SlideShare una empresa de Scribd logo
1 de 7
Descargar para leer sin conexiΓ³n
M. Ullah&C.W. Kang Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 6( Version 6), June 2014, pp.177-183
Quality Costs (IRR) Impact on Lot Size Considering Work in
Process Inventory
Misbah Ullaha
, Chang W. Kangb*
a-b
Department of Industrial and Management Engineering, ERICA Campus, Hanyang University, Ansan,
Republic of Korea
Abstract
Economic order quantity model and production quantity model assume that production processes are error
free. However, variations exist in processes which result in imperfection particularly in high machining
environments. Processes variations result in nonconformities that increase quality costs in the form of
rework, rejects and quality control techniques implementations to ensure quality product delivery. This
paper is an attempt towards development of inventory model which incorporate inspection, rework, and
rejection (IRR) quality costs in optimum lot size calculation focusing work in process inventory.
Mathematical model is derived for optimum lot size based on minimum average cost function using
analytical approach. This new developed model (GTOQIRR) assume an imperfect production environment.
Numerical examples are used to visualize the significant effect of quality cost in the proposed model in
comparison to the previously developed models. The proposed model is highly recommendable for quality
based high machining manufacturing environments considering work in process inventories.
Keywords: WIP inventory; EOQ model; quality cost; lot size; GTOQ model.
I. Introduction
During past few decades, significant attention has
been given to the area of inventory management
because of its prime importance for most
manufacturing organizations. Organizations use
economic order quantity (EOQ) and production order
quantity (POQ) models since 1913 to calculate
optimum inventory lot size considering holding and
setup costs. Researchers extended these models to
more generic and practical situations of everyday life.
However these two basic models are based on an
unrealistic assumption that perfect products are
produced every time. A number of factors affect
manufacturing processes which include machine
failure, changes in raw material supplier, tool wear
and tear, outside conditions etc. especially where
machining time is large relatively. Therefore the
phenomena of rework, reject and inspection is
common in industries like machine tool industries,
aerospace industries and glass industries etc. High
machining environments, rework, rejects and
inspection compel to introduce quality costs incurred
during these process and its ultimate impact on
inventory lot size.
Rosenblatt and Lee [1] developed EOQ model for
imperfect manufacturing environments with
conclusion that lot size reduces as imperfection is
increased. Later Sarker et al. [2] extended EOQ model
to a multistage manufacturing environment with
imperfection in processes. The developed model
assumed that rework operation is performed at the end
of each cycle and at the end of N cycle. Further
extension of such models can be realized in research
articles published in the last decade [3-12].They
realized that the impact of imperfection on inventory
lot size in addition to the importance of inspection in
real world manufacturing environments. Combined
effect of inspection with imperfections was
emphasized in few papers such as ([13] and [18]).
Furthermore, ([14], [15], [16], [17] and [19])
developed models for optimum lot size by considering
other important aspects of the real world
manufacturing systems. Researchers mainly focused
raw materials and finish goods inventories in
calculation of optimum lot size calculation. However,
the third type of inventory, work in process, in
combination with quality cost has been relatively
ignored in previous literature.
Today, industries put their efforts to reduce their
inventories cost either in the form of raw materials,
finish goods or work in process. Boucher [20]
introduced the concept of work in process inventory in
group technology based manufacturing environment
for calculation of optimum lot size. The model was
named as group technology order quantity (GTOQ)
model. Barzoki et al. [21] extended GTOQ model for
imperfect production environments under the title of
group technology order quantity model with rework
(GTOQR). Their model considered 100%
qualification of rework products. Numerical examples
were used for model comparison against previous
models. These models are recommended for group
technology work environment where machining time
and lot size are relatively large. However, increase
RESEARCH ARTICLE OPEN ACCESS
www.ijera.com 177|P a g e
M. Ullah&C.W. Kang Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 6( Version 6), June 2014, pp.177-183
machining time and higher demand rate significantly
contribute towards quality costs in processes. Higher
machining rate and large machining time results in
increased quality costs due to high tool wear and tear,
process failure etc. Increased quality costs are in the
form of rework, rejects and inspection that affect total
cost for optimum lot size calculation. This paper is an
attempt towards calculation of quality costs impact on
the optimal lot size considering imperfection in
processes. This paper incorporate quality costs due to
inspection, rework and rejection (IRR) in total cost
calculation. Inclusion of quality costs in inventory
models and its impact on lot size is lacking in previous
papers incorporating work in process inventories. The
remaining sections are listed as follow;
Section 2 explains problem statement with
assumptions and notations. In section 3, the problem is
modeled based on modeling for relevant costs
associated with this problem. Section 4 comprises
comparison of models based on numerical
computations. Conclusion is presented in the last
section.
II. Problem statement
We consider a single discrete manufacturing
setup. A machining station is followed by an
inspection process with three buffer stations in the
manufacturing workshop. Buffer stations are used for
work in process inventories i.e. raw material
(unprocessed items, waiting for operations), good
quality products and poor quality products (rejected
items).
A lot of material (Q) arrives in the workshop
processed on machining station in each cycle. Lot (Q)
is released for inspection to declare products as either
good quality products or poor quality products. As
the process is imperfect so three kinds of products are
produced during the whole cycle time.
The process produces, 𝑄𝑄𝑄𝑄10, percent of parts as poor
quality products, 𝑄𝑄𝑄𝑄1π‘Ÿπ‘Ÿ, percent of parts as reworked
and the remaining 𝑄𝑄�1 βˆ’ (𝑝𝑝10 + 𝑝𝑝1π‘Ÿπ‘Ÿ)οΏ½ as good
quality products in the first phase of the lot Q as
shown in Figure 1.
During the next phase 2, the rework items 𝑄𝑄𝑄𝑄1π‘Ÿπ‘Ÿ=
Q(𝑝𝑝20 + 𝑝𝑝21) are reprocessed and re-inspected.
𝑄𝑄𝑝𝑝21percent of products are good quality products
and, 𝑄𝑄𝑄𝑄20, percent of products are poor quality in the
phase 2 as shown in Figure 2. We assume that no
rework is needed after that. At the end of the cycle,
the following conclusion is drawn regarding the lot
size Q.
Good quality products produced,𝑄𝑄𝑝𝑝1 = 𝑄𝑄�1 βˆ’
(𝑝𝑝10 + 𝑝𝑝20)οΏ½
Poor quality products produced,𝑄𝑄𝑝𝑝0 = 𝑄𝑄(𝑝𝑝10 + 𝑝𝑝20)
Products that are rework able after phase 1,
𝑄𝑄𝑃𝑃1π‘Ÿπ‘Ÿ = 𝑄𝑄(𝑝𝑝20 + 𝑝𝑝21)
Therefore, our objective is to develop an
economic order quantity model for an imperfect
manufacturing environment taking quality costs into
consideration with work in process inventory.
Figure 1. Processing of lot size Q during phase 1
Figure 2. Processing of rework products in phase 2.
2.1. Notations
𝑑𝑑 customer demand
𝑄𝑄 lot size processed per unit cycle
𝑀𝑀𝑐𝑐 raw material cost per unit
𝑃𝑃𝑐𝑐 purchasing cost per unit of time
𝑆𝑆𝑐𝑐 setup cost per unit of time ($/unit of time)
Process
Q
i
Q(1-(p10+p1r))
Rejected: Q(p10)
Rework: Qp1r
Inspection
Process
QP1r units
Q(p21)
Rejected:Q(p20)
Rework: 0
Inspection
www.ijera.com 178|P a g e
M. Ullah&C.W. Kang Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 6( Version 6), June 2014, pp.177-183
𝑄𝑄𝑐𝑐 cost of quality per unit of time ($/unit of time)
𝐢𝐢𝑖𝑖 inspection cost
𝑅𝑅𝑅𝑅𝑐𝑐 rework cost
𝑅𝑅𝑅𝑅𝑐𝑐 rejection cost
π‘Šπ‘Šπ‘Šπ‘Šπ‘Šπ‘Šπ‘π‘ work in process holding cost
𝐼𝐼 𝐼𝐼𝑐𝑐 inventory holding cost
𝑇𝑇𝑐𝑐 total cost
𝑑𝑑𝑠𝑠 setup time
𝑑𝑑 π‘šπ‘š1 machining time in phase 1. (unit time per unit
product)
𝑑𝑑 π‘šπ‘šπ‘šπ‘š machining time per unit for rework products
𝑝𝑝10 proportion of poor quality products in phase 1
𝑝𝑝11 proportion of good quality products in phase 1
𝑝𝑝21 proportion of good quality products in phase 2
𝑝𝑝20 proportion of poor quality products in phase 2
𝑝𝑝1π‘Ÿπ‘Ÿ proportion of rework able products in phase 1
𝑝𝑝0 proportion of poor quality products at the end of
cycle
𝑝𝑝1 proportion of good quality products at the end of
cycle
𝑑𝑑𝑐𝑐 cycle time
𝑑𝑑𝑝𝑝 total processing time
𝑑𝑑 average manufacturing time for each product item
𝐺𝐺 average storage inventory
π‘Šπ‘Š average monetary value of the WIP inventory ($)
𝑖𝑖 inventory holding cost per unit of time ($/unit of
time)
𝑐𝑐 average unit value of each product (unit of money
($) per unit)
π‘˜π‘˜ rate charged per unit of cell production time
including all overheads, moving cost,
loading/unloading cost etc. (unit of money ($) per
unit of time)
2.2 Assumptions
 Shortages are not allowed
 Demand is known and pre-determined.
 Inspection is performed of all lot and
rework able products.
 Poor quality products are produced during
the rework operation.
 Rework operation is done one time only.
 No stoppage is allowed during the
manufacturing of one lot.
All parameters including demand and production rate,
setup times etc. are constants and deterministic.
III. Modeling
Modeling of the problem is based on cycle time
calculation in terms of demand rate and lot size Q in
addition to the imperfection in the production
processes.
As the objective of almost every manufacturing
setup is to meet customer demand rate (d) therefore
lot size (Q) is defined keeping customer demand in
mind. Processes undergoing imperfection results in
poor quality products at the end of each cycle.
Therefore we can have following relation for the
above mentioned scenario
𝑄𝑄(1 βˆ’ 𝑝𝑝0) = 𝑑𝑑𝑑𝑑𝑐𝑐
𝑑𝑑𝑐𝑐=
𝑄𝑄(1βˆ’π‘π‘0)
𝑑𝑑
(1)
The total processing time ( 𝑑𝑑𝑝𝑝 ) of a product
having lot size Q comprises of setup time,
manufacturing time, inspection time per unit product
(𝑑𝑑𝑖𝑖), rework time and re-inspection time.
𝑑𝑑𝑝𝑝= total processing time
𝑑𝑑𝑝𝑝= 𝑑𝑑𝑠𝑠 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 + 𝑑𝑑𝑖𝑖 𝑄𝑄 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ + 𝑑𝑑𝑖𝑖 𝑄𝑄𝑝𝑝1π‘Ÿπ‘Ÿ (2)
Therefore average processing time is given by
𝑑𝑑 =
𝑑𝑑𝑝𝑝
𝑄𝑄
𝑑𝑑 =
𝑑𝑑𝑠𝑠 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 + 𝑑𝑑𝑖𝑖 𝑄𝑄 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ + 𝑑𝑑𝑖𝑖 𝑄𝑄𝑝𝑝1π‘Ÿπ‘Ÿ
𝑄𝑄
If π‘˜π‘˜ is the average cost of production per unit of
time then the average cost added to each product unit
during its manufacturing is 𝑣𝑣 given by
𝑣𝑣 = π‘˜π‘˜π‘‘π‘‘ = π‘˜π‘˜ οΏ½
𝑑𝑑𝑠𝑠+𝑄𝑄𝑑𝑑 π‘šπ‘š1 +𝑑𝑑𝑖𝑖 𝑄𝑄+𝑄𝑄𝑑𝑑 π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ+𝑑𝑑𝑖𝑖 𝑄𝑄𝑝𝑝1π‘Ÿπ‘Ÿ
𝑄𝑄
οΏ½
We assume that 𝑀𝑀𝑐𝑐 be the material cost per unit of
product, then the average total cost added to each
product unit is given by
𝑐𝑐= 𝑀𝑀𝑐𝑐 + π‘˜π‘˜ οΏ½
𝑑𝑑𝑠𝑠+𝑄𝑄𝑑𝑑 π‘šπ‘š1 +𝑑𝑑𝑖𝑖 𝑄𝑄+𝑄𝑄𝑑𝑑 π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ+𝑑𝑑𝑖𝑖 𝑄𝑄𝑝𝑝1π‘Ÿπ‘Ÿ
𝑄𝑄
οΏ½ (3)
3.1 Cost calculation
As we know that most of the inventory models
are based on the cost minimization, therefore we are
interested to calculate associated costs with this
model. Quality costs, inventory holding cost, setup
cost and material purchasing cost are the main
components of total cost calculation. All these costs
are calculated keeping imperfection in processes
under consideration.
3.1.1 Quality cost (IRR)
Quality costs considered in this paper are mainly
associated with imperfection in processes. These
imperfection results in products failure either in the
form of rework or rejected products. Similarly,
quality control techniques including inspection exists
within the manufacturing setup which also results in
quality cost. Other kinds of quality cost related to
appraisal and prevention has been neglected.
Therefore, quality costs considered in this model are
(a) Quality costs due to products inspection (𝐢𝐢𝑖𝑖). (b)
Quality costs due to rework (πΆπΆπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿ), (c) Quality costs
due to rejection (πΆπΆπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿ).
Therefore quality cost (πΆπΆπ‘žπ‘ž) is given by
𝑄𝑄𝑐𝑐 = 𝐢𝐢𝐢𝐢+ 𝑅𝑅𝑅𝑅𝑐𝑐 + 𝑅𝑅𝑅𝑅𝑐𝑐 (4)
These costs are modeled as follow
a) Quality costs due to inspection
www.ijera.com 179|P a g e
M. Ullah&C.W. Kang Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 6( Version 6), June 2014, pp.177-183
Inspection is performed for all manufactured and
reworked products. 100% inspection of lot is carried
out when it is manufactured. Similarly reworked
products are also passed through the inspection
process. The inspection cost 𝐢𝐢𝐢𝐢 per unit of cycle time
is given by
𝐢𝐢𝐢𝐢 =
𝐼𝐼𝑐𝑐 𝑄𝑄
𝑑𝑑𝑐𝑐
+
(𝑝𝑝1π‘Ÿπ‘Ÿ)𝑄𝑄𝐼𝐼𝑐𝑐
𝑑𝑑𝑐𝑐
𝐢𝐢𝐢𝐢 =
𝐼𝐼𝑐𝑐 𝑑𝑑
(1 βˆ’ 𝑝𝑝0)
+
𝐼𝐼𝑐𝑐 𝑑𝑑(𝑝𝑝1π‘Ÿπ‘Ÿ)
(1 βˆ’ 𝑝𝑝0)
𝐢𝐢𝐢𝐢 = �
𝐼𝐼𝑐𝑐 𝑑𝑑
1βˆ’π‘π‘0
οΏ½ (1 + 𝑝𝑝1π‘Ÿπ‘Ÿ) (5)
b) Quality costs due to rework
Let 𝑝𝑝1π‘Ÿπ‘Ÿ is the percentage of rework product in a lot
size Q in a complete cycle and c be the value added to
the products in the manufacturing process. Then the
quality cost per unit time for rework items is given by
𝑅𝑅𝑅𝑅𝑐𝑐 = οΏ½
𝑛𝑛𝑝𝑝1π‘Ÿπ‘Ÿ 𝑄𝑄
𝑑𝑑𝑐𝑐
οΏ½ 𝑐𝑐 (6)
where n is the no. of times rework operation is
performed.
c) Quality costs due to rejection
Let 𝑝𝑝0 is the percentage of rejected product in a lot
size Q in a complete cycle and 𝑐𝑐 be the value added
to the products in the manufacturing process. Then
