International Journal of Performability Engineering Vol. 10, No. 6, September 2014, pp. 631-640.
© RAMS Consultants
Printed in India
Optimal Selection of Process Mean and Production Lot Size
for Newsvendor Problem
MAJID M. ALDAIHANI and M. A. DARWISH
Department of Industrial and Management Systems Engineering,
College of Engineering and Petroleum, Kuwait University,
P.O. Box 5969, 13060 Safat, Kuwait
(Received on Dec.08, 2013 and revised on March 17, 2014)
Abstract: Traditionally, the quality of product is not incorporated in the newsvendor
problem. One approach to ensure the quality of the product is through process targeting.
Selection of process mean is expected to determine production conforming rate and affects
the newsvendor decision regarding the production lot size. The purpose of this paper is to
integrate those two important issues, namely, process mean selection and production lot
size for a newsvendor who produces the item at a finite rate. The newsvendor uses raw
material to produce the item to satisfy a stochastic demand which follows a general
distribution. The amount of raw material received by an item is uncertain and follows a
normal distribution. It is assumed that the performance variable of the product has a
lower specification limit, and the items that do not conform to the specification limit are
scrapped with no salvage value. The expected total cost which consists of the following
components: acquisition cost, production cost, underage cost, and overage cost is
established. Upper and lower bounds on the optimal process mean are determined and a
simple procedure to solve the model is devised.
Sensitivity analysis is conducted to
investigate the effect of the parameters of the proposed model.
Keywords: Quality control, targeting problem, newsvendor, lot-sizing.
1.
Introduction
Optimal determination of process mean (target value) has received a considerable
attention from researchers as well as practitioners because it is important and challenging
decision for quality and manufacturing managers [3]. Usually, high process mean leads to
less number of nonconforming items, thus, it is possible that the process mean is adjusted
to high value in order to reduce the proportion of the nonconforming items. However,
high process mean may result in high production cost. On the other hand, when the
process mean is set at a low value, cost incurred due to nonconforming items will be high.
Hence, the decision of selecting process mean is based on the tradeoff between raw
material cost and the cost incurred due to nonconforming items. Another related problem
is the newsvendor problem whereas a vendor orders raw material from a supplier to
produce an item which is used to satisfy an end customer who observes a random demand
in one selling period. At the end of a selling period, the vendor may have excess
inventory or shortage depending on the production lot size and the actual demand. That
is, if it happens that the produced lot is more than the demand, the vendor ends up with
surplus that will be salvaged at a price less than the selling price, therefor, the vendor
incurs an overage cost in this case. On the other hand, when the production lot size is less
than the observed demand, the vendor will experience shortage which cause a loss of
profit and reputation of the vendor (this loss is called underage cost). Hence, the
newsvendor problem is basically a tradeoff between underage and overage costs. Clearly,
the lifecycle of the produced item is limited by the selling season duration (this type of
item is called newsvendor-type item). Usually, process mean selection and lot sizing
_____________________________________________
*Corresponding author’s email: [email protected]
631
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Majid M. Aldaihani and M. A. Darwish
decisions in a production process that manufactures a newsvendor-type item are treated
separately. However, the choice of process mean affects the conformance rate which
influences other important decisions such as the production lot size. Thus, the process
mean and lot sizing decisions should be determined jointly in order to control the total
cost associated with production process. Those two important problems, namely, the
process mean selection and lot sizing decision of a newsvendor are integrated in this
paper. It is expected that this integration leads to higher conformance rate and reduction in
scrap or reprocessing cost [3].
