This document presents a complete solution procedure for determining the optimal solutions to the EOQ and EPQ inventory models with linear and fixed backorder costs. It develops necessary and sufficient conditions for the existence of optimal solutions that satisfy the first-order conditions. It shows that if the conditions hold, the optimal solutions can be found by solving the first-order equations. If the conditions do not hold, alternative optimal solutions are identified as having zero backorders and a quantity based on demand and holding costs. The procedure improves upon prior works by providing a rigorous mathematical proof for locating and ensuring the optimal solutions.

1. Mathematical and Computer Modelling 55 (2012) 2151–2156
Contents lists available at SciVerse ScienceDirect
Mathematical and Computer Modelling
journal homepage: www.elsevier.com/locate/mcm
The complete solution procedure for the EOQ and EPQ inventory models
with linear and fixed backorder costs
Kun-Jen Chunga,b
, Leopoldo Eduardo Cárdenas-Barrónc,d,∗
a
College of Business, Chung Yuan Christian University, Chung Li, Taiwan, ROC
b
National Taiwan University of Science and Technology, Taipei, Taiwan, ROC
c
Department of Industrial and Systems Engineering, School of Engineering, Instituto Tecnológico y de Estudios Superiores de Monterrey, E. Garza Sada 2501 Sur,
C.P. 64 849, Monterrey, N.L., Mexico
d
Department of Marketing and International Business, School of Business, Instituto Tecnológico y de Estudios Superiores de Monterrey, E. Garza Sada 2501 Sur,
C.P. 64 849, Monterrey, N.L., Mexico
a r t i c l e i n f o
Article history:
Received 29 August 2011
Received in revised form 18 November
2011
Accepted 5 December 2011
Keywords:
EOQ
EPQ
Backorder costs
Convexity
a b s t r a c t
The basic EOQ/EPQ inventory models with backorders have been developed from different
perspectives by several researchers. However, the arguments to locate and guarantee the
optimal solution are not complete. This paper presents a complete and analytic solution
procedure to the EOQ/EPQ inventory models with linear and fixed backorder costs to locate
and ensure the optimal solutions. First, the sufficient and necessary conditions for the
existences of the optimal solution are developed. Second, in the case if these conditions
are not satisfied, then also the optimal solutions are identified for this situation. The final
results are two lemmas and four useful theorems to obtain the optimal solutions to both
inventory problems.
© 2012 Elsevier Ltd. All rights reserved.
1. Introduction
In 2013, the classical EOQ formula will have a century of its discovery by Harris [1] in 1913. Later the EPQ formula was
developed by Taft [2] in 1918. Perhaps the first book in inventory management was written by Raymond [3] in 1931. Since
then, many extensions of the EOQ and EPQ inventory models have been developed by researchers and academicians. Some
extensions cover deterministic and stochastic inventory models. Moreover, the researchers and academicians have shown
a huge interest in developing alternative optimization methods to derive the EOQ, EPQ and some extensions of them with a
pedagogical value.
It is worth mentioning that some optimization methods are very simple and others are too complex. For an up to date
review of different optimization methods in inventory field, see [4]. However, some of the optimization methods lack
the mathematical rigor and some of them are based on intuitive arguments. Furthermore, some of them do not have the
mathematical proof needed. In addition, all optimization methods that develop the EOQ and EPQ inventory models with
two classical backorders costs (linear and fixed) available until today are deficient or incomplete. Therefore, we find that in
both inventory models, there exists the need for developing a complete solution procedure for determining the optimal lot
size and backorders level.
Recently, an increasing interest in analyzing inventory models considering risk coherent inventories has been found
in the inventory literature. In this direction, see for example the research works of Ahmed et al. [5] and Borgonovo and
∗ Corresponding author.
E-mail addresses: kjchung@cycu.edu.tw (K.-J. Chung), lecarden@itesm.mx (L.E. Cárdenas-Barrón).
0895-7177/$ – see front matter © 2012 Elsevier Ltd. All rights reserved.
doi:10.1016/j.mcm.2011.12.051