the quality cost per unit time for rejected products is
given by
𝑅𝑅𝑅𝑅𝑐𝑐= οΏ½
𝑝𝑝0 𝑄𝑄
𝑑𝑑𝑐𝑐
οΏ½ 𝑐𝑐 (7)
Therefore, Equation (4) becomes,
𝑄𝑄𝑐𝑐= οΏ½
𝐼𝐼𝑐𝑐 𝑑𝑑
1βˆ’π‘π‘0
οΏ½ (1 + 𝑝𝑝1π‘Ÿπ‘Ÿ) + οΏ½
𝑛𝑛𝑝𝑝1π‘Ÿπ‘Ÿ 𝑄𝑄
𝑑𝑑𝑐𝑐
οΏ½ 𝑐𝑐 + οΏ½
𝑝𝑝0 𝑄𝑄
𝑑𝑑𝑐𝑐
οΏ½ 𝑐𝑐
𝑄𝑄𝑐𝑐= οΏ½
𝐼𝐼𝑐𝑐 𝑑𝑑
1βˆ’π‘ƒπ‘ƒπ‘π‘
οΏ½ (1 + 𝑝𝑝1π‘Ÿπ‘Ÿ) + οΏ½
𝑐𝑐𝑐𝑐
1βˆ’π‘π‘0
οΏ½ (𝑛𝑛𝑛𝑛1π‘Ÿπ‘Ÿ + 𝑝𝑝0) (8)
3.1.2 Inventory holding cost
Silver et al. [22] introduced following
relationship for calculation of inventory holding cost
per unit of time.
𝐼𝐼 𝐼𝐼𝑐𝑐= 𝑖𝑖𝑖𝑖𝑖𝑖, (9)
Here 𝐺𝐺 = average inventory over each cycle by the
cycle period.
𝑐𝑐 = average unit monetary value for each item.
𝑖𝑖 = carrying charge.
The average holding inventory (𝐺𝐺) per unit cycle time
is given as
𝐺𝐺 =
1
2
𝑄𝑄(1βˆ’π‘π‘0)𝑑𝑑𝑐𝑐
𝑑𝑑𝑐𝑐
=
1
2
𝑄𝑄(1 βˆ’ 𝑝𝑝0) (10)
Given Equations (3) and (10), the inventory holding
cost per unit of time will be
𝐼𝐼 𝐼𝐼𝑐𝑐=
1
2
𝑖𝑖 ��𝑀𝑀𝑐𝑐 +
π‘˜π‘˜ οΏ½
𝑑𝑑𝑠𝑠+𝑄𝑄𝑑𝑑 π‘šπ‘š1 +𝑑𝑑𝑖𝑖 𝑄𝑄+𝑄𝑄𝑑𝑑 π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ+𝑑𝑑𝑖𝑖 𝑄𝑄𝑝𝑝1π‘Ÿπ‘Ÿ
𝑄𝑄
οΏ½οΏ½ 𝑄𝑄(1 βˆ’ 𝑝𝑝0)οΏ½ (11)
3.1.3 Work in process cost
Based on the same concepts as [21] and [22], the
work in process inventory holding cost is calculated
by:
π‘Šπ‘Šπ‘Šπ‘Šπ‘Šπ‘Šπ‘π‘ = π‘–π‘–π‘Šπ‘Š, (12)
where 𝑖𝑖 = it is the carrying charge per unit of time
π‘Šπ‘Š = average work in process inventory
π‘Šπ‘Šcan be computed by summation of following types
of inventories in the workshop.
(a). The unprocessed product item waiting for
operation on machine versus manufacturing time 𝑑𝑑𝑝𝑝
during each cycle. (b). Good quality products
inventory over time. (c). Rejected parts inventory
over time.
Hence the average value of the total work in process
inventory will be equal to
π‘Šπ‘Š = οΏ½
1/2𝑄𝑄𝑑𝑑 𝑝𝑝
𝑑𝑑𝑐𝑐
οΏ½ 𝑀𝑀𝑐𝑐 + οΏ½
1
2
𝑄𝑄( 1βˆ’π‘π‘0)𝑑𝑑 𝑝𝑝
𝑑𝑑𝑐𝑐
οΏ½ 𝑐𝑐 + οΏ½
1
2
𝑄𝑄𝑝𝑝0 𝑑𝑑 𝑝𝑝
𝑑𝑑𝑐𝑐
οΏ½ 𝑐𝑐
π‘Šπ‘Š =
1
2
𝑄𝑄𝑑𝑑 𝑝𝑝
𝑑𝑑𝑐𝑐
(𝑀𝑀𝑐𝑐 + (1 βˆ’ 𝑝𝑝0)𝑐𝑐 + 𝑝𝑝0 𝑐𝑐)
π‘Šπ‘Š =
1
2
οΏ½
𝑑𝑑
1βˆ’π‘π‘0
οΏ½ (𝑑𝑑𝑠𝑠 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 + 𝑑𝑑𝑖𝑖 𝑄𝑄 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ + 𝑑𝑑𝑖𝑖 𝑄𝑄𝑝𝑝1π‘Ÿπ‘Ÿ)(𝑀𝑀𝑐𝑐 + 𝑐𝑐)
Given Equation (3), the average value of total work in process inventory will be
π‘Šπ‘Š=
1
2
οΏ½
𝑑𝑑
1βˆ’π‘π‘0
οΏ½ οΏ½2𝑀𝑀𝑐𝑐 +
π‘˜π‘˜π‘‘π‘‘π‘ π‘ 
𝑄𝑄
+ π‘˜π‘˜π‘‘π‘‘ π‘šπ‘š1 + π‘˜π‘˜π‘‘π‘‘ π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ+𝑑𝑑𝑖𝑖 π‘˜π‘˜ + 𝑑𝑑𝑖𝑖 π‘˜π‘˜π‘π‘1π‘Ÿπ‘Ÿ οΏ½ (𝑑𝑑𝑠𝑠 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 + 𝑑𝑑𝑖𝑖 𝑄𝑄 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ + 𝑑𝑑𝑖𝑖 𝑄𝑄𝑝𝑝1π‘Ÿπ‘Ÿ) (13)
Therefore given Equations (12) and (13), the average carrying charge of the work in process inventory will be
π‘Šπ‘Šπ‘Šπ‘Šπ‘Šπ‘Šπ‘π‘ =
1
2
οΏ½
𝑑𝑑𝑑𝑑
1βˆ’π‘π‘0
οΏ½ οΏ½2𝑀𝑀𝑐𝑐 +
π‘˜π‘˜π‘‘π‘‘π‘ π‘ 
𝑄𝑄
+ π‘˜π‘˜π‘‘π‘‘ π‘šπ‘š1 + π‘˜π‘˜π‘‘π‘‘ π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ+𝑑𝑑𝑖𝑖 π‘˜π‘˜ + 𝑑𝑑𝑖𝑖 π‘˜π‘˜π‘π‘1π‘Ÿπ‘Ÿ οΏ½ (𝑑𝑑𝑠𝑠 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 + 𝑑𝑑𝑖𝑖 𝑄𝑄 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ + 𝑑𝑑𝑖𝑖 𝑄𝑄𝑝𝑝1π‘Ÿπ‘Ÿ) (14)
3.1.4Material Cost
Material cost per unit of cycle time (𝑃𝑃𝑐𝑐) is given by the following equation
𝑃𝑃𝑐𝑐 = 𝑀𝑀𝑐𝑐 𝑄𝑄 𝑑𝑑𝑐𝑐⁄
We can write in terms of demand(𝑑𝑑) is
𝑃𝑃𝑐𝑐 =
𝑀𝑀𝑐𝑐 𝑑𝑑
(1βˆ’π‘π‘0)
(15)
www.ijera.com 180|P a g e
M. Ullah&C.W. Kang Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 6( Version 6), June 2014, pp.177-183
3.1.5Setup cost
Machines and inspection setup take place only once during whole lot manufacturing. We assume that no extra
setup is required for rework operations.
It can be modeled as follow
𝑆𝑆𝑐𝑐 =
𝐴𝐴
𝑑𝑑𝑐𝑐
=
𝐴𝐴𝐴𝐴
𝑄𝑄(1βˆ’π‘π‘0)
(16)
Where A is the setup cost. It is a product of set up time per cycle (𝑑𝑑𝑠𝑠) and the rate charged by the production
cycle (π‘˜π‘˜).
3.2 Total cost per unit of time
The total cost per unit of time are given by the following equation:
𝑇𝑇𝑐𝑐=𝑄𝑄𝑐𝑐 + 𝐼𝐼 𝐼𝐼𝑐𝑐+ π‘Šπ‘Šπ‘Šπ‘Šπ‘Šπ‘Šπ‘π‘ + 𝑃𝑃𝑐𝑐 + 𝑆𝑆𝑐𝑐 (17)
Given equations (4), (5), (10), (13), and (16) in Eq. (17)
𝑇𝑇𝑐𝑐= οΏ½
𝐼𝐼𝑐𝑐 𝑑𝑑
1βˆ’π‘ƒπ‘ƒπ‘π‘
οΏ½ (1 + 𝑝𝑝1π‘Ÿπ‘Ÿ) + οΏ½
𝑐𝑐𝑐𝑐
1βˆ’π‘π‘0
οΏ½ (𝑛𝑛𝑛𝑛1π‘Ÿπ‘Ÿ + 𝑝𝑝0) +
1
2
𝑖𝑖 ��𝑀𝑀𝑐𝑐 + π‘˜π‘˜ οΏ½
𝑑𝑑𝑠𝑠+𝑄𝑄𝑑𝑑 π‘šπ‘š1 +𝑑𝑑𝑖𝑖 𝑄𝑄+𝑄𝑄𝑑𝑑 π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ+𝑑𝑑𝑖𝑖 𝑄𝑄𝑝𝑝1π‘Ÿπ‘Ÿ
𝑄𝑄
οΏ½οΏ½ 𝑄𝑄(1 βˆ’ 𝑝𝑝0)οΏ½
+
1
2
οΏ½
𝑑𝑑𝑑𝑑
1βˆ’π‘π‘0
οΏ½ οΏ½2𝑀𝑀𝑐𝑐 +
π‘˜π‘˜π‘‘π‘‘π‘ π‘ 
𝑄𝑄
+ π‘˜π‘˜π‘‘π‘‘ π‘šπ‘š1 + π‘˜π‘˜π‘‘π‘‘ π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ+𝑑𝑑𝑖𝑖 π‘˜π‘˜ + 𝑑𝑑𝑖𝑖 π‘˜π‘˜π‘π‘1π‘Ÿπ‘Ÿ οΏ½ (𝑑𝑑𝑠𝑠 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 + 𝑑𝑑𝑖𝑖 𝑄𝑄 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ + 𝑑𝑑𝑖𝑖 𝑄𝑄𝑝𝑝1π‘Ÿπ‘Ÿ)+
𝑀𝑀𝑐𝑐 𝑑𝑑
(1βˆ’π‘π‘0)
+
𝐴𝐴𝐴𝐴
𝑄𝑄(1βˆ’π‘π‘0)
(18)
3.3 Optimum lot size
Equation (18) gives average cost function considering quality cost. Optimum lot size can be calculated based on
average cost minimization. Therefore by taking 1st
derivative of the function w.r.t. Q and equating to 0 i.e.
𝑑𝑑𝑇𝑇𝑐𝑐
𝑑𝑑𝑑𝑑
= 0
and sufficient condition that the equation must hold and satisfy for optimum lot size calculation is given by
𝑑𝑑2 𝑇𝑇𝑐𝑐
𝑑𝑑𝑄𝑄2 > 0
Hence the optimum lot size is given by
GTOQIRR = (19)
οΏ½
2𝐴𝐴 𝑑𝑑 + 𝑖𝑖𝑑𝑑𝑑𝑑(𝑑𝑑𝑠𝑠)2 + 2 π‘˜π‘˜π‘‘π‘‘π‘ π‘  𝑑𝑑(𝑛𝑛𝑛𝑛1π‘Ÿπ‘Ÿ + 𝑝𝑝0)
𝑖𝑖(1 βˆ’ 𝑝𝑝0)2οΏ½(𝑀𝑀𝑐𝑐 + π‘˜π‘˜π‘‘π‘‘ π‘šπ‘š1 + 𝑑𝑑𝑖𝑖 π‘˜π‘˜) + (π‘˜π‘˜π‘‘π‘‘ π‘šπ‘šπ‘Ÿπ‘Ÿ + 𝑑𝑑𝑖𝑖 π‘˜π‘˜)𝑝𝑝1π‘Ÿπ‘ŸοΏ½ + 𝑑𝑑𝑖𝑖 οΏ½οΏ½(𝑑𝑑 π‘šπ‘š1 + 𝑑𝑑 π‘šπ‘šπ‘Ÿπ‘Ÿ 𝑝𝑝1π‘Ÿπ‘Ÿ)(2𝑀𝑀𝑐𝑐 + π‘˜π‘˜π‘‘π‘‘ π‘šπ‘š1 + π‘˜π‘˜π‘‘π‘‘ π‘šπ‘šπ‘Ÿπ‘Ÿ 𝑝𝑝1π‘Ÿπ‘Ÿ)οΏ½ + (𝑑𝑑𝑖𝑖(1 + 𝑝𝑝1 π‘Ÿπ‘Ÿ)(2𝑀𝑀𝑐𝑐 + 𝑑𝑑𝑖𝑖 π‘˜π‘˜ + 2π‘˜π‘˜π‘‘π‘‘ π‘šπ‘š1 + 2π‘˜π‘˜π‘‘π‘‘ π‘šπ‘šπ‘šπ‘š 𝑝𝑝1π‘Ÿπ‘Ÿ + 𝑑𝑑𝑖𝑖 π‘˜π‘˜π‘π‘1π‘Ÿπ‘Ÿ)οΏ½
The addition of the term β€œ2 π‘˜π‘˜π‘‘π‘‘π‘ π‘  𝑑𝑑(𝑛𝑛𝑛𝑛1π‘Ÿπ‘Ÿ + 𝑝𝑝0) ” represents the quality cost (IRR) impact on the optimum lot size.
Where β€˜n’ represents the no. of times rework operation is performed.𝑝𝑝1π‘Ÿπ‘Ÿ represents the proportion of rework
components produced during the process and 𝑝𝑝0 represnest the rejected products during the production process.
It can be observed that increased in imperfection, number of rework operations increased quality costs and
optimal lot size.
IV. Results
The proposed model is compared with the most well-known EOQ model and GTOQ model developed by
Boucher [20]. Five different cases data is taken from a US tool manufacturing company, initially introduced by
Boucher [20]. Few assumptions were made regarding data calculation e.g. percentage of rejects 𝑝𝑝0 = 20%, the
percentage of rework 𝑝𝑝1π‘Ÿπ‘Ÿ= 5%,𝑑𝑑 π‘šπ‘šπ‘šπ‘š= 5% of actual machining time 𝑑𝑑 π‘šπ‘š1 and k= 3000 ($/year) in all five cases. It
is assumed that rework operation is performed only once.
Table 1. Optimum lot size using GTOQ, EOQ and proposed model. (assume, 𝑖𝑖 = 35%
Case
No.
𝑑𝑑𝑖𝑖(min/
unit)
A ($/unit)
d(units
/year)
𝑀𝑀𝑐𝑐
(USD/ unit)
𝑑𝑑𝑠𝑠*(min/
setup)
𝑑𝑑 π‘šπ‘š1* (min/
unit)
EOQ GTOQ
GTOQ
IRR
1 20 14.349 77 5.63 574 100 28 26 34
2 20 12.75 233 1.57 510 32 85 81 96
3 20 12.951 580 1.42 518 87 109 87 98
4 20 14.349 1877 1.64 574 67 216 135 139
5 20 17.274 5361 1.12 691 41 497 255 233
www.ijera.com 181|P a g e
M. Ullah&C.W. Kang Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 6( Version 6), June 2014, pp.177-183
It can be observed that the response of all three models is different to all cases. Response of the proposed
model is significant in comparison to the previous developed model (EOQ and GTOQ) as imperfection and
quality costs are taken into consideration.
The impact of quality cost (IRR) has also been highlighted in the table 2 below. We assume 𝑖𝑖 = 35 %, 𝑀𝑀𝑐𝑐 =
1($/unit), d = 14000 (units/year), π‘˜π‘˜ = 7000 ($/year), 𝑑𝑑𝑠𝑠= 0.0017(years/unit), 𝑑𝑑 π‘šπ‘š1= 0.12 (mints/unit), 𝑑𝑑 π‘šπ‘šπ‘šπ‘š= 5%
(π‘šπ‘š1), A =11.9 ($/unit).It can be observed that as imperfection in processes increase, quality costs shoots up and
optimal lot size is increased. This impact of imperfection on quality cost and optimum lot size has also been
shown in Fig.3. Quality costs are almost negligible when there processes are perfect and goes on increasing as
imperfection in the form of rework and rejected products increases.
Table 2.Impact of imperfection on quality cost in inventory model.
Case No. 𝑝𝑝𝑏𝑏(%) 𝑝𝑝1π‘Ÿπ‘Ÿ(%) GTOQIRR Setup cost WIP cost Finish goods cost Quality cost (IRR)
1 0 0 943 176.63 17.81 471.61 0.01
2 5 5 1037 169.05 20.01 492.76 1511.78
3 10 10 1139 162.57 22.59 512.40 3189.48
4 15 15 1248 157.00 25.64 530.57 5062.92
5 20 20 1368 152.22 29.28 547.23 7169.20
6 25 25 1499 148.14 33.67 562.29 9555.25
Fig. 3 Change in optimal lot size and quality cost (IRR) with change in Imperfection
V. Conclusion
The importance of quality in manufacturing
setups cannot be neglected in general and particularly
in environments where machining work is large
relatively. Boucher [20] realized in his model that
GTOQ is good for high machining environments.
This increased machining ultimately impact quality
costs as well. Impact of quality costs on lot size has
not been considered till now while developing such
models for work in process inventories. Different
inventory models are developed during last few
decades to solve the real world problems and
optimize the lot size as per actual scenarios. However
work in process based inventory model are lacking
the component of quality costs in their previously
developed models especially when the imperfection
and high machining rate of processes are considered.
This paper is an attempt towards the development of
such model for imperfect process considering IRR
quality cost component. The proposed model is of
significance for manufacturing environments having
concentration on quality costs. It is more generalized
and incorporate all aspects of the previously
developed models. This research can be further
extended by estimating the effect of shortages and
machine breakdowns on lot size focusing work in
process inventory.
VI. Acknowledgment:
The author would like to express his gratitude to
all anonymous editors and referees for their valuable
comments.
www.ijera.com 182|P a g e
M. Ullah&C.W. Kang Int. Journal of Engineering Research and Applications www.ijera.com
ISSN : 2248-9622, Vol. 4, Issue 6( Version 6), June 2014, pp.177-183
References
[1] Rosenblatt, M.J. and Lee, H.L.(1986),
Economic production cycle with imperfect
production processes. IIE Transaction, 18,
pp. 48-55.
[2] Sarker, B.R., Jamal, A.M.M., and Mondal, S.
(2007) Optimum batch size in a multi stage
production system with rework consideration.
European Journal of Operational Research,
184 (3). pp. 915- 929.
[3] Gupta, T. and Chakraborty, S. (1984)
Looping a multistage production system.
International Journal of Production
Research, 222 (2), pp. 299-311.
[4] Jamal A.M.M., Sarker, B.R., and Mondal, S.
(2004)Optimum manufacturing batch size
with rework process at a single stage
production system. Computers and Industrial
Engineering, 47 (1), pp. 77-89.
[5] Biswas, P. and Sarker, B.R. (2008). Optimal
batch quantity models for a lean production
system with in-cycle rework and scrap.
International Journal of Production
Research, 46 (23), pp. 6585-6610.
[6] Zhang, X. and Gerchak, Y. (1990). Joint lot
sizing and inspection policy in an EOQ model
with random yield. IIE Transaction, 22, pp.
41-47.
[7] Salameh, M.K. and Jaber, M.Y. (2000).
Economic production quantity model for
items with imperfect quality. International
Journal of Production Economics, 64 (1),
pp.59-64.
[8] Ben Daya, M. and Rahim, A. (2003).Optimal
lot sizing, quality improvement and inspection
errors for multistage production systems.
International Journal of Production
Research, 41 (1), pp. 65-79.
[9] Krishnamurthy, C. and Panayappan, S.
(2012). An EPQ Model with imperfect
production systems with rework of regular
production and sales return. American
Journal of Operation Research, 2, pp. 225-
234.
[10] Yoo, S. H., Kim, D.S., and Park, M.
(2012).Inventory models for imperfect
production and inspection processes with
various inspection options under one-time
and continuous improvement investment.
Journal of Computers and Operation
Research, 39 (9), pp. 2001-2015.
[11] Ojha, D., Sarker, B. R., and Biswas, P.
(2007). An optimal batch size for an imperfect
production system with quality assurance and
rework. International Journal of Production
Research, 45 (14), pp. 3191-3214.
[12] Wee, H.M., Wang, W.T., and Yang P.C.
(2013). A production quantity model for
imperfect quality items with shortage and
screening constraint. International Journal of
Production Research, 51 (6), pp. 1869-1884.
[13] Muddah, B., Salameh, M.K., and Karameh,
G.M. (2009). Lot sizing with random yield
and different qualities”. Applied
Mathematical Modeling, 33 (4), pp. 1997-
2009.
[15] Porteus, E. (1986). Optimal lot-sizing,
process quality improvement and setup cost
reduction. Operations Research, 34 (1), pp.
137–144.
[16] Chung, K.J. (2011). The economic production
quantity with rework process in supply chain
management. Computers and Mathematics
with Applications, 62 (6), pp. 2547-2550.
[17] Khan, M., Jaber M.K., and Bonney, M.
(2011).An economic order quantity (EOQ) for
items with imperfect quality and inspection
errors. International Journal of Production
Economics, 133 (1), pp. 113-118.
[18] Colledani, M. and Tolio, T. (2009)
Performance evaluation of production system
monitored by statistical process control and
off-line inspections. International Journal of
Production Economics. 120(2), pp. 348-367.
2009
[19] Krishnamoorthi, C. and Panayappan, S.
(2012). An EPQ model with reworking of
imperfect production items and shortages.
International journal of Management
Sciences and Engineering Management, 7
(3), pp. 220-228.
[20] Boucher, T.O. (1984). Lot sizing in group
technology production system. International
Journal of Production Research, 22 (1), pp.
85-93.
[21] Barzoki, M.R., Jahanbazi, M., and Bijari, M.
(2011). Effects of imperfect products on lot
sizing with work in process inventory.
Applied Mathematics and Computation, 217
(21), pp. 8328-8336.
[22] Silver, E.A., Pyke, D.F., and Peterson, R.,
1998. Inventory Management and Production
Planning and Scheduling, 3rd
Ed., Wiley,
New York.
www.ijera.com 183|P a g e