In the past, integrated targeting-production problem has been addressed by many
researchers. For instance, [7] determined the optimal process mean and lot size for
integrated targeting-inventory model when the process mean is constant during the
production cycle and the demand is deterministic. This model is generalized by [9] where
production lot size, raw material procurement policy, and the targeting problem are
included in the model. Also, [1] extended the work of [9] by relaxing the assumption of
constant process mean, instead, they considered the process mean drifts with time. They
determined optimal setting of the process mean and lot sizing decisions. However, they
assumed that the demand is deterministic. Another important work is attributed to [3]
who developed an integrated targeting-inventory model for a two-layer supply chain.
Later, [2] relaxed the assumption of deterministic demand in [9]. They assumed that the
demand follows a normal distribution and producer is subjected to service level constraint.
Moreover, [10] presented a new philosophy for process adjustment based on the
application of feed-forward control. Furthermore, [4] integrated the targeting problem
with inventory decisions when the demand is uncertain. This model simultaneously finds
the optimal decisions regarding process mean, production lot size, and reorder point in (Q,
R) continuous review model. Recently, [5] integrated the process mean selection and
maintenance schedule for imperfect production process. They found a closed form
solution to the problem.
In all of these studies, the lifecycle of the produced item is considered infinite.
However, products with short lifecycles become more common due to the advancement in
technology. As a result, researchers and practitioners paid much focus to operations
management issues related to newsvendor-type products in recent years. For example,
[11] studied the effects of demand uncertainty on the optimal pricing and lot size
decisions in the classical newsvendor model. Further, [8] developed a quality modeling
system that supports production process development and quality control. The proposed
system depends on the main interaction between process variables and quality attributes.
Moreover, [6] modeled the newsvendor problem with the objective of minimizing the perunit cost instead of the cost per season. They include the holding cost, setup cost,
underage cost as well as overage cost in the model.
In this paper, we integrate the targeting problem with production decisions of a
producer who manufactures a newsvendor-type product. The demand observed by the
newsvendor is assumed to follow a general distribution. In addition, the amount of raw
material received by a produced item is also a random variable follows a normal
distribution. An item is considered of acceptable quality if the amount of raw material
received by the item is more than a lower specification limit, otherwise, the item is
nonconforming to specification and scraped with no salvage value. We develop a model
for this integrated problem and devise a simple procedure that finds the optimal process
mean and production run size. Upper and lower bounds on the optimal process mean are
also established. We further conduct sensitivity analysis on the model’s key parameters.
The organization of this paper is as follows: Section 2 presents the statement of the
Optimal Selection of Process Mean and Production Lot Size for Newsvendor Problem
633
problem and notation used in developing the model. Then, the model is established in the
in Section 3. In Section 4, solution method and bounds on the process mean are
developed. Sensitivity analysis is presented in Section 5. Finally, the paper is concluded
in Section 6.
2.
Problem Statement and Notation
A newsvendor orders raw material from a supplier to produce an item in lots of size Q.
The amount of raw material required to produce a lot of size Q is Q R which is equal to
µ R Q where µ R is the process mean setting which can be adjusted by the producer. Let W
be a random variable represents the amount of raw material a produced item receives, we
assume in this paper that W follows a normal distribution with adjustable mean µ R and a
constant standard deviation σ R . The item is considered nonconforming to specifications if
the quality characteristic W is less than a lower specification limit L. Let Φ denote the
standard normal cumulative distribution function, thus, the probability that a produced
item is nonconforming p is given by:
 L − µR 

