Este artíciculo habla sobre economía y costos de oportunidad

1. Int J Adv Manuf Technol (2009) 45:382–388
DOI 10.1007/s00170-009-1959-1
ORIGINAL ARTICLE
A method to exchange the demand of products
for cost improvement
Sanjay Sharma
Received: 18 October 2007 / Accepted: 3 February 2009 / Published online: 24 February 2009
# Springer-Verlag London Limited 2009
Abstract In a multiproduct manufacturing environment, the facilities optimally. However, when most of the firms
actual demands of various products are either available, or achieve this level, there is loss of competitive edge, and
these are expected. There are situations when demand of a further cost reduction becomes necessary. In such a scenario,
product can be substituted with that of another. In the context an examination of significant parameters is essential.
of cyclic manufacture, all the items are produced in an optimal Demand management is critical nowadays, and therefore, a
cycle time, and the production facility runs at certain cost method is explored in the present paper to exchange the
level. The total cost consists of the facility setup cost, demand of products for cost improvement in certain cases.
inventory carrying costs, and the manufacturing time cost In a continuous production, single standard product is
for the basic case. The total cost is optimized. For the purpose manufactured in large quantities. Even if the type of
of total cost improvement, a method is presented in which the product is similar, it can be produced in a wide variety of
demand of a product is exchanged with that of another item in sizes. For instance, in a tube or pipe manufacturing
the group. The basic model without backorders is analyzed industry, these are in different diameters/thicknesses. In a
first. Then, it is extended for an inclusion of shortages that are job shop/batch production also, several items are processed
either completely backlogged or partially. In addition to the in a cycle time. For example, if the cycle time is 3 months
cost components discussed before, shortage costs are included or 0.25 year, all items/product varieties are manufactured in
in the total cost for this case. Finally, after a discussion of idle the cycle time. This is called as common cycle time. If the
time costs, these are also included briefly in the formulation of production rate of an item is, say 300 U per month, and the
the total cost. The proposed methods are useful for imple- demand rate is 100 U per month, the production time in a
mentation in a variety of industrial or business situations in the cycle time of 3 months will be 1 month, i.e., 3×(100/300).
context of internal benchmarking or gradual improvement. Benefits can be achieved by synchronizing production
activities sequentially in a cycle time [3]. A relevant cost
Keywords Multi-item cyclic manufacture . Demand rate . needs to be estimated/modeled for the concerning produc-
Production time . Idle time costs tion environment. For example, if shortages are not
allowed, the shortage costs will not become a component
of the total relevant cost. After an optimization of the total
1 Introduction relevant cost, a common cycle time is usually obtained in
which all the items in a family are produced. A generalized
In the manufacturing firms, one or more products are made production cost is used [1] including shop floor index, the
in certain cycle time. In order to become competitive, the value of which lies in the range 0–1. The generalized
progressive firms are expected to run their production production cost is obtained as the multiplication of fixed
production cost and a factor that is an exponential order of
the ratio of production rate to demand rate of an item.
S. Sharma (*)
In the context of modeling process, the rate of manufac-
National Institute of Industrial Engineering (NITIE),
Vihar Lake, Mumbai 400087, India ture and demand rate are among significant input parameters.
