than the static price is charged but the price is
decreased as time increases and becomes lower than the static price through the end of
the cycle. The reason for this is that, at the beginning, there is a higher demand for the
fresh products and charging a higher price is optimal. However, as time increases the
demand decreases as the products’ freshness decrease and lower prices are charged to
maximize the profits with this lower demand function.
Conclusion
In this study, we analyze the coordinated pricing and inventory decisions for perishable
products in a deterministic setting in which the demand not only depends on the price but
also on the freshness of the products. In addition, the products in inventory are assumed
to decay at a certain rate which adds another dimension to the problem. We derive
explicit results for the optimal order size, optimal number of times the prices should be
changed and the optimal price values at each time for the single product case and we
propose a heuristic to approximate the optimal solution in the multi product model.
Through numerical experiments, we compare the results with the dynamic pricing policy
with the single pricing case in which the price is not allowed to change throughout the
selling period. We observe that substantial benefits can be obtained by changing the
prices throughout the selling period.
Our model can be extended in several ways. Firstly, different types of demand
functions or decay processes can be analyzed in detail and explicit results can be obtained.
In addition, this problem can be analyzed in the stochastic setting in which the demand
and also the decaying process is random. Even though our results can form a basis for
developing coordinated pricing and inventory policies that can be also applied in
stochastic environments, the performances of such policies need to be investigated and
effective policies need to be developed for stochastic systems.
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Acknowledgements
We are grateful to Tubitak for its support to our work through Tubitak 1001 Grant
#111M533.
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