ORIGINAL ARTICLE

A method to exchange the demand of productsfor 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, theactual demands of various products are either available, orthese are expected. There are situations when demand of aproduct can be substituted with that of another. In the contextof cyclic manufacture, all the items are produced in an optimalcycle time, and the production facility runs at certain costlevel. The total cost consists of the facility setup cost,inventory carrying costs, and the manufacturing time costfor the basic case. The total cost is optimized. For the purposeof total cost improvement, a method is presented in which thedemand of a product is exchanged with that of another item inthe group. The basic model without backorders is analyzedfirst. Then, it is extended for an inclusion of shortages that areeither completely backlogged or partially. In addition to thecost components discussed before, shortage costs are includedin the total cost for this case. Finally, after a discussion of idletime costs, these are also included briefly in the formulation ofthe total cost. The proposed methods are useful for imple-mentation in a variety of industrial or business situations in thecontext of internal benchmarking or gradual improvement.

Keywords Multi-item cyclic manufacture . Demand rate .

Production time . Idle time costs

1 Introduction

In the manufacturing firms, one or more products are madein certain cycle time. In order to become competitive, theprogressive firms are expected to run their production

facilities optimally. However, when most of the firmsachieve this level, there is loss of competitive edge, andfurther cost reduction becomes necessary. In such a scenario,an examination of significant parameters is essential.Demand management is critical nowadays, and therefore, amethod is explored in the present paper to exchange thedemand of products for cost improvement in certain cases.

In a continuous production, single standard product ismanufactured in large quantities. Even if the type ofproduct is similar, it can be produced in a wide variety ofsizes. For instance, in a tube or pipe manufacturingindustry, these are in different diameters/thicknesses. In ajob shop/batch production also, several items are processedin a cycle time. For example, if the cycle time is 3 monthsor 0.25 year, all items/product varieties are manufactured inthe cycle time. This is called as common cycle time. If theproduction rate of an item is, say 300 U per month, and thedemand rate is 100 U per month, the production time in acycle time of 3 months will be 1 month, i.e., 3×(100/300).Benefits can be achieved by synchronizing productionactivities sequentially in a cycle time [3]. A relevant costneeds to be estimated/modeled for the concerning produc-tion environment. For example, if shortages are notallowed, the shortage costs will not become a componentof the total relevant cost. After an optimization of the totalrelevant cost, a common cycle time is usually obtained inwhich all the items in a family are produced. A generalizedproduction cost is used [1] including shop floor index, thevalue of which lies in the range 0–1. The generalizedproduction cost is obtained as the multiplication of fixedproduction cost and a factor that is an exponential order ofthe ratio of production rate to demand rate of an item.

In the context of modeling process, the rate of manufac-ture and demand rate are among significant input parameters.Manufacturing rate is considered to be a decision variable

Int J Adv Manuf Technol (2009) 45:382–388DOI 10.1007/s00170-009-1959-1

S. Sharma (*)National Institute of Industrial Engineering (NITIE),Vihar Lake, Mumbai 400087, Indiae-mail: s_nsit@rediffmail.com

[8]. Shortages are included in the production system. Thesemay be backordered completely/partially. Various cases areanalyzed [5–7, 9] for single/multi-item scenario. Thedemand rate per year or an annual demand needs to beadjusted in order to incorporate partial or fractional back-ordering situation. For a single product case, the demandincrease is included in different context [2, 4] consideringdemand function with respect to time. As it will be discussedlater, a quite different approach is presented in this paper inthe context of multiproduct manufacturing environment.This is expected to be useful in certain situations of businesswhen more or less stable product demands exist.

In the traditional production/manufacturing setup, thedemand is analyzed solely as an input parameter. In thepresent paper, the demands are being viewed in an uncon-ventional manner. For instance, several production lines run inparallel in the pharmaceutical industries. Whether it ismultiple or single production line, a batch production isusually adopted. After certain development or value addition,the management wishes to promote the improved product(which may be patented in a different name) at the cost ofsimilar (more or less for medicinal purpose) matured product.However, the improved product is at least presently in lowerdemand because of either the availability of a familiar maturedproduct at higher demand level or lack of awareness. Thismayalso be due to purely psychological or emotional reasonsattached to a familiar product. As the aggregate demand ismore or less uniform for similar types of products, theproduction strategy may be based on a conscious anticipateddemand swapping. Further, there should be a strong justifica-tion if it yields into the total relevant cost reduction.