e-mail: s_nsit@rediffmail.com Manufacturing rate is considered to be a decision variable

2. Int J Adv Manuf Technol (2009) 45:382–388 383
[8]. Shortages are included in the production system. These With the purpose of an internal benchmarking/improve-
may be backordered completely/partially. Various cases are ment activities, it seems reasonable to consider an appro-
analyzed [5–7, 9] for single/multi-item scenario. The priate item whose demand is to be interchanged by any
demand rate per year or an annual demand needs to be other remaining item in the group. The present paper is
adjusted in order to incorporate partial or fractional back- divided into nine sections. Assumptions and notations are
ordering situation. For a single product case, the demand provided in the “Assumptions” section, followed by
increase is included in different context [2, 4] considering methodology in the “Methodology” section. Mathematical
demand function with respect to time. As it will be discussed formulation for the basic problem is dealt with in the
later, a quite different approach is presented in this paper in “Mathematical formulation” section followed by an illus-
the context of multiproduct manufacturing environment. trative numerical example in the “Illustrative example”
This is expected to be useful in certain situations of business section. Shortages are included in the “Extension for
when more or less stable product demands exist. shortages” section with the assumption that all the shortage
In the traditional production/manufacturing setup, the quantities will be backordered completely. This assumption
demand is analyzed solely as an input parameter. In the is relaxed in the “Partial backlogging” section. An idle time
present paper, the demands are being viewed in an uncon- cost is introduced in the “Incorporating an idle time cost”
ventional manner. For instance, several production lines run in section for this approach, and finally, the concluding
parallel in the pharmaceutical industries. Whether it is remarks are provided in the “Concluding remarks” section.
multiple or single production line, a batch production is
usually adopted. After certain development or value addition,
the management wishes to promote the improved product 2 Assumptions
(which may be patented in a different name) at the cost of
similar (more or less for medicinal purpose) matured product. An industrial organization is engaged in the production of
However, the improved product is at least presently in lower multiple items in a common cycle time. The manufacturing
demand because of either the availability of a familiar matured facility is being run conventionally in an optimum manner. It
product at higher demand level or lack of awareness. This may is often difficult to obtain information for benchmarking
also be due to purely psychological or emotional reasons purpose particularly at the production facility level. With the
attached to a familiar product. As the aggregate demand is aim of a gradual improvement, an intentional search is made
more or less uniform for similar types of products, the to exchange the demand of an item (strategically selected by
production strategy may be based on a conscious anticipated the management) with another appropriate item in the family
demand swapping. Further, there should be a strong justifica- for any potential cost reduction. A business environment of
tion if it yields into the total relevant cost reduction. stable demand exists in general. The proposed method
In oligopoly, few firms dominate the market. While in considers an exact interchange of the demand level of two
the monopolistic competition, many firms are active in items because it is in the interest of the organization to
satisfying the market demands. Whether it is monopolistic maintain a similar aggregate demand for the whole family of
competition or oligopoly, each progressive firm in the items.
industrial sector would run their production operations at a In addition to the above, the following assumptions are
certain optimum level. There is continuous pressure to also made:
adopt a kind of internal benchmarking and improve the
1. The facility is set-up for a family of items, and
production/operational cost further. In a planning period, it
therefore, the facility setup cost is included in the
is possible to substitute the demand of an item by another
formulation. As the individual item setup time is not
suitable item in the product family. The firm may have
relevant in the present context, it is ignored.
invested in product development activities. It would like to
2. All the items are manufactured in a common cycle time.
exchange the lower demand of new product with higher
3. Shortages may or may not be allowed.
demand of an old matured product, and the firm manage-
4. In case shortages are allowed, these may be backordered
ment is confident of getting it consumed as a substitute in
completely/partially depending on the situation.
the market. In yet another situation, a factory may face
5. An idle time exists usually in a common cycle time. If the
quality problems related to the input item of a product, and
idle time costs are significant, these may be incorporated
it wants to exchange the demand of such a product with
in the modeling process depending on the case.
another in the short-run. In many cases, contribution per
unit is almost similar for the products in a family. It is an Based on these assumptions, a formulation is first made
interesting approach to explore the possibility concerning for the basic production situation. Then the shortages are
the exchange of demand of items and examine the effects incorporated with complete backordering. This is extended
on total relevant cost. for a fractional backordering case. The idle time cost is

3. 384 Int J Adv Manuf Technol (2009) 45:382–388
further discussed briefly with its inclusion in the suggested Compute the existing cost, E
method.
2.1 Notation Select Dk from the set Di , i≠j
∝ Shop floor index lying usually in the range (0≤ ∝ <1). No
Exchange Dj with Dk
Ai Setup cost for item i.
bi A faction of shortage quantity which is not
backordered for product i. ∑(Di/Pi)< 1
c Fixed production cost per year.
c1 Idle time cost per year.