MΓ‘s contenido relacionado

La actualidad mΓ‘s candente

IRJET- A Study on Productivity of Concreting Work in Building Construction in...
IRJET- A Study on Productivity of Concreting Work in Building Construction in...IRJET- A Study on Productivity of Concreting Work in Building Construction in...
IRJET- A Study on Productivity of Concreting Work in Building Construction in...IRJET Journal
Β 
IRJET- Optimization of Cycle Time by using Various Techniques: A Review
IRJET- Optimization of Cycle Time by using Various Techniques: A ReviewIRJET- Optimization of Cycle Time by using Various Techniques: A Review
IRJET- Optimization of Cycle Time by using Various Techniques: A ReviewIRJET Journal
Β 
Cost Accounting
Cost AccountingCost Accounting
Cost AccountingSameera Khan
Β 
IRJET- Analysis of Constructions Productivity based on Progress Payment Certi...
IRJET- Analysis of Constructions Productivity based on Progress Payment Certi...IRJET- Analysis of Constructions Productivity based on Progress Payment Certi...
IRJET- Analysis of Constructions Productivity based on Progress Payment Certi...IRJET Journal
Β 
Ms ii unit ppt by DURGA PRASAD NAVULLA (jntuk)
Ms ii unit ppt by DURGA PRASAD NAVULLA (jntuk)Ms ii unit ppt by DURGA PRASAD NAVULLA (jntuk)
Ms ii unit ppt by DURGA PRASAD NAVULLA (jntuk)Dr. Durgaprasad Navulla
Β 
IRJET-Productivity Improvement in a Pressure Vessel using Lean Principles
IRJET-Productivity Improvement in a Pressure Vessel using Lean PrinciplesIRJET-Productivity Improvement in a Pressure Vessel using Lean Principles
IRJET-Productivity Improvement in a Pressure Vessel using Lean PrinciplesIRJET Journal
Β 
Production process
Production processProduction process
Production processBoss Hsiang
Β 
manufacturing operations
manufacturing operationsmanufacturing operations
manufacturing operationsAnshu Singh
Β 
INDUSTRIAL ENGINEERING 2 nd module
INDUSTRIAL ENGINEERING 2 nd moduleINDUSTRIAL ENGINEERING 2 nd module
INDUSTRIAL ENGINEERING 2 nd moduleJim Alex
Β 
Production and Operations Management
Production and Operations ManagementProduction and Operations Management
Production and Operations ManagementKate Sevilla
Β 
Job Order, Process, and ABC Costing
Job Order, Process, and ABC CostingJob Order, Process, and ABC Costing
Job Order, Process, and ABC CostingPaola Dawn Guerra
Β 
Om lect 04_a(r0-aug08)_facility location & layout_mms_sies
Om lect 04_a(r0-aug08)_facility location & layout_mms_siesOm lect 04_a(r0-aug08)_facility location & layout_mms_sies
Om lect 04_a(r0-aug08)_facility location & layout_mms_siesvideoaakash15
Β 
Process costing
Process costingProcess costing
Process costingacctg2012
Β 
Introduction to Production Planning & Control & Manufacturing.pptx
Introduction to Production Planning & Control & Manufacturing.pptxIntroduction to Production Planning & Control & Manufacturing.pptx
Introduction to Production Planning & Control & Manufacturing.pptxTarek Erin
Β 
Swayamacademicwriting
SwayamacademicwritingSwayamacademicwriting
SwayamacademicwritingSunil P Kumar
Β 
Work Study- Methods Study
Work Study- Methods StudyWork Study- Methods Study
Work Study- Methods StudyJoseph Konnully
Β 