2. 2152 K.-J. Chung, L.E. Cárdenas-Barrón / Mathematical and Computer Modelling 55 (2012) 2151–2156
Peccati [6]. Borgonovo and Peccati [6] propose a new approach to calculate the differences between different inventory
policies determined by several risk aversion attitudes using different risk measures.
Another research direction is to perform a good sensitivity analysis of inventory models. For example, Cárdenas-
Barrón [4] made a sensitivity analysis for the EOQ and EPQ inventory models without/with backorders considering two
backorder costs (linear and fixed). On the other hand, it is important to point out that another type of sensitivity analysis
to the EOQ/EPQ inventory models with backorders can be done in line of the works of Borgonovo and Peccati [7],
Borgonovo [8,9], and Borgonovo and Peccati [10] just to name a few.
As we was mentioned before, there are several researchers have developed the basic EOQ/EPQ inventory models with
linear and fixed backorder costs using alternative solutions procedures. See for example the works of Hadley and Whitin [11],
Johnson and Montgomery [12], Sphicas, [13], Omar et al. [14], and Cárdenas-Barrón [4] just to name a few. However, their
solution procedures to locate and guarantee the optimal solution are not of all complete.
Perhaps, Hadley and Whitin [11, pp. 52], and Johnson and Montgomery [12, pp. 26–33] were the first researchers to
treat the EOQ/EPQ inventory models with two backorder costs, a linear and a fixed cost per unit. Their solution procedure is
based on calculus and solved the system equations resulting from the first-order conditions. Recently, Sphicas [13] indicates
that Johnson and Montgomery [12]’s arguments to locate the optimal solutions are not complete. Therefore, Sphicas [13]
adopts an algebraic approach to demonstrate how a relatively complex model can be fully analyzed without derivatives and
obtained the explicit identification of the EOQ/EPQ inventory models.
In this respect, our research work basically improves Johnson and Montgomery [12] and Sphicas [13] works to adopt an
analytic approach based on calculus to locate and ensure the optimal solutions of the EOQ/EPQ inventory models. The main
results of this paper are fourfold:
(A) This paper develops the sufficient and necessary conditions for the existences of solutions ( ¯Qi, ¯Si) satisfying the first-
order conditions of the annual total relevant cost TCi(Q , S) (i = 1, 2), respectively.
(B) If the sufficient and necessary conditions for the respective i mentioned in (A) hold, then TCi(Q , S) is convex and the
optimal solution (Q ∗
i , S∗
i ) of TCi(Q , S) is equal to ( ¯Qi, ¯Si) for i = 1, 2.
(C) If the sufficient and necessary conditions for the respective i mentioned in (A) do not hold, then the optimal solutions
are (Q ∗
1 , S∗
1 ) = (

2KD
h
, 0) or (Q ∗
2 , S∗
2 ) = (

2KD
h(1−D/P)
, 0).
(D) This paper improves Johnson and Montgomery [12] and Sphicas [13] works.
The remainder of the paper is organized as follows. Section 2 presents the complete and analytic solution procedure to
locate and guarantee the optimal solutions to both inventory models. In Sections 3 and 4 some discussions and conclusions
are given.
2. The derivation of EOQ and EPQ inventory models
We call the EOQ and EPQ inventory models model with two backorder costs as Model I and Model II respectively. In the
analysis of both inventory models, the following notation and assumptions are defined below:
Notation:
Q Size of order for Model (I) (size of production for Model (II)) in units
S Size of backorders in units
TC1(Q , S) Total annual cost function of Q and S for Model (I) in $ per year
TC2(Q , S) Total annual cost function of Q and S for Model (II) in $ per year
D Demand rate in units per unit of time
P Production rate in units per unit of time for Model (II)
K Ordering cost per order/setup
h Holding cost per unit, per unit of time
p Backorder cost per unit, per unit of time (linear backorder cost)
π Backorder cost per unit (fixed backorder cost)
(Q ∗
i , S∗
i ) The optimal solution of TCi(Q , S) for i = 1, 2.
Assumptions:
(1) demand rate is constant and known over the horizon planning;
(2) production rate is constant and known over the horizon planning;
(3) the production rate must be greater that demand rate, i.e. P > D;
(4) Backorders are allowed and all backorders are satisfied;
(5) Two type backorders cost are considered: linear backorder cost (backorder cost is applied to average backorders) and
fixed cost (backorder cost is applied to maximum backorder level allowed);
(6) The model is for only one product;
(7) The planning horizon is infinite.