p = Φ
 σR 
(1)
We assume that nonconforming items which are equal to pQ are screened and scraped
with no salvage value while conforming items which are equal to (1- p)Q are used to
satisfy end customers.
It is assumed that the demand observed by newsvendor is a random variable D which
follows a general probability distribution function f(x) with a mean of µ and a standard
deviation σ. In case of low demand seasons, an overage cost C o is incurred for each
unsold item. On the other hand, when demand is high, newsvendor incurs an underage
cost C u for each unsatisfied demand. The following notation is used in developing the
model:
W
L
µR
σR
Q
QR
Cu
Co
CR
p
Β
D
f(x)
µ
Amount of material received by an item,
lower specification limit,
expected value of W,
standard deviation of W,
production lot size,
lot size of raw material,
underage cost associated with each demand that cannot be met,
overage cost associated with each unsold item
unit material cost,
fraction of nonconforming items,
cost of producing one item,
a random variable represents demand on newsvendor,
probability distribution function of D,
expected value of D,
634
σ
Majid M. Aldaihani and M. A. Darwish
standard deviation of D,
3.
Model Development
The objective of the model is to determine the optimal process mean (µ R ) and the lot size
(Q) such that the expected total cost of the newsvendor (ETC) is minimized. The
components of ETC are itemized as follows:
1. Acquisition Cost: The amount of raw material needed to produce a lot of size Q is
µ R Q, thus, the acquisition cost in one selling period (TC R ) is given by:
(2)
TC R = CR µ RQ
where C R is the unit material cost.
2. Production Cost: The direct production cost is assumed to be a linear function of
number of items produced [1] and [9]. Assume that β is cost of producing one item,
hence, the production total cost per selling period (TC P ) is:
(3)
TCP = β Q
3. Cost Due to Demand Uncertainty: Two costs are incurred due to uncertainty in
demand, the first of which is when the produced quantity is more than the anticipated
demand, in this case, an overage cost in one selling period (OC) is incurred. It is
worth mentioning that the produced lot of size Q contains (1-p)Q conforming items
that will be used to satisfy the demand, thus overage cost in one selling period is
given by:
(1− p ) Q
∫ ((1 − p)Q − x) f ( x)dx
OC = Co
(4)
0
The other cost is when the anticipated demand is lower than the actual demand, then
shortage will occur. In this case, newsvendor incurs an underage cost (UC) in one selling
period which can be set out as follows:
∞
UC = Cu
∫ ( x − (1 − p)Q) f ( x)dx
(5)
(1− p ) Q
Thus, from (4) and (5), the cost associated with randomness in demand (RDC) is:
RDC = Co
(1− p ) Q
∞
∞
(1− p ) Q
∫ ((1 − p)Q − x) f ( x)dx + Cu
∫ ( x − (1 − p)Q) f ( x)dx
(6)
Thus, from (2), (3), and (6) the expected total cost of the newsvendor (ETC) is:
(1− p ) Q
ETC = ( β + CR µ R )Q + Co
∫ ((1 − p)Q − x) f ( x)dx +
∞
∞
Cu
∫ ( x − (1 − p)Q) f ( x)dx
(7)
(1− p ) Q
The decision variables in this model are Q and µ R . It is important to mention that the
probability of producing nonconforming container p in Equation (7) is not a constant but a
function of the process mean µ R as shown by Equation (1).
Optimal Selection of Process Mean and Production Lot Size for Newsvendor Problem
4.
635
Method of Solution and Bounds on Process Mean
In this section, we find the values of µ R and Q that minimize ETC (obtained by
differentiating ETC with respect to µ R and Q and setting the result to 0). This leads to the
following equations:
(1− p ) Q
Cu (1 − p ) − ( β + C R µ R )
(8)
∫ f ( x)dx =
(Co + Cu )(1 − p )
0
(1− p ) Q
∫
f ( x ) dx =
0
Cu
σR
CR
σR −
C o + Cu
Co + Cu φ (u )
(9)
where u = (L-µ R )/σ R . Derivation of Equations (8) and (9) is shown in Appendix A.
Solving (8) and (9) yields:
φ (u )
− C Rσ R = 0
ψ (µ R ) = (β + CR µ R )
1 − Φ (u )
(10)
Note that Equation (10) has only one decision variable µ R , thus it can be solved by
Bisection Method or any method based on functional evaluations. We then can use
Equation (8) or (9) to find the optimal Q. Using Bisection Method requires a lower limit
and an upper limit on µ R . We establish those limits in Appendix B and they are as
follows:
L +σR
 1
ln 
2π