La actualidad mΓ‘s candente (20)

IRJET- A Study on Productivity of Concreting Work in Building Construction in...
IRJET- A Study on Productivity of Concreting Work in Building Construction in...IRJET- A Study on Productivity of Concreting Work in Building Construction in...
IRJET- A Study on Productivity of Concreting Work in Building Construction in...
Β 
IRJET- Optimization of Cycle Time by using Various Techniques: A Review
IRJET- Optimization of Cycle Time by using Various Techniques: A ReviewIRJET- Optimization of Cycle Time by using Various Techniques: A Review
IRJET- Optimization of Cycle Time by using Various Techniques: A Review
Β 
Cost Accounting
Cost AccountingCost Accounting
Cost Accounting
Β 
IRJET- Analysis of Constructions Productivity based on Progress Payment Certi...
IRJET- Analysis of Constructions Productivity based on Progress Payment Certi...IRJET- Analysis of Constructions Productivity based on Progress Payment Certi...
IRJET- Analysis of Constructions Productivity based on Progress Payment Certi...
Β 
Ms ii unit ppt by DURGA PRASAD NAVULLA (jntuk)
Ms ii unit ppt by DURGA PRASAD NAVULLA (jntuk)Ms ii unit ppt by DURGA PRASAD NAVULLA (jntuk)
Ms ii unit ppt by DURGA PRASAD NAVULLA (jntuk)
Β 
A012210104
A012210104A012210104
A012210104
Β 
IRJET-Productivity Improvement in a Pressure Vessel using Lean Principles
IRJET-Productivity Improvement in a Pressure Vessel using Lean PrinciplesIRJET-Productivity Improvement in a Pressure Vessel using Lean Principles
IRJET-Productivity Improvement in a Pressure Vessel using Lean Principles
Β 
Production process
Production processProduction process
Production process
Β 
manufacturing operations
manufacturing operationsmanufacturing operations
manufacturing operations
Β 
A Case Study on Productivity Improvement of Assembly line using VSM Methodology
A Case Study on Productivity Improvement of Assembly line using VSM MethodologyA Case Study on Productivity Improvement of Assembly line using VSM Methodology
A Case Study on Productivity Improvement of Assembly line using VSM Methodology
Β 
INDUSTRIAL ENGINEERING 2 nd module
INDUSTRIAL ENGINEERING 2 nd moduleINDUSTRIAL ENGINEERING 2 nd module
INDUSTRIAL ENGINEERING 2 nd module
Β 
Production and Operations Management
Production and Operations ManagementProduction and Operations Management
Production and Operations Management
Β 
Ep 08
Ep 08Ep 08
Ep 08
Β 
Operation Process
Operation ProcessOperation Process
Operation Process
Β 
Job Order, Process, and ABC Costing
Job Order, Process, and ABC CostingJob Order, Process, and ABC Costing
Job Order, Process, and ABC Costing
Β 
Om lect 04_a(r0-aug08)_facility location & layout_mms_sies
Om lect 04_a(r0-aug08)_facility location & layout_mms_siesOm lect 04_a(r0-aug08)_facility location & layout_mms_sies
Om lect 04_a(r0-aug08)_facility location & layout_mms_sies
Β 
Process costing
Process costingProcess costing
Process costing
Β 
Introduction to Production Planning & Control & Manufacturing.pptx
Introduction to Production Planning & Control & Manufacturing.pptxIntroduction to Production Planning & Control & Manufacturing.pptx
Introduction to Production Planning & Control & Manufacturing.pptx
Β 
Swayamacademicwriting
SwayamacademicwritingSwayamacademicwriting
Swayamacademicwriting
Β 
Work Study- Methods Study
Work Study- Methods StudyWork Study- Methods Study
Work Study- Methods Study
Β 

Destacado

The power of being LinkedIn 2014 by J.R. Atkins
The power of being LinkedIn 2014 by J.R. AtkinsThe power of being LinkedIn 2014 by J.R. Atkins
The power of being LinkedIn 2014 by J.R. AtkinsJ.R. Atkins, MBA, MDiv
Β 
Stockholm a scene for creativity - Presentation for MICEboard
Stockholm a scene for creativity - Presentation for MICEboardStockholm a scene for creativity - Presentation for MICEboard
Stockholm a scene for creativity - Presentation for MICEboardMICEboard
Β 
ГлянСц β„–13 (Π½ΠΎΡΠ±Ρ€ΡŒ 2012)
ГлянСц β„–13 (Π½ΠΎΡΠ±Ρ€ΡŒ 2012)ГлянСц β„–13 (Π½ΠΎΡΠ±Ρ€ΡŒ 2012)
ГлянСц β„–13 (Π½ΠΎΡΠ±Ρ€ΡŒ 2012)gorodche
Β 
Π’ΠΈΠ΄ Π½Π° ΠΆΠΈΡ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ Π² Испании
Π’ΠΈΠ΄ Π½Π° ΠΆΠΈΡ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ Π² Π˜ΡΠΏΠ°Π½ΠΈΠΈΠ’ΠΈΠ΄ Π½Π° ΠΆΠΈΡ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ Π² Испании
Π’ΠΈΠ΄ Π½Π° ΠΆΠΈΡ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ Π² ИспанииDan Group Immigration
Β 
A importΓ’ncia da leitura
A importΓ’ncia da leituraA importΓ’ncia da leitura
A importΓ’ncia da leituraKΓͺnia Machado
Β 
HOJITA EVANGELIO DOMINGO XX CICLO A SERIE
HOJITA EVANGELIO  DOMINGO XX CICLO A SERIEHOJITA EVANGELIO  DOMINGO XX CICLO A SERIE
HOJITA EVANGELIO DOMINGO XX CICLO A SERIENelson GΓ³mez
Β 
Ab04603165169
Ab04603165169Ab04603165169
Ab04603165169IJERA Editor
Β 
Uu no 7_th_2004 sumber daya air
Uu no 7_th_2004 sumber daya airUu no 7_th_2004 sumber daya air
Uu no 7_th_2004 sumber daya airSei Enim
Β 
Book OratΓ³rio Eco Clube
Book OratΓ³rio Eco ClubeBook OratΓ³rio Eco Clube
Book OratΓ³rio Eco ClubeDanilo Reis
Β 
7 ΰΈͺาฑัญ ΰΈ„ΰΈ“ΰΈ΄ΰΈ•
7 ΰΈͺาฑัญ ΰΈ„ΰΈ“ΰΈ΄ΰΈ•7 ΰΈͺาฑัญ ΰΈ„ΰΈ“ΰΈ΄ΰΈ•
7 ΰΈͺาฑัญ คณิตฝฝ' ฝน
Β 
Portfolio 2010
Portfolio 2010Portfolio 2010
Portfolio 2010RobbyDesigns
Β 
ΞŒΟ„Ξ±Ξ½ ΞΏ ΠΡρσέας συνάντησΡ τον Ωρίωνα: ΞœΞ­ΟΞΏΟ‚ Ξ’Ξ„
ΞŒΟ„Ξ±Ξ½ ΞΏ ΠΡρσέας συνάντησΡ τον Ωρίωνα: ΞœΞ­ΟΞΏΟ‚ Ξ’Ξ„ ΞŒΟ„Ξ±Ξ½ ΞΏ ΠΡρσέας συνάντησΡ τον Ωρίωνα: ΞœΞ­ΟΞΏΟ‚ Ξ’Ξ„
ΞŒΟ„Ξ±Ξ½ ΞΏ ΠΡρσέας συνάντησΡ τον Ωρίωνα: ΞœΞ­ΟΞΏΟ‚ Ξ’Ξ„ HIOTELIS IOANNIS
Β 
Aa04602180190
Aa04602180190Aa04602180190
Aa04602180190IJERA Editor
Β 
Catalogo
CatalogoCatalogo
Catalogogeisajph
Β 
Linux Y Sus Equivalencias
Linux Y Sus EquivalenciasLinux Y Sus Equivalencias
Linux Y Sus Equivalenciasjggc
Β 

Destacado (20)

The power of being LinkedIn 2014 by J.R. Atkins
The power of being LinkedIn 2014 by J.R. AtkinsThe power of being LinkedIn 2014 by J.R. Atkins
The power of being LinkedIn 2014 by J.R. Atkins
Β 
F046052832
F046052832F046052832
F046052832
Β 
Wethebiocloud.patrocinadores.2013
Wethebiocloud.patrocinadores.2013Wethebiocloud.patrocinadores.2013
Wethebiocloud.patrocinadores.2013
Β 
Stockholm a scene for creativity - Presentation for MICEboard
Stockholm a scene for creativity - Presentation for MICEboardStockholm a scene for creativity - Presentation for MICEboard
Stockholm a scene for creativity - Presentation for MICEboard
Β 
ГлянСц β„–13 (Π½ΠΎΡΠ±Ρ€ΡŒ 2012)
ГлянСц β„–13 (Π½ΠΎΡΠ±Ρ€ΡŒ 2012)ГлянСц β„–13 (Π½ΠΎΡΠ±Ρ€ΡŒ 2012)
ГлянСц β„–13 (Π½ΠΎΡΠ±Ρ€ΡŒ 2012)
Β 
Π’ΠΈΠ΄ Π½Π° ΠΆΠΈΡ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ Π² Испании
Π’ΠΈΠ΄ Π½Π° ΠΆΠΈΡ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ Π² Π˜ΡΠΏΠ°Π½ΠΈΠΈΠ’ΠΈΠ΄ Π½Π° ΠΆΠΈΡ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ Π² Испании
Π’ΠΈΠ΄ Π½Π° ΠΆΠΈΡ‚Π΅Π»ΡŒΡΡ‚Π²ΠΎ Π² Испании
Β 
A importΓ’ncia da leitura
A importΓ’ncia da leituraA importΓ’ncia da leitura
A importΓ’ncia da leitura
Β 
HOJITA EVANGELIO DOMINGO XX CICLO A SERIE
HOJITA EVANGELIO  DOMINGO XX CICLO A SERIEHOJITA EVANGELIO  DOMINGO XX CICLO A SERIE
HOJITA EVANGELIO DOMINGO XX CICLO A SERIE
Β 
RECETA DE COMIDA
RECETA  DE COMIDARECETA  DE COMIDA
RECETA DE COMIDA
Β 
Ab04603165169
Ab04603165169Ab04603165169
Ab04603165169
Β 
Uu no 7_th_2004 sumber daya air
Uu no 7_th_2004 sumber daya airUu no 7_th_2004 sumber daya air
Uu no 7_th_2004 sumber daya air
Β 
C046011620
C046011620C046011620
C046011620
Β 
Book OratΓ³rio Eco Clube
Book OratΓ³rio Eco ClubeBook OratΓ³rio Eco Clube
Book OratΓ³rio Eco Clube
Β 
7 ΰΈͺาฑัญ ΰΈ„ΰΈ“ΰΈ΄ΰΈ•
7 ΰΈͺาฑัญ ΰΈ„ΰΈ“ΰΈ΄ΰΈ•7 ΰΈͺาฑัญ ΰΈ„ΰΈ“ΰΈ΄ΰΈ•
7 ΰΈͺาฑัญ ΰΈ„ΰΈ“ΰΈ΄ΰΈ•
Β 
Portfolio 2010
Portfolio 2010Portfolio 2010
Portfolio 2010
Β 
ΞŒΟ„Ξ±Ξ½ ΞΏ ΠΡρσέας συνάντησΡ τον Ωρίωνα: ΞœΞ­ΟΞΏΟ‚ Ξ’Ξ„
ΞŒΟ„Ξ±Ξ½ ΞΏ ΠΡρσέας συνάντησΡ τον Ωρίωνα: ΞœΞ­ΟΞΏΟ‚ Ξ’Ξ„ ΞŒΟ„Ξ±Ξ½ ΞΏ ΠΡρσέας συνάντησΡ τον Ωρίωνα: ΞœΞ­ΟΞΏΟ‚ Ξ’Ξ„
ΞŒΟ„Ξ±Ξ½ ΞΏ ΠΡρσέας συνάντησΡ τον Ωρίωνα: ΞœΞ­ΟΞΏΟ‚ Ξ’Ξ„
Β 
Aa04602180190
Aa04602180190Aa04602180190
Aa04602180190
Β 
Catalogo
CatalogoCatalogo
Catalogo
Β 
Linux Y Sus Equivalencias
Linux Y Sus EquivalenciasLinux Y Sus Equivalencias
Linux Y Sus Equivalencias
Β 
Case de Sucesso Symnetics: Pfizer
Case de Sucesso Symnetics: PfizerCase de Sucesso Symnetics: Pfizer
Case de Sucesso Symnetics: Pfizer
Β 