3. K.-J. Chung, L.E. Cárdenas-Barrón / Mathematical and Computer Modelling 55 (2012) 2151–2156 2153
2.1. Model (I): the EOQ model with two backorder costs
Based on the above notation and assumptions, the well known total annual cost function TC1(Q , S) for the EOQ model
with two backorder costs is expressed as follows:
TC1(Q , S) =
KD
Q
+
h(Q − S)2
2Q
+
pS2
2Q
+
πDS
Q
(1)
for all Q > 0 and S ≥ 0.
So, from Eq. (1), one can obtain the followings results
∂TC1(Q , S)
∂Q
=
−2KD + hQ 2
− (h + p)S2
− 2πDS
2Q 2
(2)
∂2
TC1(Q , S)
∂Q 2
=
2KD + (h + p)S2
+ 2πDS
Q 3
> 0 (3)
∂TC1(Q , S)
∂S
=
−hQ + (h + p)S + πD
Q
(4)
∂2
TC1(Q , S)
∂S2
=
h + p
Q
> 0 (5)
and
∂2
TC1(Q , S)
∂Q ∂S
=
−(h + p)S − πD
Q 2
. (6)
Let H(Q , S) denote the determinant of the Hessian of TC1(Q , S). Then, this is given by
H(Q , S) =
∂2
TC1(Q , S)
∂Q 2
·
∂2
TC1(Q , S)
∂S2
−

∂2
TC1(Q , S)
∂Q ∂S
2
=
2KD(h + p) − π2
D2
Q 4
. (7)
It is well known that the first-order conditions of a minimum for TC1(Q , S) are given by
∂TC1(Q , S)
∂Q
= 0 (8)
and
∂TC1(Q , S)
∂S
= 0. (9)
Let ( ¯Q1, ¯S1) denote the solution of the simultaneous equations (8) and (9). Eqs. (8) and (9) reveal that
hQ 2
= 2KD + (h + p)S2
+ 2πDS (10)
and
hQ = (h + p)S + πD. (11)
Solving Eqs. (10) and (11) simultaneously for ¯Q1 and ¯S1, we get the following solutions
¯S1 =
h ¯Q1 − πD
h + p
, (12)
and
¯Q1 =

2KD(h + p) − π2D2
hp
. (13)
Avrial [15, pp. 92] presents a Theorem 4.31 that explains that if TC1(Q , S) is convex then the optimal solution is (Q ∗
1 , S∗
1 ) =
( ¯Q1, ¯S1). However, it is worth mentioning that the solution ( ¯Q1, ¯S1) does not exist if the following situations occur
2KD(h + p) − π2
D2
hp
≤ 0, (14)

4. 2154 K.-J. Chung, L.E. Cárdenas-Barrón / Mathematical and Computer Modelling 55 (2012) 2151–2156
or
¯S1 =
h ¯Q1 − πD
h + p
< 0. (15)
To overcome Eq. (15), substituting (13) into (12) to make ¯S1 ≥ 0, we have
2KDh ≥ π2
D2
. (16)
Eq. (16) will imply that the following result holds.
Lemma 1. If 2KDh ≥ π2
D2
, then
(i) 2KD(h + p) − π2
D2
> 0. (17)
(ii) TC1(Q , S) is convex.
Proof. (i) If 2KDh ≥ π2
D2
, then (i) holds.
(ii) Eqs. (3), (5), (7) and (i) imply that TC1(Q , S) is convex.
Theorem 1. The solution ( ¯Q1, ¯S1) satisfying the first-order conditions for TC1(Q , S) exists if and only if 2KDh ≥ π2
D2
.
Notice that the algebraic approach of Sphicas [13] cannot produce the above result. He only proposed an intuitive
argument that lacks of a rigorous mathematical proof.
2.1.1. The solution procedure for Model (I)
From Theorem 1, there are two cases to occur:
Case (A): 2KDh ≥ π2
D2
.
Case (A) implies that TC1(Q , S) is convex on Q > 0 and S ≥ 0. The first-order conditions for a minimum imply
that
¯S1 =
h ¯Q1 − πD
h + p
(18)
and
¯Q1 =

2KD(h + p) − π2D2
hp
. (19)
Theorem 4.31 [15, pp. 92] implies (Q ∗
1 , S∗
1 ) = ( ¯Q1, ¯S1).
Case (B): 2KDh < π2
D2
Case (B) implies that there are three cases to occur:
(i) 2KD(h + p) − π2
D2
> 0.
In this case, Eq. (19) is well-defined. Substituting (19) into (18), we get ¯S1 < 0. Hence, ( ¯Q1, ¯S1) does not exist.
(ii) 2KD(h + p) − π2
D2
= 0.
In this case, ¯Q1 = 0 and ¯S1 = −πD
h+p
< 0. Evidently, ( ¯Q1, ¯S1) does not exist.
(iii) 2KD(h + p) − π2
D2
< 0.
In this case, ¯Q1 is not well-defined. Eq. (18) implies that ¯S1 does not exist. Obviously, ( ¯Q1, ¯S1) does not exist.
Therefore, under Case (B), situations (i)–(iii) conclude that if Q > 0 and S > 0, then (Q , S) is never the optimal solution of
TC1(Q , S) on Q > 0 and S ≥ 0. So, if the optimal solution of TC1(Q , S) on Q > 0 and S ≥ 0 exists, then S∗
1 = 0. Consequently,
we have the following result.
Theorem 2. (A) If 2KDh ≥ π2
D2
, then the optimal solution (Q ∗
1 , S∗
1 ) of TC1(Q , S) on Q > 0 and S ≥ 0 can be determined by
Eqs. (13) and (12), respectively.
(B) If 2KDh < π2
D2
, then S∗
1 = 0 and Q ∗
1 =