In oligopoly, few firms dominate the market. While inthe monopolistic competition, many firms are active insatisfying the market demands. Whether it is monopolisticcompetition or oligopoly, each progressive firm in theindustrial sector would run their production operations at acertain optimum level. There is continuous pressure toadopt a kind of internal benchmarking and improve theproduction/operational cost further. In a planning period, itis possible to substitute the demand of an item by anothersuitable item in the product family. The firm may haveinvested in product development activities. It would like toexchange the lower demand of new product with higherdemand of an old matured product, and the firm manage-ment is confident of getting it consumed as a substitute inthe market. In yet another situation, a factory may facequality problems related to the input item of a product, andit wants to exchange the demand of such a product withanother in the short-run. In many cases, contribution perunit is almost similar for the products in a family. It is aninteresting approach to explore the possibility concerningthe exchange of demand of items and examine the effectson total relevant cost.

With the purpose of an internal benchmarking/improve-ment activities, it seems reasonable to consider an appro-priate item whose demand is to be interchanged by anyother remaining item in the group. The present paper isdivided into nine sections. Assumptions and notations areprovided in the “Assumptions” section, followed bymethodology in the “Methodology” section. Mathematicalformulation for the basic problem is dealt with in the“Mathematical formulation” section followed by an illus-trative numerical example in the “Illustrative example”section. Shortages are included in the “Extension forshortages” section with the assumption that all the shortagequantities will be backordered completely. This assumptionis relaxed in the “Partial backlogging” section. An idle timecost is introduced in the “Incorporating an idle time cost”section for this approach, and finally, the concludingremarks are provided in the “Concluding remarks” section.

2 Assumptions

An industrial organization is engaged in the production ofmultiple items in a common cycle time. The manufacturingfacility is being run conventionally in an optimum manner. Itis often difficult to obtain information for benchmarkingpurpose particularly at the production facility level. With theaim of a gradual improvement, an intentional search is madeto exchange the demand of an item (strategically selected bythe management) with another appropriate item in the familyfor any potential cost reduction. A business environment ofstable demand exists in general. The proposed methodconsiders an exact interchange of the demand level of twoitems because it is in the interest of the organization tomaintain a similar aggregate demand for the whole family ofitems.

In addition to the above, the following assumptions arealso made:

1. The facility is set-up for a family of items, andtherefore, the facility setup cost is included in theformulation. As the individual item setup time is notrelevant in the present context, it is ignored.

2. All the items are manufactured in a common cycle time.3. Shortages may or may not be allowed.4. In case shortages are allowed, these may be backordered

completely/partially depending on the situation.5. An idle time exists usually in a common cycle time. If the

idle time costs are significant, these may be incorporatedin the modeling process depending on the case.

Based on these assumptions, a formulation is first madefor the basic production situation. Then the shortages areincorporated with complete backordering. This is extendedfor a fractional backordering case. The idle time cost is

Int J Adv Manuf Technol (2009) 45:382–388 383

further discussed briefly with its inclusion in the suggestedmethod.

2.1 Notation

∝ Shop floor index lying usually in the range (0≤∝<1).

Ai Setup cost for item i.bi A faction of shortage quantity which is not

backordered for product i.c Fixed production cost per year.c1 Idle time cost per year.Di Annual demand for item i.E Total relevant cost.E1 Total cost after exchange of the demand rate of two

items.Hi Inventory carrying cost for an item i per unit-year.j An item whose demand rate is desired to be

exchanged with another appropriate item.Ji Shortage quantities for a product i.k Selected another appropriate item whose demand rate

would be exchanged with that of item j.Ki Annual shortage cost per unit for a product i.n Number of items in the group.Pi Production rate per year for item i.T Common cycle time in year.

3 Methodology

From a family of n items, an item j is selected by themanagement whose demand rate is to be exchanged by thatof another appropriate item k among the remaining items.Figure 1 represents the process of exchange of demand rates.

The production time is TPni¼1

Di=Pið Þ in a cycle time T, and

in order to have a feasible schedule, the production timeshould be less than T, i.e.,

Pni¼1

Di=Pið Þ < 1. In the iterative

process of exchange (Fig. 2), Dj is exchanged by Dk suchthat the constraint on total production time is satisfied.

All the remaining items can be considered one at a time.However, the conditions are developed next in order tohave a small subset of items to make the search procedureconvenient.