Similar a Ac04606177183

A production - Inventory model with JIT setup cost incorporating inflation an...
A production - Inventory model with JIT setup cost incorporating inflation an...A production - Inventory model with JIT setup cost incorporating inflation an...
A production - Inventory model with JIT setup cost incorporating inflation an...IJMER
Β 
Line Balancing and facility optimization of Machine Shop with Work Study and ...
Line Balancing and facility optimization of Machine Shop with Work Study and ...Line Balancing and facility optimization of Machine Shop with Work Study and ...
Line Balancing and facility optimization of Machine Shop with Work Study and ...IRJET Journal
Β 
IRJET- Value Stream Mapping (VSM) – A Case Study in Manufacturing Facility
IRJET- Value Stream Mapping (VSM) – A Case Study in Manufacturing FacilityIRJET- Value Stream Mapping (VSM) – A Case Study in Manufacturing Facility
IRJET- Value Stream Mapping (VSM) – A Case Study in Manufacturing FacilityIRJET Journal
Β 
An EPQ Model With Unit Production Cost And Set-Up Cost As Functions Of Produc...
An EPQ Model With Unit Production Cost And Set-Up Cost As Functions Of Produc...An EPQ Model With Unit Production Cost And Set-Up Cost As Functions Of Produc...
An EPQ Model With Unit Production Cost And Set-Up Cost As Functions Of Produc...Stephen Faucher
Β 
Optimization of Imperfect Manufacturing Systems: Interference Analysis Betwee...
Optimization of Imperfect Manufacturing Systems: Interference Analysis Betwee...Optimization of Imperfect Manufacturing Systems: Interference Analysis Betwee...
Optimization of Imperfect Manufacturing Systems: Interference Analysis Betwee...GuyRichardKibouka
Β 
IRJET- A Case Study for Overall Equipment Effectiveness Improved in Manufactu...
IRJET- A Case Study for Overall Equipment Effectiveness Improved in Manufactu...IRJET- A Case Study for Overall Equipment Effectiveness Improved in Manufactu...
IRJET- A Case Study for Overall Equipment Effectiveness Improved in Manufactu...IRJET Journal
Β 
A MULTIPLE PRODUCTION SETUPS INVENTORY MODEL FOR IMPERFECT ITEMS CONSIDERING ...
A MULTIPLE PRODUCTION SETUPS INVENTORY MODEL FOR IMPERFECT ITEMS CONSIDERING ...A MULTIPLE PRODUCTION SETUPS INVENTORY MODEL FOR IMPERFECT ITEMS CONSIDERING ...
A MULTIPLE PRODUCTION SETUPS INVENTORY MODEL FOR IMPERFECT ITEMS CONSIDERING ...orajjournal
Β 
Manufacturing Lead Time Reduction in Monoblock (SWJ) Pump Industry
Manufacturing Lead Time Reduction in Monoblock (SWJ) Pump IndustryManufacturing Lead Time Reduction in Monoblock (SWJ) Pump Industry
Manufacturing Lead Time Reduction in Monoblock (SWJ) Pump IndustryIRJET Journal
Β 
Manufacturing Lead Time Reduction in Monoblock (SWJ) Pump Industry [irjet-v4 ...
Manufacturing Lead Time Reduction in Monoblock (SWJ) Pump Industry [irjet-v4 ...Manufacturing Lead Time Reduction in Monoblock (SWJ) Pump Industry [irjet-v4 ...
Manufacturing Lead Time Reduction in Monoblock (SWJ) Pump Industry [irjet-v4 ...PERUMALSAMY M
Β 
Estimation of Manufacturing Costs in the Early Stages of Product Development ...
Estimation of Manufacturing Costs in the Early Stages of Product Development ...Estimation of Manufacturing Costs in the Early Stages of Product Development ...
Estimation of Manufacturing Costs in the Early Stages of Product Development ...IJMER
Β 
A Review On Application Of Defect Analysis And Optimization For Quality Contr...
A Review On Application Of Defect Analysis And Optimization For Quality Contr...A Review On Application Of Defect Analysis And Optimization For Quality Contr...
A Review On Application Of Defect Analysis And Optimization For Quality Contr...IRJET Journal
Β 
Solving Multi-level, Multi-product and Multi-period Lot Sizing and Scheduling...
Solving Multi-level, Multi-product and Multi-period Lot Sizing and Scheduling...Solving Multi-level, Multi-product and Multi-period Lot Sizing and Scheduling...
Solving Multi-level, Multi-product and Multi-period Lot Sizing and Scheduling...Editor IJCATR
Β 
The Performance Analysis of a Fettling Shop Using Simulation
The Performance Analysis of a Fettling Shop Using SimulationThe Performance Analysis of a Fettling Shop Using Simulation
The Performance Analysis of a Fettling Shop Using SimulationIOSR Journals
Β 
IRJET- Value Engineering: Better Way of Implementing Conventional Methods
IRJET- Value Engineering: Better Way of Implementing Conventional MethodsIRJET- Value Engineering: Better Way of Implementing Conventional Methods
IRJET- Value Engineering: Better Way of Implementing Conventional MethodsIRJET Journal
Β 
Case Study: Mass Production in Small Scale Industries
Case Study: Mass Production in Small Scale IndustriesCase Study: Mass Production in Small Scale Industries
Case Study: Mass Production in Small Scale Industriespaperpublications3
Β 
Case Study: Mass Production in Small Scale Industries
Case Study: Mass Production in Small Scale IndustriesCase Study: Mass Production in Small Scale Industries
Case Study: Mass Production in Small Scale Industriespaperpublications3
Β 
Reducing Manufacturing Cost through Value Stream Mapping
Reducing Manufacturing Cost through Value Stream MappingReducing Manufacturing Cost through Value Stream Mapping
Reducing Manufacturing Cost through Value Stream MappingEditor IJCATR
Β 
Case Study: Mass Production in Small Scale Industries
Case Study: Mass Production in Small Scale IndustriesCase Study: Mass Production in Small Scale Industries
Case Study: Mass Production in Small Scale Industriespaperpublications3
Β 

Similar a Ac04606177183 (20)

A production - Inventory model with JIT setup cost incorporating inflation an...
A production - Inventory model with JIT setup cost incorporating inflation an...A production - Inventory model with JIT setup cost incorporating inflation an...
A production - Inventory model with JIT setup cost incorporating inflation an...
Β 
I012536474
I012536474I012536474
I012536474
Β 
Line Balancing and facility optimization of Machine Shop with Work Study and ...
Line Balancing and facility optimization of Machine Shop with Work Study and ...Line Balancing and facility optimization of Machine Shop with Work Study and ...
Line Balancing and facility optimization of Machine Shop with Work Study and ...
Β 
IRJET- Value Stream Mapping (VSM) – A Case Study in Manufacturing Facility
IRJET- Value Stream Mapping (VSM) – A Case Study in Manufacturing FacilityIRJET- Value Stream Mapping (VSM) – A Case Study in Manufacturing Facility
IRJET- Value Stream Mapping (VSM) – A Case Study in Manufacturing Facility
Β 
An EPQ Model With Unit Production Cost And Set-Up Cost As Functions Of Produc...
An EPQ Model With Unit Production Cost And Set-Up Cost As Functions Of Produc...An EPQ Model With Unit Production Cost And Set-Up Cost As Functions Of Produc...
An EPQ Model With Unit Production Cost And Set-Up Cost As Functions Of Produc...
Β 
Optimization of Imperfect Manufacturing Systems: Interference Analysis Betwee...
Optimization of Imperfect Manufacturing Systems: Interference Analysis Betwee...Optimization of Imperfect Manufacturing Systems: Interference Analysis Betwee...
Optimization of Imperfect Manufacturing Systems: Interference Analysis Betwee...
Β 
IRJET- A Case Study for Overall Equipment Effectiveness Improved in Manufactu...
IRJET- A Case Study for Overall Equipment Effectiveness Improved in Manufactu...IRJET- A Case Study for Overall Equipment Effectiveness Improved in Manufactu...
IRJET- A Case Study for Overall Equipment Effectiveness Improved in Manufactu...
Β 
A MULTIPLE PRODUCTION SETUPS INVENTORY MODEL FOR IMPERFECT ITEMS CONSIDERING ...
A MULTIPLE PRODUCTION SETUPS INVENTORY MODEL FOR IMPERFECT ITEMS CONSIDERING ...A MULTIPLE PRODUCTION SETUPS INVENTORY MODEL FOR IMPERFECT ITEMS CONSIDERING ...
A MULTIPLE PRODUCTION SETUPS INVENTORY MODEL FOR IMPERFECT ITEMS CONSIDERING ...
Β 
Manufacturing Lead Time Reduction in Monoblock (SWJ) Pump Industry
Manufacturing Lead Time Reduction in Monoblock (SWJ) Pump IndustryManufacturing Lead Time Reduction in Monoblock (SWJ) Pump Industry
Manufacturing Lead Time Reduction in Monoblock (SWJ) Pump Industry
Β 
Manufacturing Lead Time Reduction in Monoblock (SWJ) Pump Industry [irjet-v4 ...
Manufacturing Lead Time Reduction in Monoblock (SWJ) Pump Industry [irjet-v4 ...Manufacturing Lead Time Reduction in Monoblock (SWJ) Pump Industry [irjet-v4 ...
Manufacturing Lead Time Reduction in Monoblock (SWJ) Pump Industry [irjet-v4 ...
Β 
Estimation of Manufacturing Costs in the Early Stages of Product Development ...
Estimation of Manufacturing Costs in the Early Stages of Product Development ...Estimation of Manufacturing Costs in the Early Stages of Product Development ...
Estimation of Manufacturing Costs in the Early Stages of Product Development ...
Β 
A Review On Application Of Defect Analysis And Optimization For Quality Contr...
A Review On Application Of Defect Analysis And Optimization For Quality Contr...A Review On Application Of Defect Analysis And Optimization For Quality Contr...
A Review On Application Of Defect Analysis And Optimization For Quality Contr...
Β 
Solving Multi-level, Multi-product and Multi-period Lot Sizing and Scheduling...
Solving Multi-level, Multi-product and Multi-period Lot Sizing and Scheduling...Solving Multi-level, Multi-product and Multi-period Lot Sizing and Scheduling...
Solving Multi-level, Multi-product and Multi-period Lot Sizing and Scheduling...
Β 
The Performance Analysis of a Fettling Shop Using Simulation
The Performance Analysis of a Fettling Shop Using SimulationThe Performance Analysis of a Fettling Shop Using Simulation
The Performance Analysis of a Fettling Shop Using Simulation
Β 
IRJET- Value Engineering: Better Way of Implementing Conventional Methods
IRJET- Value Engineering: Better Way of Implementing Conventional MethodsIRJET- Value Engineering: Better Way of Implementing Conventional Methods
IRJET- Value Engineering: Better Way of Implementing Conventional Methods
Β 
O0123190100
O0123190100O0123190100
O0123190100
Β 
Case Study: Mass Production in Small Scale Industries
Case Study: Mass Production in Small Scale IndustriesCase Study: Mass Production in Small Scale Industries
Case Study: Mass Production in Small Scale Industries
Β 
Case Study: Mass Production in Small Scale Industries
Case Study: Mass Production in Small Scale IndustriesCase Study: Mass Production in Small Scale Industries
Case Study: Mass Production in Small Scale Industries
Β 
Reducing Manufacturing Cost through Value Stream Mapping
Reducing Manufacturing Cost through Value Stream MappingReducing Manufacturing Cost through Value Stream Mapping
Reducing Manufacturing Cost through Value Stream Mapping
Β 
Case Study: Mass Production in Small Scale Industries
Case Study: Mass Production in Small Scale IndustriesCase Study: Mass Production in Small Scale Industries
Case Study: Mass Production in Small Scale Industries
Β 

Último

Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...
Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...
Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...Will Schroeder
Β 
COMPUTER 10 Lesson 8 - Building a Website
COMPUTER 10 Lesson 8 - Building a WebsiteCOMPUTER 10 Lesson 8 - Building a Website
COMPUTER 10 Lesson 8 - Building a Websitedgelyza
Β 
IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019
IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019
IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019IES VE
Β 
KubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCost
KubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCostKubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCost
KubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCostMatt Ray
Β 
UiPath Studio Web workshop series - Day 6
UiPath Studio Web workshop series - Day 6UiPath Studio Web workshop series - Day 6
UiPath Studio Web workshop series - Day 6DianaGray10
Β 
UiPath Platform: The Backend Engine Powering Your Automation - Session 1
UiPath Platform: The Backend Engine Powering Your Automation - Session 1UiPath Platform: The Backend Engine Powering Your Automation - Session 1
UiPath Platform: The Backend Engine Powering Your Automation - Session 1DianaGray10
Β 
NIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopNIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopBachir Benyammi
Β 
UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...
UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...
UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...UbiTrack UK
Β 
Salesforce Miami User Group Event - 1st Quarter 2024
Salesforce Miami User Group Event - 1st Quarter 2024Salesforce Miami User Group Event - 1st Quarter 2024
Salesforce Miami User Group Event - 1st Quarter 2024SkyPlanner
Β 
20230202 - Introduction to tis-py
20230202 - Introduction to tis-py20230202 - Introduction to tis-py
20230202 - Introduction to tis-pyJamie (Taka) Wang
Β 
UiPath Studio Web workshop series - Day 8
UiPath Studio Web workshop series - Day 8UiPath Studio Web workshop series - Day 8
UiPath Studio Web workshop series - Day 8DianaGray10
Β 
The Data Metaverse: Unpacking the Roles, Use Cases, and Tech Trends in Data a...
The Data Metaverse: Unpacking the Roles, Use Cases, and Tech Trends in Data a...The Data Metaverse: Unpacking the Roles, Use Cases, and Tech Trends in Data a...
The Data Metaverse: Unpacking the Roles, Use Cases, and Tech Trends in Data a...Aggregage
Β 
activity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdf
activity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdf
activity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdfJamie (Taka) Wang
Β 
UiPath Community: AI for UiPath Automation Developers
UiPath Community: AI for UiPath Automation DevelopersUiPath Community: AI for UiPath Automation Developers
UiPath Community: AI for UiPath Automation DevelopersUiPathCommunity
Β 
Nanopower In Semiconductor Industry.pdf
Nanopower  In Semiconductor Industry.pdfNanopower  In Semiconductor Industry.pdf
Nanopower In Semiconductor Industry.pdfPedro Manuel
Β 
Building AI-Driven Apps Using Semantic Kernel.pptx
Building AI-Driven Apps Using Semantic Kernel.pptxBuilding AI-Driven Apps Using Semantic Kernel.pptx
Building AI-Driven Apps Using Semantic Kernel.pptxUdaiappa Ramachandran
Β 
Building Your Own AI Instance (TBLC AI )
Building Your Own AI Instance (TBLC AI )Building Your Own AI Instance (TBLC AI )
Building Your Own AI Instance (TBLC AI )Brian Pichman
Β 
Igniting Next Level Productivity with AI-Infused Data Integration Workflows
Igniting Next Level Productivity with AI-Infused Data Integration WorkflowsIgniting Next Level Productivity with AI-Infused Data Integration Workflows
Igniting Next Level Productivity with AI-Infused Data Integration WorkflowsSafe Software
Β 
VoIP Service and Marketing using Odoo and Asterisk PBX
VoIP Service and Marketing using Odoo and Asterisk PBXVoIP Service and Marketing using Odoo and Asterisk PBX
VoIP Service and Marketing using Odoo and Asterisk PBXTarek Kalaji
Β 
AI Fame Rush Review – Virtual Influencer Creation In Just Minutes
AI Fame Rush Review – Virtual Influencer Creation In Just MinutesAI Fame Rush Review – Virtual Influencer Creation In Just Minutes
AI Fame Rush Review – Virtual Influencer Creation In Just MinutesMd Hossain Ali
Β 