 Cu
C o + Cu

C − σ C
R
R
 R




2


min L + σ R



 ≤ µR ≤


 1
ln 
2π


 Cu

C
 R




2
 C −β C −β
, o
, u
CR
CR







(11)
We have to indicate that the demand is assumed to follow a normal distribution in
deriving Equation (11). The normality assumption is the most common in the literature of
newsvendor problem. A simple solution procedure that solves the model can be as
follows: First we determine the bounds in Equation (11), we then find the optimal µ R by
solving Equation (10) and the optimal Q can then be determined from Equation (9). It is
worth mentioning that Equation (10) can be solved by any numerical method, for
example, Bisection Method, Newton-Raphson Method, Chord Method, etc. Because it is
simple, we used Bisection Method to find the results in the next section.
5.
Sensitivity Analysis
In this section, we perform numerical analysis for the model presented in this paper. We
consider a dairy plant orders raw milk from a farm at a cost of 20 cents per liter. The
plant processes the raw milk to produce containers of powdered milk at a cost of $4 per
container. The amount of powdered milk in a container is a random variable that follows
normal distribution with a standard deviation of 0.6 Kg. The plant is subjected to a
random demand that follows a normal distribution with a mean of 200 containers and
standard deviation of 50 containers. It is estimated that a shortage cost of $20 per
container is observed due to lost sale, and if a container is not sold before the expiration
date, a cost of $10 per container is incurred. It is also estimated that 10 liters of raw milk
636
Majid M. Aldaihani and M. A. Darwish
is needed to produce 1 kg of powdered milk. Thus, C o =$10/container, C u =$20/container,
C R =$2/container, σ R =0.6 Kg, L=1 Kg, µ=200 containers, σ=50 containers, and
β=$4/container. Unless specified otherwise, the above data is used to study the effects of
C R , σ R , β, and L on the optimal solution. We have to point out that some parameters are
not easy to estimate in practice, thus an expert opinion must be sought in this case.
Another important point is that we tried many randomly selected sets of basic data, we
observed that the behavior of the model is the same regardless of the selected basic data.
5.1 Effect of Unit Material Cost
In order to reduce the increasing material cost, the optimal target value is expected to
decrease, this is shown in Figure 1. Even though, the probability of producing a
nonconforming item is increased which in turn affects the cost associated with
nonconforming items, the cost of material becomes too high and thus the process mean is
decreased because the extra cost due to nonconforming items is not compensated by the
increase in nonconforming items. Moreover, Figure 1 shows that the lot size is decreased
when material cost is high. This, in fact, is because the material cost becomes more
significant than underage cost and consequently low lot sizes are more economical.
5.2 Effect of Variation of the Amount of Material Received by an Item
It is known that increasing the standard deviation of the amount of material received by an
item at a fixed process mean, leads to higher number of defective items. Thus, it is
expected that as standard deviation of the amount of material received by an item
increases, the optimal process mean increases to lower the number of defective items.
This trend is shown in Figure 2, however, unexpectedly, the process mean decreases when
the standard deviation is too high. This can be attributed to the fact that after a certain σR,
the increase in µR will not help in reducing the expected total cost. Thus, the model
suggests that if the variation is too high, it is optimal to reduce the process mean. Figure 2
shows that the lot size is always increasing with standard deviation of the amount of fill.
This is true because the number produced conforming items will be less when σR is high,
thus the lot size increases.
Figure 1: Effect of Unit Material Cost on Optimal Process Mean and Lot Size.
Optimal Selection of Process Mean and Production Lot Size for Newsvendor Problem
637
Figure 2: Effect of Raw Material Standard Deviation on Optimal Process Mean and Lot Size.
5.3 Effect of Unit Production Cost and Lower Specification Limit
Figure 3 shows that when production cost is high, the process mean is set high at the
optimal solution. This is due to the fact that high production cost makes the item so
valuable to be lost because it does not receive appropriate amount of raw material. Since
the standard deviation of the amount of fill is fixed, the only way to guarantee high
conformance rate is by increasing the process mean. Furthermore, Figure 3 shows that
the production lot size decreases with β because the cost of a shortage is, relatively
speaking, low when unit production cost is high.
Figure 3: Effect of Production Cost on Optimal Process Mean and Lot Size.
When the standard deviation and mean of the amount of raw material an item receives
are fixed, the lower specification limit affects the probability of producing a
nonconforming item. Thus, if the lower specification limit is high, the optimal amount of