4 Mathematical formulation

A generalized production cost is c Pi=Dið Þa per year, and asthe manufacturing time for an item i is (Di/Pi), the annualmanufacturing time cost for an item i is c Di=Pið Þ1�a. Withthe inclusion of this cost component, a total relevant costfor the basic model without shortages,

E ¼ cXni¼1

Di=Pið Þ1�a þ 1

T

Xni¼1

Ai þ T

2

Xni¼1

DiHi 1� Di=Pið Þ

ð1ÞThe optimal cycle time can be obtained by differentiatingEq. 1 with respect to T and equating to 0. The optimalvalues (T* and subsequently E*) can easily be obtained as,

T* ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Pni¼1

Ai

Pni¼1

DiHi 1� Di=Pið Þ

vuuuuut ð2Þ

and E* ¼ cXni¼1

Di=Pið Þ1�a

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

Xni¼1

Ai

" # Xni¼1

DiHi 1� Di=Pið Þ" #vuut

ð3Þ

Di, i≠j

D1

D2

.

.

.

Dk

.

.

.

Dn

Dj

Fig. 1 Exchanging the demandrate

No

∑(Di/Pi)< 1

Yes

Compute the existing cost, E

Select Dk from the set Di , i≠j

Exchange Dj with Dk

Compute the revised cost

Retain the minimum cost along with corresponding exchange and implement

Fig. 2 An iterative process of demand exchange

384 Int J Adv Manuf Technol (2009) 45:382–388

With reference to Eq. 3, the components concerning itemj and item k are separated from the remaining items. Afterexchanging Dj and Dk, the total optimal cost,

E�1 ¼ c

Xni 6¼ji 6¼k

Di=Pið Þ1�a þ Dk

�Pj

� �1�a þ Dj

�Pk

� �1�a

2664

3775

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

Xni¼1

Ai

" # Xni 6¼ji 6¼k

DiHi 1� Di=Pið Þf g þ DkHj 1� Dk

�Pj

� �þ DjHk 1� Dj

�Pk

� �2664

3775

vuuuuut

ð4Þ

Subtracting Eqs. (4) from (3), any potential cost improvement,

E� � E�1 ¼ c Dj

�Pj

� �1�a þ Dk=Pkð Þ1�a � Dk

�Pj

� �1�a � Dj

�Pk

� �1�ah i

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Xni¼1

Ai

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni¼1

DiHi 1� Di=Pið Þf gs

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni6¼ji 6¼k

DiHi 1� Di=Pið Þf g þ DkHj 1� Dk

�Pj

� �þ DjHk 1� Dj

�Pk

� �vuut

2666664

3777775

ð5Þ

Equation (5) has two components, the first component iscertain to be positive if,

Dj

�Pj

� �1�aþ Dk=Pkð Þ1�a> Dk

�Pj

� �1�aþ Dj

�Pk

� �1�a ð6Þ

The second component is certain to be positive if,

DjHj 1� Dj

�Pj

� �þ DkHk 1� Dk=Pkð Þ > DkHj 1� Dk

�Pj

� �þ DjHk 1� Dj

�Pk

� �or Dj � Dk

� �Hj � Hk

� �þ HkPk

� Hj

Pj

� �Dj þ Dk

� �h i> 0

ð7Þ

There is a guaranteed cost improvement if the conditions 6and 7 are satisfied. The entire feasible remaining itemdemand rate can be exchanged if it is difficult to draw anyconclusion with the use of conditions 6 and 7.

5 Illustrative example

Table 1 shows the input parameters concerning two items.As it is a simple numerical example for illustration purpose,Pni 6¼ji6¼k

DiHi 1� Di=Pið Þ ¼ 0 andPni 6¼ji 6¼k

Di=Pið Þ1�a¼ 0.

Using the relevant parameters for the basic case, i.e.,without shortages,

Pni¼1

Di=Pið Þ ¼ 0:955 < 1, and the feasibledata are ensured.

Table 1 Input parameters

Item

1 2

Annual demand Di 400 300

Annual production rate Pi 720 750

Setup cost, Ai ($) 100 150

Annual carrying cost Hi ($ per unit) 13 5

Annual shortage cost Ki ($ per unit) 120 80

Fraction bi 0.2 0.3

c=$9,000 per year; α=0.2

Int J Adv Manuf Technol (2009) 45:382–388 385

From Eq. 3, the total relevant cost, E*=$11,214.88.Now, let j=1 and k=2. After exchanging Dj with Dk,Pn

i¼1Di=Pið Þ ¼ 0:95, and the feasibility is ensured.From condition 6, 1.105>1.101.From condition 7, 2.78>0.As the both conditions are satisfied, there is a guaranteed

cost improvement with the implementation of the proposedmethod.