Último (20)

Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...
Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...
Apres-Cyber - The Data Dilemma: Bridging Offensive Operations and Machine Lea...
Β 
COMPUTER 10 Lesson 8 - Building a Website
COMPUTER 10 Lesson 8 - Building a WebsiteCOMPUTER 10 Lesson 8 - Building a Website
COMPUTER 10 Lesson 8 - Building a Website
Β 
IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019
IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019
IESVE Software for Florida Code Compliance Using ASHRAE 90.1-2019
Β 
KubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCost
KubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCostKubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCost
KubeConEU24-Monitoring Kubernetes and Cloud Spend with OpenCost
Β 
UiPath Studio Web workshop series - Day 6
UiPath Studio Web workshop series - Day 6UiPath Studio Web workshop series - Day 6
UiPath Studio Web workshop series - Day 6
Β 
UiPath Platform: The Backend Engine Powering Your Automation - Session 1
UiPath Platform: The Backend Engine Powering Your Automation - Session 1UiPath Platform: The Backend Engine Powering Your Automation - Session 1
UiPath Platform: The Backend Engine Powering Your Automation - Session 1
Β 
NIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 WorkshopNIST Cybersecurity Framework (CSF) 2.0 Workshop
NIST Cybersecurity Framework (CSF) 2.0 Workshop
Β 
UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...
UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...
UWB Technology for Enhanced Indoor and Outdoor Positioning in Physiological M...
Β 
Salesforce Miami User Group Event - 1st Quarter 2024
Salesforce Miami User Group Event - 1st Quarter 2024Salesforce Miami User Group Event - 1st Quarter 2024
Salesforce Miami User Group Event - 1st Quarter 2024
Β 
20230202 - Introduction to tis-py
20230202 - Introduction to tis-py20230202 - Introduction to tis-py
20230202 - Introduction to tis-py
Β 
UiPath Studio Web workshop series - Day 8
UiPath Studio Web workshop series - Day 8UiPath Studio Web workshop series - Day 8
UiPath Studio Web workshop series - Day 8
Β 
The Data Metaverse: Unpacking the Roles, Use Cases, and Tech Trends in Data a...
The Data Metaverse: Unpacking the Roles, Use Cases, and Tech Trends in Data a...The Data Metaverse: Unpacking the Roles, Use Cases, and Tech Trends in Data a...
The Data Metaverse: Unpacking the Roles, Use Cases, and Tech Trends in Data a...
Β 
activity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdf
activity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdf
activity_diagram_combine_v4_20190827.pdfactivity_diagram_combine_v4_20190827.pdf
Β 
UiPath Community: AI for UiPath Automation Developers
UiPath Community: AI for UiPath Automation DevelopersUiPath Community: AI for UiPath Automation Developers
UiPath Community: AI for UiPath Automation Developers
Β 
Nanopower In Semiconductor Industry.pdf
Nanopower  In Semiconductor Industry.pdfNanopower  In Semiconductor Industry.pdf
Nanopower In Semiconductor Industry.pdf
Β 
Building AI-Driven Apps Using Semantic Kernel.pptx
Building AI-Driven Apps Using Semantic Kernel.pptxBuilding AI-Driven Apps Using Semantic Kernel.pptx
Building AI-Driven Apps Using Semantic Kernel.pptx
Β 
Building Your Own AI Instance (TBLC AI )
Building Your Own AI Instance (TBLC AI )Building Your Own AI Instance (TBLC AI )
Building Your Own AI Instance (TBLC AI )
Β 
Igniting Next Level Productivity with AI-Infused Data Integration Workflows
Igniting Next Level Productivity with AI-Infused Data Integration WorkflowsIgniting Next Level Productivity with AI-Infused Data Integration Workflows
Igniting Next Level Productivity with AI-Infused Data Integration Workflows
Β 
VoIP Service and Marketing using Odoo and Asterisk PBX
VoIP Service and Marketing using Odoo and Asterisk PBXVoIP Service and Marketing using Odoo and Asterisk PBX
VoIP Service and Marketing using Odoo and Asterisk PBX
Β 
AI Fame Rush Review – Virtual Influencer Creation In Just Minutes
AI Fame Rush Review – Virtual Influencer Creation In Just MinutesAI Fame Rush Review – Virtual Influencer Creation In Just Minutes
AI Fame Rush Review – Virtual Influencer Creation In Just Minutes
Β 