Yes
Di Annual demand for item i.
Compute the revised cost
E Total relevant cost.
E1 Total cost after exchange of the demand rate of two
Retain the minimum cost along with corresponding exchange and implement
items.
Hi Inventory carrying cost for an item i per unit-year.
j An item whose demand rate is desired to be Fig. 2 An iterative process of demand exchange
exchanged with another appropriate item.
Ji Shortage quantities for a product i.
k Selected another appropriate item whose demand rate All the remaining items can be considered one at a time.
would be exchanged with that of item j. However, the conditions are developed next in order to
Ki Annual shortage cost per unit for a product i. have a small subset of items to make the search procedure
n Number of items in the group. convenient.
Pi Production rate per year for item i.
T Common cycle time in year.
4 Mathematical formulation
A generalized production cost is cðPi =Di Þa per year, and as
the manufacturing time for an item i is (Di/Pi), the annual
manufacturing time cost for an item i is cðDi =Pi Þ1Àa . With
3 Methodology the inclusion of this cost component, a total relevant cost
for the basic model without shortages,
From a family of n items, an item j is selected by the
management whose demand rate is to be exchanged by that X
n
1X n
TX n
E¼c ðDi =Pi Þ1Àa þ Ai þ Di Hi ð1 À Di =Pi Þ
of another appropriate item k among the remaining items. T i¼1 2 i¼1
i¼1
Figure 1 represents the process of exchange of demand rates.
Pn ð1Þ
The production time is T ðDi =Pi Þ in a cycle time T, and
i¼1
in order to have a feasible schedule, the production time The optimal cycle time can be obtained by differentiating
Pn
should be less than T, i.e., ðDi =Pi Þ < 1. In the iterative Eq. 1 with respect to T and equating to 0. The optimal
i¼1
process of exchange (Fig. 2), Dj is exchanged by Dk such values (T* and subsequently E*) can easily be obtained as,
that the constraint on total production time is satisfied.
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u P n
u 2 Ai
u
u
T * ¼ uP i¼1
Fig. 1 Exchanging the demand D1 ð2Þ
rate t n
D2 Di Hi ð1 À Di =Pi Þ
i¼1
.
.
Dj . X
n
Dk and E * ¼ c ðDi =Pi Þ1Àa ð3Þ
. i¼1
.
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
. u " n #" #
Dn u X X n
þ t2 Ai Di Hi ð1 À Di =Pi Þ
Di, i≠j i¼1 i¼1

4. Int J Adv Manuf Technol (2009) 45:382–388 385
With reference to Eq. 3, the components concerning item
j and item k are separated from the remaining items. After
exchanging Dj and Dk, the total optimal cost,
2 3
6Xn À Á1Àa À Á1Àa 7
E1 ¼ c6
Ã
4 ðDi =Pi Þ1Àa þ Dk Pj þ Dj Pk 7
5 ð4Þ
i6¼j
i6¼k
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 3
u
u
u X #6X n n À Á À Á7
u
þ u2 Ai 6 4 fDi Hi ð1 À Di =Pi Þg þ Dk Hj 1 À Dk Pj þ Dj Hk 1 À Dj Pk 7 5
t
i¼1 i6¼j
i6¼k
Subtracting Eqs. (4) from (3), any potential cost improvement,
hÀ Á À Á1Àa À Á1Àa i
E Ã À E1 ¼ c Dj Pj
à 1Àa
þ ðDk =Pk Þ1Àa À Dk Pj À Dj Pk ð5Þ
2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
Pn
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6 fDi Hi ð1 À Di =Pi Þg 7
X 6n i¼1 7
6 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7
þ 2 A i 6 uP 7
6 Àu fD H ð1 À D =P Þg þ D H À1 À D P Á þ D H À1 À D P Á 7
n
i¼1 4 t i i i i k j k j j k j k 5
i6¼j
i6¼k
Equation (5) has two components, the first component is The second component is certain to be positive if,
certain to be positive if,
À Á1Àa À Á1Àa À Á1Àa
Dj Pj þðDk =Pk Þ1Àa Dk Pj þ Dj Pk ð6Þ
À Á À Á À Á
Dj Hj 1 À Dj hPj þ Dk Hk ð1 Dk =PÞ Dk Hj 1 À Dk Pj þ Dj Hk 1 À Dj Pk
À k
À Á À Á H À Ái ð7Þ
or Dj À Dk Hj À Hk þ Hkk À Pjj Dj þ Dk 0
P
There is a guaranteed cost improvement if the conditions 6
and 7 are satisfied. The entire feasible remaining item
demand rate can be exchanged if it is difficult to draw any Table 1 Input parameters
conclusion with the use of conditions 6 and 7.