2KD
h
.
The above arguments reveal that the optimal solution (Q ∗
1 , S∗
1 ) for Model (I) by using our analytical approach is consistent
with that of Sphicas [13]. Furthermore, if ¯Q1 is not well-defined, Johnson and Montgomery [12] propose in an intuitive way
that the optimal policy is to permit no backorders (S∗
1 = 0). Theorem 2(B) demonstrates that their proposal is true. Also,
Theorem 2(B) tells us that under this situation the better to do is to use the basic EOQ model without backorders.

5. K.-J. Chung, L.E. Cárdenas-Barrón / Mathematical and Computer Modelling 55 (2012) 2151–2156 2155
2.1.2. Model (II): the EPQ model with two backorder costs
Recall that Q and P denote size of production and rate of production in units per unit of time for Model (II), respectively.
In this case the well known total annual cost function TC2(Q , S) is expressed as follows:
TC2(Q , S) =
KD
Q
+
h

Q

1 − D
P

− S
2
2Q

1 − D
P
 +
pS2
2Q

1 − D
P
 +
πDS
Q
. (20)
Dividing the above total cost by (1 − D/P), then we have
TC2(Q , S)
1 − D
P
=

K/

1 − D
P

D
Q
+
h

Q − S/

1 − D
P
2
2Q
+
p

S/

1 − D
P
2
2Q
+
πD

S/

1 − D
P

Q
=
¯KD
¯Q
+
h( ¯Q − ¯S)2
2 ¯Q
+
p(¯S)2
2 ¯Q
+
πD¯S
¯Q
(21)
where
¯K = K/

1 −
D
P

, ¯Q = Q , ¯S = S/

1 −
D
P

and TC2( ¯Q , ¯S) =
TC2(Q , S)
1 − D
P
.
Let
TC2( ¯Q , ¯S) =
¯KD
¯Q
+
h( ¯Q − ¯S)2
2 ¯Q
+
p(¯S)2
2 ¯Q
+
πD¯S
¯Q
. (22)
Eq. (22) reveals that the form of the function TC2( ¯Q , ¯S) to be minimized is identical to Eq. (1). Obviously, Lemma 1,
Theorems 1 and 2 can be modified as Lemma 2, Theorems 3 and 4, respectively.
Lemma 2. If 2KDh/(1 − D/P) ≥ π2
D2
, then
(i) 2KD(h + p)/(1 − D/P) − π2
D2
> 0.
(ii) TC2(Q , S) is convex.
Theorem 3. The solution ( ¯Q2, ¯S2) satisfying the first-order conditions for TC2(Q , S) exists if and only if 2KDh/(1−D/P) ≥ π2
D2
.
Theorem 4. (A) If 2KDh/(1 − D/P) ≥ π2
D2
, then the optimal solution (Q ∗
2 , S∗
2 ) of TC2(Q , S) on Q > 0 and S ≥ 0 will be
expressed as follows:
Q ∗
2 =

2KDh/(1 − D/P) − π2D2
hp
(23)
S∗
2 =
(hQ ∗
2 − πD)(1 − D/P)
h + p
. (24)
(B) If 2KDh/(1 − D/P) < π2
D2
, then the optimal solution (Q ∗
2 , S∗
2 ) of TC2(Q , S) on Q > 0 and S ≥ 0 will be Q ∗
2 =