Ac04606177183

  • 1. M. Ullah&C.W. Kang Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 6( Version 6), June 2014, pp.177-183 Quality Costs (IRR) Impact on Lot Size Considering Work in Process Inventory Misbah Ullaha , Chang W. Kangb* a-b Department of Industrial and Management Engineering, ERICA Campus, Hanyang University, Ansan, Republic of Korea Abstract Economic order quantity model and production quantity model assume that production processes are error free. However, variations exist in processes which result in imperfection particularly in high machining environments. Processes variations result in nonconformities that increase quality costs in the form of rework, rejects and quality control techniques implementations to ensure quality product delivery. This paper is an attempt towards development of inventory model which incorporate inspection, rework, and rejection (IRR) quality costs in optimum lot size calculation focusing work in process inventory. Mathematical model is derived for optimum lot size based on minimum average cost function using analytical approach. This new developed model (GTOQIRR) assume an imperfect production environment. Numerical examples are used to visualize the significant effect of quality cost in the proposed model in comparison to the previously developed models. The proposed model is highly recommendable for quality based high machining manufacturing environments considering work in process inventories. Keywords: WIP inventory; EOQ model; quality cost; lot size; GTOQ model. I. Introduction During past few decades, significant attention has been given to the area of inventory management because of its prime importance for most manufacturing organizations. Organizations use economic order quantity (EOQ) and production order quantity (POQ) models since 1913 to calculate optimum inventory lot size considering holding and setup costs. Researchers extended these models to more generic and practical situations of everyday life. However these two basic models are based on an unrealistic assumption that perfect products are produced every time. A number of factors affect manufacturing processes which include machine failure, changes in raw material supplier, tool wear and tear, outside conditions etc. especially where machining time is large relatively. Therefore the phenomena of rework, reject and inspection is common in industries like machine tool industries, aerospace industries and glass industries etc. High machining environments, rework, rejects and inspection compel to introduce quality costs incurred during these process and its ultimate impact on inventory lot size. Rosenblatt and Lee [1] developed EOQ model for imperfect manufacturing environments with conclusion that lot size reduces as imperfection is increased. Later Sarker et al. [2] extended EOQ model to a multistage manufacturing environment with imperfection in processes. The developed model assumed that rework operation is performed at the end of each cycle and at the end of N cycle. Further extension of such models can be realized in research articles published in the last decade [3-12].They realized that the impact of imperfection on inventory lot size in addition to the importance of inspection in real world manufacturing environments. Combined effect of inspection with imperfections was emphasized in few papers such as ([13] and [18]). Furthermore, ([14], [15], [16], [17] and [19]) developed models for optimum lot size by considering other important aspects of the real world manufacturing systems. Researchers mainly focused raw materials and finish goods inventories in calculation of optimum lot size calculation. However, the third type of inventory, work in process, in combination with quality cost has been relatively ignored in previous literature. Today, industries put their efforts to reduce their inventories cost either in the form of raw materials, finish goods or work in process. Boucher [20] introduced the concept of work in process inventory in group technology based manufacturing environment for calculation of optimum lot size. The model was named as group technology order quantity (GTOQ) model. Barzoki et al. [21] extended GTOQ model for imperfect production environments under the title of group technology order quantity model with rework (GTOQR). Their model considered 100% qualification of rework products. Numerical examples were used for model comparison against previous models. These models are recommended for group technology work environment where machining time and lot size are relatively large. However, increase RESEARCH ARTICLE OPEN ACCESS www.ijera.com 177|P a g e
  • 2. M. Ullah&C.W. Kang Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 6( Version 6), June 2014, pp.177-183 machining time and higher demand rate significantly contribute towards quality costs in processes. Higher machining rate and large machining time results in increased quality costs due to high tool wear and tear, process failure etc. Increased quality costs are in the form of rework, rejects and inspection that affect total cost for optimum lot size calculation. This paper is an attempt towards calculation of quality costs impact on the optimal lot size considering imperfection in processes. This paper incorporate quality costs due to inspection, rework and rejection (IRR) in total cost calculation. Inclusion of quality costs in inventory models and its impact on lot size is lacking in previous papers incorporating work in process inventories. The remaining sections are listed as follow; Section 2 explains problem statement with assumptions and notations. In section 3, the problem is modeled based on modeling for relevant costs associated with this problem. Section 4 comprises comparison of models based on numerical computations. Conclusion is presented in the last section. II. Problem statement We consider a single discrete manufacturing setup. A machining station is followed by an inspection process with three buffer stations in the manufacturing workshop. Buffer stations are used for work in process inventories i.e. raw material (unprocessed items, waiting for operations), good quality products and poor quality products (rejected items). A lot of material (Q) arrives in the workshop processed on machining station in each cycle. Lot (Q) is released for inspection to declare products as either good quality products or poor quality products. As the process is imperfect so three kinds of products are produced during the whole cycle time. The process produces, 𝑄𝑄𝑄𝑄10, percent of parts as poor quality products, 𝑄𝑄𝑄𝑄1π‘Ÿπ‘Ÿ, percent of parts as reworked and the remaining 𝑄𝑄�1 βˆ’ (𝑝𝑝10 + 𝑝𝑝1π‘Ÿπ‘Ÿ)οΏ½ as good quality products in the first phase of the lot Q as shown in Figure 1. During the next phase 2, the rework items 𝑄𝑄𝑄𝑄1π‘Ÿπ‘Ÿ= Q(𝑝𝑝20 + 𝑝𝑝21) are reprocessed and re-inspected. 𝑄𝑄𝑝𝑝21percent of products are good quality products and, 𝑄𝑄𝑄𝑄20, percent of products are poor quality in the phase 2 as shown in Figure 2. We assume that no rework is needed after that. At the end of the cycle, the following conclusion is drawn regarding the lot size Q. Good quality products produced,𝑄𝑄𝑝𝑝1 = 𝑄𝑄�1 βˆ’ (𝑝𝑝10 + 𝑝𝑝20)οΏ½ Poor quality products produced,𝑄𝑄𝑝𝑝0 = 𝑄𝑄(𝑝𝑝10 + 𝑝𝑝20) Products that are rework able after phase 1, 𝑄𝑄𝑃𝑃1π‘Ÿπ‘Ÿ = 𝑄𝑄(𝑝𝑝20 + 𝑝𝑝21) Therefore, our objective is to develop an economic order quantity model for an imperfect manufacturing environment taking quality costs into consideration with work in process inventory. Figure 1. Processing of lot size Q during phase 1 Figure 2. Processing of rework products in phase 2. 2.1. Notations 𝑑𝑑 customer demand 𝑄𝑄 lot size processed per unit cycle 𝑀𝑀𝑐𝑐 raw material cost per unit 𝑃𝑃𝑐𝑐 purchasing cost per unit of time 𝑆𝑆𝑐𝑐 setup cost per unit of time ($/unit of time) Process Q i Q(1-(p10+p1r)) Rejected: Q(p10) Rework: Qp1r Inspection Process QP1r units Q(p21) Rejected:Q(p20) Rework: 0 Inspection www.ijera.com 178|P a g e
  • 3. M. Ullah&C.W. Kang Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 6( Version 6), June 2014, pp.177-183 𝑄𝑄𝑐𝑐 cost of quality per unit of time ($/unit of time) 𝐢𝐢𝑖𝑖 inspection cost 𝑅𝑅𝑅𝑅𝑐𝑐 rework cost 𝑅𝑅𝑅𝑅𝑐𝑐 rejection cost π‘Šπ‘Šπ‘Šπ‘Šπ‘Šπ‘Šπ‘π‘ work in process holding cost 𝐼𝐼 𝐼𝐼𝑐𝑐 inventory holding cost 𝑇𝑇𝑐𝑐 total cost 𝑑𝑑𝑠𝑠 setup time 𝑑𝑑 π‘šπ‘š1 machining time in phase 1. (unit time per unit product) 𝑑𝑑 π‘šπ‘šπ‘šπ‘š machining time per unit for rework products 𝑝𝑝10 proportion of poor quality products in phase 1 𝑝𝑝11 proportion of good quality products in phase 1 𝑝𝑝21 proportion of good quality products in phase 2 𝑝𝑝20 proportion of poor quality products in phase 2 𝑝𝑝1π‘Ÿπ‘Ÿ proportion of rework able products in phase 1 𝑝𝑝0 proportion of poor quality products at the end of cycle 𝑝𝑝1 proportion of good quality products at the end of cycle 𝑑𝑑𝑐𝑐 cycle time 𝑑𝑑𝑝𝑝 total processing time 𝑑𝑑 average manufacturing time for each product item 𝐺𝐺 average storage inventory π‘Šπ‘Š average monetary value of the WIP inventory ($) 𝑖𝑖 inventory holding cost per unit of time ($/unit of time) 𝑐𝑐 average unit value of each product (unit of money ($) per unit) π‘˜π‘˜ rate charged per unit of cell production time including all overheads, moving cost, loading/unloading cost etc. (unit of money ($) per unit of time) 2.2 Assumptions  Shortages are not allowed  Demand is known and pre-determined.  Inspection is performed of all lot and rework able products.  Poor quality products are produced during the rework operation.  Rework operation is done one time only.  No stoppage is allowed during the manufacturing of one lot. All parameters including demand and production rate, setup times etc. are constants and deterministic. III. Modeling Modeling of the problem is based on cycle time calculation in terms of demand rate and lot size Q in addition to the imperfection in the production processes. As the objective of almost every manufacturing setup is to meet customer demand rate (d) therefore lot size (Q) is defined keeping customer demand in mind. Processes undergoing imperfection results in poor quality products at the end of each cycle. Therefore we can have following relation for the above mentioned scenario 𝑄𝑄(1 βˆ’ 𝑝𝑝0) = 𝑑𝑑𝑑𝑑𝑐𝑐 𝑑𝑑𝑐𝑐= 𝑄𝑄(1βˆ’π‘π‘0) 𝑑𝑑 (1) The total processing time ( 𝑑𝑑𝑝𝑝 ) of a product having lot size Q comprises of setup time, manufacturing time, inspection time per unit product (𝑑𝑑𝑖𝑖), rework time and re-inspection time. 𝑑𝑑𝑝𝑝= total processing time 𝑑𝑑𝑝𝑝= 𝑑𝑑𝑠𝑠 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 + 𝑑𝑑𝑖𝑖 𝑄𝑄 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ + 𝑑𝑑𝑖𝑖 𝑄𝑄𝑝𝑝1π‘Ÿπ‘Ÿ (2) Therefore average processing time is given by 𝑑𝑑 = 𝑑𝑑𝑝𝑝 𝑄𝑄 𝑑𝑑 = 𝑑𝑑𝑠𝑠 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 + 𝑑𝑑𝑖𝑖 𝑄𝑄 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ + 𝑑𝑑𝑖𝑖 𝑄𝑄𝑝𝑝1π‘Ÿπ‘Ÿ 𝑄𝑄 If π‘˜π‘˜ is the average cost of production per unit of time then the average cost added to each product unit during its manufacturing is 𝑣𝑣 given by 𝑣𝑣 = π‘˜π‘˜π‘‘π‘‘ = π‘˜π‘˜ οΏ½ 𝑑𝑑𝑠𝑠+𝑄𝑄𝑑𝑑 π‘šπ‘š1 +𝑑𝑑𝑖𝑖 𝑄𝑄+𝑄𝑄𝑑𝑑 π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ+𝑑𝑑𝑖𝑖 𝑄𝑄𝑝𝑝1π‘Ÿπ‘Ÿ 𝑄𝑄 οΏ½ We assume that 𝑀𝑀𝑐𝑐 be the material cost per unit of product, then the average total cost added to each product unit is given by 𝑐𝑐= 𝑀𝑀𝑐𝑐 + π‘˜π‘˜ οΏ½ 𝑑𝑑𝑠𝑠+𝑄𝑄𝑑𝑑 π‘šπ‘š1 +𝑑𝑑𝑖𝑖 𝑄𝑄+𝑄𝑄𝑑𝑑 π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ+𝑑𝑑𝑖𝑖 𝑄𝑄𝑝𝑝1π‘Ÿπ‘Ÿ 𝑄𝑄 οΏ½ (3) 3.1 Cost calculation As we know that most of the inventory models are based on the cost minimization, therefore we are interested to calculate associated costs with this model. Quality costs, inventory holding cost, setup cost and material purchasing cost are the main components of total cost calculation. All these costs are calculated keeping imperfection in processes under consideration. 3.1.1 Quality cost (IRR) Quality costs considered in this paper are mainly associated with imperfection in processes. These imperfection results in products failure either in the form of rework or rejected products. Similarly, quality control techniques including inspection exists within the manufacturing setup which also results in quality cost. Other kinds of quality cost related to appraisal and prevention has been neglected. Therefore, quality costs considered in this model are (a) Quality costs due to products inspection (𝐢𝐢𝑖𝑖). (b) Quality costs due to rework (πΆπΆπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿ), (c) Quality costs due to rejection (πΆπΆπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿ). Therefore quality cost (πΆπΆπ‘žπ‘ž) is given by 𝑄𝑄𝑐𝑐 = 𝐢𝐢𝐢𝐢+ 𝑅𝑅𝑅𝑅𝑐𝑐 + 𝑅𝑅𝑅𝑅𝑐𝑐 (4) These costs are modeled as follow a) Quality costs due to inspection www.ijera.com 179|P a g e