Item
1 2
5 Illustrative example
Annual demand Di 400 300
Table 1 shows the input parameters concerning two items. Annual production rate Pi 720 750
As it is a simple numerical example for illustration purpose, Setup cost, Ai ($) 100 150
Pn P
n
Di Hi ð1 À Di =Pi Þ ¼ 0 and ðDi =Pi Þ1Àa ¼ 0. Annual carrying cost Hi ($ per unit) 13 5
i6¼j i6¼j
i6¼k i6¼k Annual shortage cost Ki ($ per unit) 120 80
Using the relevant parameters for the basic case, i.e., Fraction bi
Pn 0.2 0.3
without shortages, ðDi =Pi Þ ¼ 0:955 1, and the feasible
i¼1
data are ensured. c=$9,000 per year; α=0.2

5. 386 Int J Adv Manuf Technol (2009) 45:382–388
From Eq. 3, the total relevant cost, E* =$11,214.88.
Now, let j=1 and k=2. After exchanging Dj with Dk,
Pn
ðDi =Pi Þ ¼ 0:95, and the feasibility is ensured. Vi
i¼1 Production
From condition 6, 1.1051.101. inventory
Pi – D i Di
From condition 7, 2.780.
As the both conditions are satisfied, there is a guaranteed
0
cost improvement with the implementation of the proposed Time
method.
With the use of Eq. 4, a reduced total relevant cost after
Ji
demand exchange, E1* =$11,177.19. T
Fig. 3 The production cycle with shortages
6 Extension for shortages
TDi Hi ð1 À Di =Pi Þ
Substituting optimal Ji ¼ ð11Þ
Quite often, the shortages are included in a manufacturing ðHi þ Ki Þ
system. These are assumed to be completely backordered at
present. Figure 3 shows this kind of environment. X
n
1 Xn
T X Di Hi Ki ð1 À Di =Pi Þ
n
Since the shortages exist for a period Ji =ðPi À Di Þþ E¼c ðDi =Pi Þ1Àa þ Ai þ
T i¼1 2 iÀ1 ðHi þ Ki Þ
ðJi =Di Þ, the annual shortage cost for an item i, i¼1
h i ð12Þ
¼ Ji ðPi ÀDi Þ þ Dii Ki
2
Ji J
T
The optimal values can be obtained as,
P Ki Ji2
n
and the total annual shortage cost ¼ 2T1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Di ð1ÀDi =Pi Þ u P n
i¼1 u 2 Ai
u
ð8Þ u
T * ¼ uP
i¼1
ð13Þ
Now, the maximum inventory level, Vi ¼ ðPi ÀiDi ÞTDi =Pi À Ji
h
t n
½Di Hi Ki ð1 À Di =Pi Þ=ðHi þ Ki ÞŠ
and the annual carrying cost ¼ Vi T À ðPi ÀDi Þ À Dii Hi Substitut-
2
Ji J
T i¼1
ing Vi, the total annual carrying cost,
T Xn Xn
1 Xn
Hi Ji2
¼ Di Hi ð1 À Di =Pi Þ À Hi J i þ
2 i¼1 i¼1
2T i¼1 Di ð1 À Di =Pi Þ
P
n
ð9Þ and E * ¼ c ðDi =Pi Þ1Àa
i¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
! n !