2KD
h(1−D/P)
and S∗
2 = 0.
The above arguments reveal that the optimal solution (Q ∗
2 , S∗
2 )for Model (II) by using our analytical approach is consistent
with that by using Sphicas [13]. But, our approach is more formal and does not depend on intuitive arguments. Also,
Theorem 4(B) establishes that under this situation the better to do is to use the basic EPQ model without backorders.
3. Discussions
There are three directions to be used to make the following discussions among Johnson and Montgomery [12],
Sphicas [13] and this paper. These directions are: (i) the optimal solution, (ii) processes of derivations to find the optimal
solutions, and (iii) the convexity. They are discussed as follows:
(i) The optimal solution: Sections 2.1.1 and 2.1.2 in this paper reveal that the optimal solution (Q ∗
, S∗
) for Model (I) and
(II) by using our approach are consistent with those by using the algebraic approach of Sphicas [13]. Sphicas [13] indicates
that if
√
2KDh < πD <
√
2KD(h + p), Johnson and Montgomery [12] cannot obtain the optimal solution since S∗
< 0. This
is a shortcoming of Johnson and Montgomery [12]. However, our approach complements the shortcoming of Johnson and
Montgomery [12] and obtains the same optimal solutions as those of Sphicas [13] in a formal mathematical way.
(ii) Processes of derivations to find the optimal solutions: Facing the optimal solution for a problem with an objective
function to be minimized or maximized, the standard approach is to use calculus to explore the functional behaviors
(such as continuous, increasing, decreasing, convex or concave) to locate the optimal solution. Although Johnson and
Montgomery [12] adopted calculus to find the optimal solutions, they did not explicitly identify the two distinct cases:

6. 2156 K.-J. Chung, L.E. Cárdenas-Barrón / Mathematical and Computer Modelling 55 (2012) 2151–2156
(a) 2KDh ≥ π2
D2
for Model (I) (or 2KDh/(1 − D
P
) ≥ π2
D2
for Model (II))
(b) 2KDh < π2
D2
for Model (I) (or 2KDh/(1 − D
P
) < π2
D2
for Model (II)). Basically, Johnson and Montgomery [12]’s
arguments about the optimal solution are not complete. Sphicas [13] overcomes the shortcomings of Johnson and
Montgomery [12] to use the algebraic approach to get the full analysis about locating the optimal solution without
derivatives and obtain the explicit identification of the two cases. The results of Sphicas [13] are correct and interesting;
however, Sphicas [13] is rather trick such that the process of derivation to locate the optimal solutions is not easy.
Furthermore, he uses some intuitive arguments.
The analytic approach of this paper is not only to overcome the shortcomings of Johnson and Montgomery [12] but also
to get the same optimal solutions as those obtained by Sphicas [13] with a rigorous mathematical form. Furthermore, the
processes of the derivation to locate the optimal solution appear as easy to do by using the algebraic approach adopted by
Sphicas [13].
(iii) The convexity: Hadley and Whitin [11] show that the total annual cost function of the basic EOQ (economic order
quantity) without backorders model is convex. Rachamadugu [16] demonstrates that the discounted total annual cost
function is also convex. Chiu et al. [17] presents a better proof of convexity of a long-run average cost function for an
inventory system with stochastic machine breakdown and rework process. Based on the above truths, in fact, if the total
annual cost function of the inventory model is convex, it is easier to find the optimal solution by using the convexity property.
Therefore, it is worth to study the convexity of the total annual cost function of the inventory model to locate the optimal
solution. Consequently, the exploration of the convexity of the total annual cost function is still one of the main research
topics of the inventory models. In fact, the algebraic approach discussed in [13] cannot be used to explore the convexities of
Model (I) and (II). However, Lemmas 1 and 2 in this paper give the explorations of the convexities of TCi(Q , S) for i = 1, 2.
On the other hand, Theorems 1 and 3 in this paper obtain sufficient and necessary conditions for the existences of solutions
( ¯Qi, ¯Si) (i = 1, 2) satisfying the first-order conditions of the total annual cost function of Models (I) and (II), respectively.
Basically, the results of Theorems 1 and 3 cannot be achieved by the algebraic approach discussed in [13] as well.
4. Conclusions
This paper uses an analytic approach to examine the EOQ/EPQ inventory models with two backorder costs and derives
two lemmas and four useful theorems to obtain the optimal solution to both inventory models. Basically, this paper is not
only to overcome the shortcomings of Johnson and Montgomery [12] but also to improve Sphicas [13]. Finally, our analytic
solution procedure is complete and easy. It is important to remark that the EOQ/EPQ models analyzed in this paper are
deterministic. However, our solution procedure can be used in developing more complex inventory models.
Acknowledgments
The second author was supported partially by the School of Business and the Tecnológico de Monterrey research fund
numbers CAT128 and CAT185.
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