fill will be high as shown in Figure 4. This, in turn, leads to low lot size as depicted in
Figure 4.
Figure 4: Effect of Lower Specification Limit on Optimal Process Mean and Lot Size.
638
6.
Majid M. Aldaihani and M. A. Darwish
Conclusion
Quality and production issues are integrated in this paper. In particular, the optimal
selection of process mean and lot-sizing decisions are jointly integrated. The producer
uses raw material to produce a newsvendor-type item to satisfy a stochastic demand which
follows a general distribution. Furthermore, the amount of raw material received by an
item is a random variable follows a normal distribution. High quality products are used to
satisfy end customers and items with low quality are scraped with no salvage value. A
lower specification limit on a quality characteristic is used to evaluate the quality of an
item.
The expected total cost is established and it consists of the following components:
acquisition cost, production cost, underage cost, and overage cost. We then determined
the values of process mean and lot size that minimize expected total cost. Moreover, upper
and lower bounds on the optimal process mean are developed and a simple solution
method is established.
The results show that the production decisions are greatly affected by the setting of the
process mean. In particular, the raw material cost, the standard deviation of the amount of
fill, production cost, and the lower specification limit have significant impact on the on
both the optimal process mean and production lot size.
Acknowledgment: The authors are grateful to the anonymous referees for their
constructive comments and helpful suggestions. They would like also to acknowledge
Kuwait University.
References
[1].
Al-Fawzan, M.A., and M. Hariga. An Integrated Inventory-Targeting Problem with Time
Dependent Process Mean. Production Planning and Control, 2002; 13(1): 11-16.
[2]. Alkhedher, M. and M.A. Darwish. Optimal Process Mean for a Stochastic Inventory Model
under Service Level Constraint. International Journal of Operational Research, 2012; 18(3):
346-363.
[3]. Darwish, M.A. Economic Selection of Process Mean for Single-Vendor Single-Buyer Supply
Chain. European Journal of Operational Research, 2009; 199(1): 162-169.
[4]. Darwish, M. A., F. Abdelmalek and M. Alkhedher. Optimal Decisions for Integrated
Targeting-Inventory Problem under Stochastic Demand. European Journal of Operational
Research, 2013; 226(3): 481-490.
[5]. Darwish, M.A., Abdulrahman Alenezi, and S.K. Goyal. Simultaneous Determination of
Maintenance Schedule and Process Mean of a Production System. Submitted to International
Journal of Productivity and Quality Management, 2014.
[6]. Darwish, M.A., Abdulrahman Alenezi, and Majid Aldaihani. Per-Unit Cost Minimization of
the Newsvendor Problem. Accepted to International Journal of Industrial and Systems
Engineering, 2013.
[7]. Gong, L., J. Roan, and K. Tang. Process Mean Determination with Quantity Discounts in
Raw Material Cost. Decision Sciences, 1988; 29(1): 271-302.
[8]. Kazmer, D. and L. Zhu. A Product Quality and Process Feasibility Modeling System.
International Journal of Performability Engineering, 2012; 8(6): 615-624.
[9]. Roan, J., L. Gong, and K. Tang. Joint Determination of Process Mean, Production Run Size
and Material Order Quantity for a Container-Filling Process. International Journal of
Production Economics, 2000; 63(3): 303-317.
[10]. Shi, L. and K.C. Kapur. Process Monitoring and Feedforward Control for Proactive Quality
Improvement. International Journal of Performability Engineering, 2012; 8(6): 601-614.
[11]. Xu, M., Y. Chen, and X. Xu. The Effect of Demand Uncertainty in a Price-Setting
Newsvendor Model. European Journal of Operational Research, 2010; 207(2): 946–957.
Optimal Selection of Process Mean and Production Lot Size for Newsvendor Problem
639
Appendix A
In this appendix, we show how Equations (8) and (9) are derived. From the expected total cost given by
Equation (7), we obtain:
∂ETC
= ( β + c R µ R ) + Co (1 − p )
∂Q
(1− p ) Q
∫ f ( x)dx − C
∞
u
(1 − p )
0
∫ f ( x)dx = 0
(1− p ) Q
This equation can be simplified as follows:
(1− p ) Q
∫ f ( x)dx =
0
Cu (1 − p ) − ( β + C R µ R )
Cu
(β + CR µ R )
=
−
(Co + Cu )(1 − p )
Co + Cu (Co + Cu )(1 − p )
After taking the derivative with respect to μ, we obtain:
(1− p ) Q
∫ f ( x)dx = C
0
Cu
σR
CR
σR −
+
+
φ
C
C
C
(u )
o
u
o
u
and the proof is complete.
Appendix B
We establish the upper and lower bounds on process mean µ R here. We use the following fact:
(1− p ) Q
0≤
∫ f ( x)dx ≤ 1
(B1)
0
We consider the following four cases:
(1− p ) Q
Case I: Using
∫ f ( x)dx ≥ 0 from Equation (B1) and Equation (8), we obtain:
0
Cu (1 − p ) − ( β + C R µ R )
≥0
(Co + Cu )(1 − p )
Equation (B2) can be rewritten as follows:
p ≤1−
(B2)
β + CR µ R
(B3)
Cu
β
C
+
However, Equation (B3) reveals that
R µR
≤ 1 , which yields
Cu
C −β
µR ≤ u
CR
(B4)
Case II: Using Equation (B1) and Equation (9), we get:
Cu
σR
CR
σR −
≥0
C o + Cu
Co + Cu φ (u )
(B5)
But
φ ( z) =
1
− z2
1
e 2
2π
(B6)
From Equations (B5) and (B6), we obtain:
2
L −σ R
 Cu