Adding the Eqs. 8, 9, and the remaining cost components, P n P
þ 2 Ai Di Hi Ki ð1 À Di =Pi Þ=ðHi þ Ki Þ
X
n
1X n
1 X ðHi þ Ki ÞJi2
n
i¼1 i¼1
E¼c ðDi =Pi Þ1Àa þ Ai þ
i¼1
T i¼1 2T i¼1 Di ð1 À Di =Pi Þ ð14Þ
T X
n X
n
þ Di Hi ð1 À Di =Pi Þ À Hi Ji With the swapping of Dj and Dk,
2 i¼1 i¼1
ð10Þ
2 3
* 6Xn À Á1Àa À Á1Àa 7
E1 ¼ c 6
4 ðDi =Pi Þ1Àa þ Dk Pj þ D j Pk 7
5
i6¼j
i6¼k
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 3
u
u
u X #6X n n À ÁÀ Á À Á 7
u
þ u2 Ai 6 4 fDi Hi Ki ð1 À Di =Pi Þ=ðHi þ Ki Þg þ Dk Hj Kj 1 À Dk Pj Hj þ Kj þ Dj Hk Kk 1 À Dj Pk ðHk þ Kk Þ7 5
t
i¼1 i6¼j
i6¼k
ð15Þ

6. Int J Adv Manuf Technol (2009) 45:382–388 387
Following the procedure discussed in the “Mathematical obtained. The first condition is similar to 6. The second
formulation” section, the relevant conditions can be condition is obtained as,
( )#
À Á Hj Kj Hk Kk À Á Hk Kk Hj Kj
Dj À Dk À ÁÀ þ Dj þ Dk À À Á 0 ð16Þ
Hj þ Kj ðHk þ Kk Þ Pk ðHk þ Kk Þ Pj Hj þ Kj
With the input parameters of Table 1, the condition 6 is advertising costs apportioned for unit product and loss of
already satisfied. profit among other factors, an explicit computation for
From condition 16, 1.2090. contribution of the lost units of product is not necessary. A
As the both conditions (6) and (16) are satisfied, there is suitable parameter for relevant cost is assumed for all the
certain cost improvement using the proposed approach. shortage quantities whether these are backlogged or not. An
From Eq. 14, E*=$ 11,158.62. annual demand needs to be adjusted in order to incorporate
Ã
The reduced relevant cost from Eq. 15, E1 ¼ $11; 121:22. the partial backordering.
The corresponding costs are also lower than that From Eq. 8, the annual shortage quantity can be obtained
obtained in the previous section. This can be justified by as,
observing Eqs. 3 and 14. As Ki =ðHi þ Ki Þ is less than 1, the
X
n
Ji2
relevant costs are lower with relaxation of the constraint ¼
that the backorders would not be allowed. i¼1
2TDi ð1 À Di =Pi Þ
A fraction bi of the shortage quantity is not backordered,
and therefore, the annual manufacturing cost,
7 Partial backlogging
X 1
n !1Àa
bi Ji2
In a real-world situation, a portion of the shortage quantities ¼c Di À
P1Àa
i¼1 i
2TDi ð1 À Di =Pi Þ
may not be backordered. A particular customer may switch
over to another competitive firm in the industry. However, Equation 10 can now be adjusted as follows for this
with the advertising among other efforts, a new customer situation,
can replace the old one, at a later date. In case where the
shortage costs are estimated to be a good representation of
X 1 !1Àa
n
bi Ji2 1 Xn
1 X ðHi þ Ki ÞJi2
n
T X
n Xn
E¼c Di À þ Ai þ þ Di Hi ð1 À Di =Pi Þ À H i Ji ð17Þ
P1Àa
i¼1 i
2TDi ð1 À Di =Pi Þ T i¼1 2T i¼1 Di ð1 À Di =Pi Þ 2 i¼1 i¼1
Mathematical/analytical procedure as discussed before, 7.1 Specific case
cannot be followed for the optimization of Eq. 17. However,
conventional search process such as univariate method can α=0 in a specific case, and the Eq. 17 can be written as,
be implemented conveniently for any real data set.