  • 4. M. Ullah&C.W. Kang Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 6( Version 6), June 2014, pp.177-183 Inspection is performed for all manufactured and reworked products. 100% inspection of lot is carried out when it is manufactured. Similarly reworked products are also passed through the inspection process. The inspection cost 𝐢𝐢𝐢𝐢 per unit of cycle time is given by 𝐢𝐢𝐢𝐢 = 𝐼𝐼𝑐𝑐 𝑄𝑄 𝑑𝑑𝑐𝑐 + (𝑝𝑝1π‘Ÿπ‘Ÿ)𝑄𝑄𝐼𝐼𝑐𝑐 𝑑𝑑𝑐𝑐 𝐢𝐢𝐢𝐢 = 𝐼𝐼𝑐𝑐 𝑑𝑑 (1 βˆ’ 𝑝𝑝0) + 𝐼𝐼𝑐𝑐 𝑑𝑑(𝑝𝑝1π‘Ÿπ‘Ÿ) (1 βˆ’ 𝑝𝑝0) 𝐢𝐢𝐢𝐢 = οΏ½ 𝐼𝐼𝑐𝑐 𝑑𝑑 1βˆ’π‘π‘0 οΏ½ (1 + 𝑝𝑝1π‘Ÿπ‘Ÿ) (5) b) Quality costs due to rework Let 𝑝𝑝1π‘Ÿπ‘Ÿ is the percentage of rework product in a lot size Q in a complete cycle and c be the value added to the products in the manufacturing process. Then the quality cost per unit time for rework items is given by 𝑅𝑅𝑅𝑅𝑐𝑐 = οΏ½ 𝑛𝑛𝑝𝑝1π‘Ÿπ‘Ÿ 𝑄𝑄 𝑑𝑑𝑐𝑐 οΏ½ 𝑐𝑐 (6) where n is the no. of times rework operation is performed. c) Quality costs due to rejection Let 𝑝𝑝0 is the percentage of rejected product in a lot size Q in a complete cycle and 𝑐𝑐 be the value added to the products in the manufacturing process. Then the quality cost per unit time for rejected products is given by 𝑅𝑅𝑅𝑅𝑐𝑐= οΏ½ 𝑝𝑝0 𝑄𝑄 𝑑𝑑𝑐𝑐 οΏ½ 𝑐𝑐 (7) Therefore, Equation (4) becomes, 𝑄𝑄𝑐𝑐= οΏ½ 𝐼𝐼𝑐𝑐 𝑑𝑑 1βˆ’π‘π‘0 οΏ½ (1 + 𝑝𝑝1π‘Ÿπ‘Ÿ) + οΏ½ 𝑛𝑛𝑝𝑝1π‘Ÿπ‘Ÿ 𝑄𝑄 𝑑𝑑𝑐𝑐 οΏ½ 𝑐𝑐 + οΏ½ 𝑝𝑝0 𝑄𝑄 𝑑𝑑𝑐𝑐 οΏ½ 𝑐𝑐 𝑄𝑄𝑐𝑐= οΏ½ 𝐼𝐼𝑐𝑐 𝑑𝑑 1βˆ’π‘ƒπ‘ƒπ‘π‘ οΏ½ (1 + 𝑝𝑝1π‘Ÿπ‘Ÿ) + οΏ½ 𝑐𝑐𝑐𝑐 1βˆ’π‘π‘0 οΏ½ (𝑛𝑛𝑛𝑛1π‘Ÿπ‘Ÿ + 𝑝𝑝0) (8) 3.1.2 Inventory holding cost Silver et al. [22] introduced following relationship for calculation of inventory holding cost per unit of time. 𝐼𝐼 𝐼𝐼𝑐𝑐= 𝑖𝑖𝑖𝑖𝑖𝑖, (9) Here 𝐺𝐺 = average inventory over each cycle by the cycle period. 𝑐𝑐 = average unit monetary value for each item. 𝑖𝑖 = carrying charge. The average holding inventory (𝐺𝐺) per unit cycle time is given as 𝐺𝐺 = 1 2 𝑄𝑄(1βˆ’π‘π‘0)𝑑𝑑𝑐𝑐 𝑑𝑑𝑐𝑐 = 1 2 𝑄𝑄(1 βˆ’ 𝑝𝑝0) (10) Given Equations (3) and (10), the inventory holding cost per unit of time will be 𝐼𝐼 𝐼𝐼𝑐𝑐= 1 2 𝑖𝑖 ��𝑀𝑀𝑐𝑐 + π‘˜π‘˜ οΏ½ 𝑑𝑑𝑠𝑠+𝑄𝑄𝑑𝑑 π‘šπ‘š1 +𝑑𝑑𝑖𝑖 𝑄𝑄+𝑄𝑄𝑑𝑑 π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ+𝑑𝑑𝑖𝑖 𝑄𝑄𝑝𝑝1π‘Ÿπ‘Ÿ 𝑄𝑄 οΏ½οΏ½ 𝑄𝑄(1 βˆ’ 𝑝𝑝0)οΏ½ (11) 3.1.3 Work in process cost Based on the same concepts as [21] and [22], the work in process inventory holding cost is calculated by: π‘Šπ‘Šπ‘Šπ‘Šπ‘Šπ‘Šπ‘π‘ = π‘–π‘–π‘Šπ‘Š, (12) where 𝑖𝑖 = it is the carrying charge per unit of time π‘Šπ‘Š = average work in process inventory π‘Šπ‘Šcan be computed by summation of following types of inventories in the workshop. (a). The unprocessed product item waiting for operation on machine versus manufacturing time 𝑑𝑑𝑝𝑝 during each cycle. (b). Good quality products inventory over time. (c). Rejected parts inventory over time. Hence the average value of the total work in process inventory will be equal to π‘Šπ‘Š = οΏ½ 1/2𝑄𝑄𝑑𝑑 𝑝𝑝 𝑑𝑑𝑐𝑐 οΏ½ 𝑀𝑀𝑐𝑐 + οΏ½ 1 2 𝑄𝑄( 1βˆ’π‘π‘0)𝑑𝑑 𝑝𝑝 𝑑𝑑𝑐𝑐 οΏ½ 𝑐𝑐 + οΏ½ 1 2 𝑄𝑄𝑝𝑝0 𝑑𝑑 𝑝𝑝 𝑑𝑑𝑐𝑐 οΏ½ 𝑐𝑐 π‘Šπ‘Š = 1 2 𝑄𝑄𝑑𝑑 𝑝𝑝 𝑑𝑑𝑐𝑐 (𝑀𝑀𝑐𝑐 + (1 βˆ’ 𝑝𝑝0)𝑐𝑐 + 𝑝𝑝0 𝑐𝑐) π‘Šπ‘Š = 1 2 οΏ½ 𝑑𝑑 1βˆ’π‘π‘0 οΏ½ (𝑑𝑑𝑠𝑠 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 + 𝑑𝑑𝑖𝑖 𝑄𝑄 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ + 𝑑𝑑𝑖𝑖 𝑄𝑄𝑝𝑝1π‘Ÿπ‘Ÿ)(𝑀𝑀𝑐𝑐 + 𝑐𝑐) Given Equation (3), the average value of total work in process inventory will be π‘Šπ‘Š= 1 2 οΏ½ 𝑑𝑑 1βˆ’π‘π‘0 οΏ½ οΏ½2𝑀𝑀𝑐𝑐 + π‘˜π‘˜π‘‘π‘‘π‘ π‘  𝑄𝑄 + π‘˜π‘˜π‘‘π‘‘ π‘šπ‘š1 + π‘˜π‘˜π‘‘π‘‘ π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ+𝑑𝑑𝑖𝑖 π‘˜π‘˜ + 𝑑𝑑𝑖𝑖 π‘˜π‘˜π‘π‘1π‘Ÿπ‘Ÿ οΏ½ (𝑑𝑑𝑠𝑠 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 + 𝑑𝑑𝑖𝑖 𝑄𝑄 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ + 𝑑𝑑𝑖𝑖 𝑄𝑄𝑝𝑝1π‘Ÿπ‘Ÿ) (13) Therefore given Equations (12) and (13), the average carrying charge of the work in process inventory will be π‘Šπ‘Šπ‘Šπ‘Šπ‘Šπ‘Šπ‘π‘ = 1 2 οΏ½ 𝑑𝑑𝑑𝑑 1βˆ’π‘π‘0 οΏ½ οΏ½2𝑀𝑀𝑐𝑐 + π‘˜π‘˜π‘‘π‘‘π‘ π‘  𝑄𝑄 + π‘˜π‘˜π‘‘π‘‘ π‘šπ‘š1 + π‘˜π‘˜π‘‘π‘‘ π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ+𝑑𝑑𝑖𝑖 π‘˜π‘˜ + 𝑑𝑑𝑖𝑖 π‘˜π‘˜π‘π‘1π‘Ÿπ‘Ÿ οΏ½ (𝑑𝑑𝑠𝑠 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 + 𝑑𝑑𝑖𝑖 𝑄𝑄 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ + 𝑑𝑑𝑖𝑖 𝑄𝑄𝑝𝑝1π‘Ÿπ‘Ÿ) (14) 3.1.4Material Cost Material cost per unit of cycle time (𝑃𝑃𝑐𝑐) is given by the following equation 𝑃𝑃𝑐𝑐 = 𝑀𝑀𝑐𝑐 𝑄𝑄 𝑑𝑑𝑐𝑐⁄ We can write in terms of demand(𝑑𝑑) is 𝑃𝑃𝑐𝑐 = 𝑀𝑀𝑐𝑐 𝑑𝑑 (1βˆ’π‘π‘0) (15) www.ijera.com 180|P a g e
  • 5. M. Ullah&C.W. Kang Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 6( Version 6), June 2014, pp.177-183 3.1.5Setup cost Machines and inspection setup take place only once during whole lot manufacturing. We assume that no extra setup is required for rework operations. It can be modeled as follow 𝑆𝑆𝑐𝑐 = 𝐴𝐴 𝑑𝑑𝑐𝑐 = 𝐴𝐴𝐴𝐴 𝑄𝑄(1βˆ’π‘π‘0) (16) Where A is the setup cost. It is a product of set up time per cycle (𝑑𝑑𝑠𝑠) and the rate charged by the production cycle (π‘˜π‘˜). 3.2 Total cost per unit of time The total cost per unit of time are given by the following equation: 𝑇𝑇𝑐𝑐=𝑄𝑄𝑐𝑐 + 𝐼𝐼 𝐼𝐼𝑐𝑐+ π‘Šπ‘Šπ‘Šπ‘Šπ‘Šπ‘Šπ‘π‘ + 𝑃𝑃𝑐𝑐 + 𝑆𝑆𝑐𝑐 (17) Given equations (4), (5), (10), (13), and (16) in Eq. (17) 𝑇𝑇𝑐𝑐= οΏ½ 𝐼𝐼𝑐𝑐 𝑑𝑑 1βˆ’π‘ƒπ‘ƒπ‘π‘ οΏ½ (1 + 𝑝𝑝1π‘Ÿπ‘Ÿ) + οΏ½ 𝑐𝑐𝑐𝑐 1βˆ’π‘π‘0 οΏ½ (𝑛𝑛𝑛𝑛1π‘Ÿπ‘Ÿ + 𝑝𝑝0) + 1 2 𝑖𝑖 ��𝑀𝑀𝑐𝑐 + π‘˜π‘˜ οΏ½ 𝑑𝑑𝑠𝑠+𝑄𝑄𝑑𝑑 π‘šπ‘š1 +𝑑𝑑𝑖𝑖 𝑄𝑄+𝑄𝑄𝑑𝑑 π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ+𝑑𝑑𝑖𝑖 𝑄𝑄𝑝𝑝1π‘Ÿπ‘Ÿ 𝑄𝑄 οΏ½οΏ½ 𝑄𝑄(1 βˆ’ 𝑝𝑝0)οΏ½ + 1 2 οΏ½ 𝑑𝑑𝑑𝑑 1βˆ’π‘π‘0 οΏ½ οΏ½2𝑀𝑀𝑐𝑐 + π‘˜π‘˜π‘‘π‘‘π‘ π‘  𝑄𝑄 + π‘˜π‘˜π‘‘π‘‘ π‘šπ‘š1 + π‘˜π‘˜π‘‘π‘‘ π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ+𝑑𝑑𝑖𝑖 π‘˜π‘˜ + 𝑑𝑑𝑖𝑖 π‘˜π‘˜π‘π‘1π‘Ÿπ‘Ÿ οΏ½ (𝑑𝑑𝑠𝑠 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 + 𝑑𝑑𝑖𝑖 𝑄𝑄 + 𝑄𝑄𝑑𝑑 π‘šπ‘š1 𝑝𝑝1π‘Ÿπ‘Ÿ + 𝑑𝑑𝑖𝑖 𝑄𝑄𝑝𝑝1π‘Ÿπ‘Ÿ)+ 𝑀𝑀𝑐𝑐 𝑑𝑑 (1βˆ’π‘π‘0) + 𝐴𝐴𝐴𝐴 𝑄𝑄(1βˆ’π‘π‘0) (18) 3.3 Optimum lot size Equation (18) gives average cost function considering quality cost. Optimum lot size can be calculated based on average cost minimization. Therefore by taking 1st derivative of the function w.r.t. Q and equating to 0 i.e. 𝑑𝑑𝑇𝑇𝑐𝑐 𝑑𝑑𝑑𝑑 = 0 and sufficient condition that the equation must hold and satisfy for optimum lot size calculation is given by 𝑑𝑑2 𝑇𝑇𝑐𝑐 𝑑𝑑𝑄𝑄2 > 0 Hence the optimum lot size is given by GTOQIRR = (19) οΏ½ 2𝐴𝐴 𝑑𝑑 + 𝑖𝑖𝑑𝑑𝑑𝑑(𝑑𝑑𝑠𝑠)2 + 2 π‘˜π‘˜π‘‘π‘‘π‘ π‘  𝑑𝑑(𝑛𝑛𝑛𝑛1π‘Ÿπ‘Ÿ + 𝑝𝑝0) 𝑖𝑖(1 βˆ’ 𝑝𝑝0)2οΏ½(𝑀𝑀𝑐𝑐 + π‘˜π‘˜π‘‘π‘‘ π‘šπ‘š1 + 𝑑𝑑𝑖𝑖 π‘˜π‘˜) + (π‘˜π‘˜π‘‘π‘‘ π‘šπ‘šπ‘Ÿπ‘Ÿ + 𝑑𝑑𝑖𝑖 π‘˜π‘˜)𝑝𝑝1π‘Ÿπ‘ŸοΏ½ + 𝑑𝑑𝑖𝑖 οΏ½οΏ½(𝑑𝑑 π‘šπ‘š1 + 𝑑𝑑 π‘šπ‘šπ‘Ÿπ‘Ÿ 𝑝𝑝1π‘Ÿπ‘Ÿ)(2𝑀𝑀𝑐𝑐 + π‘˜π‘˜π‘‘π‘‘ π‘šπ‘š1 + π‘˜π‘˜π‘‘π‘‘ π‘šπ‘šπ‘Ÿπ‘Ÿ 𝑝𝑝1π‘Ÿπ‘Ÿ)οΏ½ + (𝑑𝑑𝑖𝑖(1 + 𝑝𝑝1 π‘Ÿπ‘Ÿ)(2𝑀𝑀𝑐𝑐 + 𝑑𝑑𝑖𝑖 π‘˜π‘˜ + 2π‘˜π‘˜π‘‘π‘‘ π‘šπ‘š1 + 2π‘˜π‘˜π‘‘π‘‘ π‘šπ‘šπ‘šπ‘š 𝑝𝑝1π‘Ÿπ‘Ÿ + 𝑑𝑑𝑖𝑖 π‘˜π‘˜π‘π‘1π‘Ÿπ‘Ÿ)οΏ½ The addition of the term β€œ2 π‘˜π‘˜π‘‘π‘‘π‘ π‘  𝑑𝑑(𝑛𝑛𝑛𝑛1π‘Ÿπ‘Ÿ + 𝑝𝑝0) ” represents the quality cost (IRR) impact on the optimum lot size. Where β€˜n’ represents the no. of times rework operation is performed.𝑝𝑝1π‘Ÿπ‘Ÿ represents the proportion of rework components produced during the process and 𝑝𝑝0 represnest the rejected products during the production process. It can be observed that increased in imperfection, number of rework operations increased quality costs and optimal lot size. IV. Results The proposed model is compared with the most well-known EOQ model and GTOQ model developed by Boucher [20]. Five different cases data is taken from a US tool manufacturing company, initially introduced by Boucher [20]. Few assumptions were made regarding data calculation e.g. percentage of rejects 𝑝𝑝0 = 20%, the percentage of rework 𝑝𝑝1π‘Ÿπ‘Ÿ= 5%,𝑑𝑑 π‘šπ‘šπ‘šπ‘š= 5% of actual machining time 𝑑𝑑 π‘šπ‘š1 and k= 3000 ($/year) in all five cases. It is assumed that rework operation is performed only once. Table 1. Optimum lot size using GTOQ, EOQ and proposed model. (assume, 𝑖𝑖 = 35% Case No. 𝑑𝑑𝑖𝑖(min/ unit) A ($/unit) d(units /year) 𝑀𝑀𝑐𝑐 (USD/ unit) 𝑑𝑑𝑠𝑠*(min/ setup) 𝑑𝑑 π‘šπ‘š1* (min/ unit) EOQ GTOQ GTOQ IRR 1 20 14.349 77 5.63 574 100 28 26 34 2 20 12.75 233 1.57 510 32 85 81 96 3 20 12.951 580 1.42 518 87 109 87 98 4 20 14.349 1877 1.64 574 67 216 135 139 5 20 17.274 5361 1.12 691 41 497 255 233 www.ijera.com 181|P a g e
  • 6. M. Ullah&C.W. Kang Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 6( Version 6), June 2014, pp.177-183 It can be observed that the response of all three models is different to all cases. Response of the proposed model is significant in comparison to the previous developed model (EOQ and GTOQ) as imperfection and quality costs are taken into consideration. The impact of quality cost (IRR) has also been highlighted in the table 2 below. We assume 𝑖𝑖 = 35 %, 𝑀𝑀𝑐𝑐 = 1($/unit), d = 14000 (units/year), π‘˜π‘˜ = 7000 ($/year), 𝑑𝑑𝑠𝑠= 0.0017(years/unit), 𝑑𝑑 π‘šπ‘š1= 0.12 (mints/unit), 𝑑𝑑 π‘šπ‘šπ‘šπ‘š= 5% (π‘šπ‘š1), A =11.9 ($/unit).It can be observed that as imperfection in processes increase, quality costs shoots up and optimal lot size is increased. This impact of imperfection on quality cost and optimum lot size has also been shown in Fig.3. Quality costs are almost negligible when there processes are perfect and goes on increasing as imperfection in the form of rework and rejected products increases. Table 2.Impact of imperfection on quality cost in inventory model. Case No. 𝑝𝑝𝑏𝑏(%) 𝑝𝑝1π‘Ÿπ‘Ÿ(%) GTOQIRR Setup cost WIP cost Finish goods cost Quality cost (IRR) 1 0 0 943 176.63 17.81 471.61 0.01 2 5 5 1037 169.05 20.01 492.76 1511.78 3 10 10 1139 162.57 22.59 512.40 3189.48 4 15 15 1248 157.00 25.64 530.57 5062.92 5 20 20 1368 152.22 29.28 547.23 7169.20 6 25 25 1499 148.14 33.67 562.29 9555.25 Fig. 3 Change in optimal lot size and quality cost (IRR) with change in Imperfection V. Conclusion The importance of quality in manufacturing setups cannot be neglected in general and particularly in environments where machining work is large relatively. Boucher [20] realized in his model that GTOQ is good for high machining environments. This increased machining ultimately impact quality costs as well. Impact of quality costs on lot size has not been considered till now while developing such models for work in process inventories. Different inventory models are developed during last few decades to solve the real world problems and optimize the lot size as per actual scenarios. However work in process based inventory model are lacking the component of quality costs in their previously developed models especially when the imperfection and high machining rate of processes are considered. This paper is an attempt towards the development of such model for imperfect process considering IRR quality cost component. The proposed model is of significance for manufacturing environments having concentration on quality costs. It is more generalized and incorporate all aspects of the previously developed models. This research can be further extended by estimating the effect of shortages and machine breakdowns on lot size focusing work in process inventory. VI. Acknowledgment: The author would like to express his gratitude to all anonymous editors and referees for their valuable comments. www.ijera.com 182|P a g e
  • 7. M. Ullah&C.W. Kang Int. Journal of Engineering Research and Applications www.ijera.com ISSN : 2248-9622, Vol. 4, Issue 6( Version 6), June 2014, pp.177-183 References [1] Rosenblatt, M.J. and Lee, H.L.(1986), Economic production cycle with imperfect production processes. IIE Transaction, 18, pp. 48-55. [2] Sarker, B.R., Jamal, A.M.M., and Mondal, S. (2007) Optimum batch size in a multi stage production system with rework consideration. European Journal of Operational Research, 184 (3). pp. 915- 929. [3] Gupta, T. and Chakraborty, S. (1984) Looping a multistage production system. International Journal of Production Research, 222 (2), pp. 299-311. [4] Jamal A.M.M., Sarker, B.R., and Mondal, S. (2004)Optimum manufacturing batch size with rework process at a single stage production system. Computers and Industrial Engineering, 47 (1), pp. 77-89. [5] Biswas, P. and Sarker, B.R. (2008). Optimal batch quantity models for a lean production system with in-cycle rework and scrap. International Journal of Production Research, 46 (23), pp. 6585-6610. [6] Zhang, X. and Gerchak, Y. (1990). Joint lot sizing and inspection policy in an EOQ model with random yield. IIE Transaction, 22, pp. 41-47. [7] Salameh, M.K. and Jaber, M.Y. (2000). Economic production quantity model for items with imperfect quality. International Journal of Production Economics, 64 (1), pp.59-64. [8] Ben Daya, M. and Rahim, A. (2003).Optimal lot sizing, quality improvement and inspection errors for multistage production systems. International Journal of Production Research, 41 (1), pp. 65-79. [9] Krishnamurthy, C. and Panayappan, S. (2012). An EPQ Model with imperfect production systems with rework of regular production and sales return. American Journal of Operation Research, 2, pp. 225- 234. [10] Yoo, S. H., Kim, D.S., and Park, M. (2012).Inventory models for imperfect production and inspection processes with various inspection options under one-time and continuous improvement investment. Journal of Computers and Operation Research, 39 (9), pp. 2001-2015. [11] Ojha, D., Sarker, B. R., and Biswas, P. (2007). An optimal batch size for an imperfect production system with quality assurance and rework. International Journal of Production Research, 45 (14), pp. 3191-3214. [12] Wee, H.M., Wang, W.T., and Yang P.C. (2013). A production quantity model for imperfect quality items with shortage and screening constraint. International Journal of Production Research, 51 (6), pp. 1869-1884. [13] Muddah, B., Salameh, M.K., and Karameh, G.M. (2009). Lot sizing with random yield and different qualities”. Applied Mathematical Modeling, 33 (4), pp. 1997- 2009. [15] Porteus, E. (1986). Optimal lot-sizing, process quality improvement and setup cost reduction. Operations Research, 34 (1), pp. 137–144. [16] Chung, K.J. (2011). The economic production quantity with rework process in supply chain management. Computers and Mathematics with Applications, 62 (6), pp. 2547-2550. [17] Khan, M., Jaber M.K., and Bonney, M. (2011).An economic order quantity (EOQ) for items with imperfect quality and inspection errors. International Journal of Production Economics, 133 (1), pp. 113-118. [18] Colledani, M. and Tolio, T. (2009) Performance evaluation of production system monitored by statistical process control and off-line inspections. International Journal of Production Economics. 120(2), pp. 348-367. 2009 [19] Krishnamoorthi, C. and Panayappan, S. (2012). An EPQ model with reworking of imperfect production items and shortages. International journal of Management Sciences and Engineering Management, 7 (3), pp. 220-228. [20] Boucher, T.O. (1984). Lot sizing in group technology production system. International Journal of Production Research, 22 (1), pp. 85-93. [21] Barzoki, M.R., Jahanbazi, M., and Bijari, M. (2011). Effects of imperfect products on lot sizing with work in process inventory. Applied Mathematics and Computation, 217 (21), pp. 8328-8336. [22] Silver, E.A., Pyke, D.F., and Peterson, R., 1998. Inventory Management and Production Planning and Scheduling, 3rd Ed., Wiley, New York. www.ijera.com 183|P a g e