ln 
 ≤ µR ≤ L + σ R
 2π C R 
Case III: Using Equation (B1) and Equation (8), we obtain:
 Cu

ln 

 2π C R 
2
(B7)
640
Majid M. Aldaihani and M. A. Darwish
 σ C − C − C
R
u
o
u
2π σ R C R


µ R ≥ L + σ R ln 





2




(B8)
Case IV: Using Equation (B1) and Equation (8), we obtain:
µR ≤
Co − β
CR
(B9)
From B4, B7, B8 and B9, we get:
L +σR
 1
ln 
2π


 Cu
C o + Cu

C − σ C
R
R
 R




2

 ≤ µR ≤



 1

min L + σ R ln 
 2π


 Cu

 CR



2
 C −β C −β
, o
, u
CR
CR






This completes the proof.
Majid M. Aldaihani is an Associate Professor and Chairman of Industrial and
Management Systems Engineering at Kuwait University. He received his B.S. in
Petroleum Engineering from Kuwait University and his M.S. and Ph.D. in Industrial and
Systems Engineering from the University of Southern California. His research interests
focus on applied operations research. He is a Certified Educational Institution Auditor, a
recipient of Six Sigma Master Black Belt, and a recipient of Kuwait University Award of
Excellence in Teaching. He is a member of Institute for Operations Research and
Management Science (INFORMS), Institute of Industrial Engineers (IIE) and Kuwait
Society for Engineers (KSE).
Mohammed A. Darwish is an Associate Professor in Industrial and Management
Systems Engineering Department at Kuwait University. He obtained his Ph.D. in
Industrial and Manufacturing Systems Engineering in 1998 from the University of Texas
at Arlington, U.S.A. His research interests include Supply Chain Management,
Production Planning & Control, and Quality Control. He has published in several
international journals. He received the CCSE Outstanding Research Performance Award
(2007/2008). He also received the KFUPM Distinguished Teaching and Advising Award
of (2004/2005). He is a member in the Engineering honor Society (Tau Beta Pi) and
Industrial Engineering Honor Society (Alpha Pi Mu).