X
n
1 Xn
1 X ðHi þ Ki À cbi =Pi ÞJi2 T X
n n Xn
E¼c ðDi =Pi Þ þ Ai þ þ Di Hi ð1 À Di =Pi Þ À H i Ji ð18Þ
i¼1
T i¼1 2T i¼1 Di ð1 À Di =Pi Þ 2 i¼1 i¼1
The optimal relevant cost can be obtained as,
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u n # #
X
n u X X n
E* ¼ c ðD =P Þ þ t2
i i A i D H ð1 À D =P ÞðK À cb =P Þ=ðH þ K À cb =P Þ
i i i i i i i i i i i ð19Þ
i¼1 i¼1 i¼1

7. 388 Int J Adv Manuf Technol (2009) 45:382–388
As c/Pi is the unit production cost, and shortage costs are framework of organization along with several input
much greater than this in the real world, an optimality/ parameters. However, these are continuously striving for
feasibility condition, i.e., Ki c/Pi, is satisfied easily. the cost improvement. Internal benchmarking practices are
In order to exchange the demand of products, Eq. 19 can also adopted where the standards are bound to vary with
be used as a reference equation. time. A method is proposed and analyzed in which the
Consider the input data of Table 1. demand of a strategically selected item is exchanged with
From Eq. 19, E*=$9,809.46 another suitable item in the group. Analysis is first made for
After an exchange of the demands, the reduced relevant the basic case without shortages and conditions are
cost is obtained as, developed for convenience in the search of another suitable
à item. The process is illustrated with the help of a numerical
E1 ¼ $ 9; 759:21
example. Further extensions are concerning the inclusion of
shortages that may be backlogged completely or partially.
The costs are obtained at a lower level with the allowable
8 Incorporating an idle time cost backorders. However, an annual shortage cost needs to be
estimated with care considering the all relevant factors.
In the cyclic manufacture, a production activity usually takes In a production cycle time, a certain period is usually idle.
place for certain portion of the cycle time, and the remaining This idle time frequently repeats itself in case where the
Pn
portion is idle. With reference to Eq. 3, ðDi =Pi Þ is the associated manufacturing schedule is implemented. Idle time
i¼1
annual manufacturing time. After an exchange of demand, cost is introduced for the proposed method. With the inclusion
this parameter will vary. For instance, an annual manufac- of this cost, the reference equations are obtained which can be
turing time has been reduced after the exchange of demand useful for an exchange of demand. In the presence of a
in the illustrative example of the “Illustrative example” relevant situation, these are suitable for a trade-off concerning
section. This means that the idle time during the cycle has the production time and idle time among other factors.
increased. In few cases, the problems are associated with an The possibilities for demand exchange can be conve-
idle production facility such as the maintenance problems. niently explored, and depending on the business strategy, the
Consistency in the quality of a product and skills of the proposed approach may be implemented in a short-run/long-
human resources may also get affected up to some extent. run. In case of the various problems being faced by the firm,
With the occurrence of this type of problems, it seems an alternate schedule is available on the basis of certain
reasonable to introduce the idle time cost. methodology. This will help in incorporating flexibility in
the industrial system and also in the decision-making process
Consider an idle time cost per year ¼ c1 ðc1 cÞ in a variety of situations.
#
X n
Idle time cost in a year ¼ c1 1 À ðDi =Pi Þ
i¼1 References
Equation 3 can now be transformed as follows:
1. Chowdhury MR, Sarker BR (2001) Manufacturing batch size and
#
Xn
1Àa
X
n ordering policy for products with shelf lives. Int J Prod Res 39
E* ¼ c ðDi =Pi Þ þ c1 1 À ðDi =Pi Þ (7):1405–1426. doi:10.1080/00207540110052148
i¼1 i¼1 2. Giri BC, Jalan AK, Chaudhari KS (2005) An economic production
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lot size model with increasing demand, shortages and partial
u n # # backlogging. Int Trans Oper Res 12:235–245
u X X n
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