With the use of Eq. 4, a reduced total relevant cost afterdemand exchange, E1

*=$11,177.19.

6 Extension for shortages

Quite often, the shortages are included in a manufacturingsystem. These are assumed to be completely backordered atpresent. Figure 3 shows this kind of environment.

Since the shortages exist for a period Ji= Pi � Dið ÞþJi=Dið Þ, the annual shortage cost for an item i,

¼ Ji2

JiPi�Dið Þ þ Ji

Di

h iKiT

and the total annual shortage cost ¼ 12T

Pni¼1

KiJ 2iDi 1�Di=Pið Þ

ð8ÞNow, the maximum inventory level, Vi ¼ Pi � Dið ÞTDi=Pi � Ji

and the annual carrying cost ¼ Vi2 T � Ji

Pi�Dið Þ � JiDi

h iHiT Substitut-

ing Vi, the total annual carrying cost,

¼ T

2

Xni¼1

DiHi 1� Di=Pið Þ �Xni¼1

HiJi þ 1

2T

Xni¼1

HiJ 2iDi 1� Di=Pið Þ

ð9ÞAdding the Eqs. 8, 9, and the remaining cost components,

E ¼ cXni¼1

Di=Pið Þ1�a þ 1

T

Xni¼1

Ai þ 1

2T

Xni¼1

Hi þ Kið ÞJ 2iDi 1� Di=Pið Þ

þ T

2

Xni¼1

DiHi 1� Di=Pið Þ �Xni¼1

HiJi

ð10Þ

Substituting optimal Ji ¼ TDiHi 1� Di=Pið ÞHi þ Kið Þ ð11Þ

E ¼ cXni¼1

Di=Pið Þ1�a þ 1

T

Xni¼1

Ai þ T

2

Xni�1

DiHiKi 1� Di=Pið ÞHi þ Kið Þ

ð12ÞThe optimal values can be obtained as,

T* ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Pni¼1

Ai

Pni¼1

DiHiKi 1� Di=Pið Þ= Hi þ Kið Þ½ �

vuuuuut ð13Þ

and E* ¼ cPni¼1

Di=Pið Þ1�a

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

Pni¼1

Ai

� � Pni¼1

DiHiKi 1� Di=Pið Þ= Hi þ Kið Þ� �s

ð14Þ

With the swapping of Dj and Dk,

E*1 ¼ cXni6¼ji 6¼k

Di=Pið Þ1�a þ Dk

�Pj

� �1�a þ Dj

�Pk

� �1�a

2664

3775

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

Xni¼1

Ai

" # Xni 6¼ji 6¼k

DiHiKi 1� Di=Pið Þ= Hi þ Kið Þf g þ DkHjKj 1� Dk

�Pj

� ��Hj þ Kj

� �þ DjHkKk 1� Dj

�Pk

� ��Hk þ Kkð Þ

2664

3775

vuuuuutð15Þ

Vi

Productioninventory

Pi – Di Di

0 Time

Ji

T

Fig. 3 The production cycle with shortages

386 Int J Adv Manuf Technol (2009) 45:382–388

Following the procedure discussed in the “Mathematicalformulation” section, the relevant conditions can be

obtained. The first condition is similar to 6. The secondcondition is obtained as,

Dj � Dk

� � HjKj

Hj þ Kj

� � � HkKk

Hk þ Kkð Þ þ Dj þ Dk

� � HkKk

Pk Hk þ Kkð Þ �HjKj

Pj Hj þ Kj

� �( )" #

> 0 ð16Þ

With the input parameters of Table 1, the condition 6 isalready satisfied.

From condition 16, 1.209>0.As the both conditions (6) and (16) are satisfied, there is

certain cost improvement using the proposed approach.From Eq. 14, E*=$ 11,158.62.The reduced relevant cost from Eq. 15, E�

1 ¼ $11; 121:22.The corresponding costs are also lower than that

obtained in the previous section. This can be justified byobserving Eqs. 3 and 14. As Ki= Hi þ Kið Þ is less than 1, therelevant costs are lower with relaxation of the constraintthat the backorders would not be allowed.

7 Partial backlogging

In a real-world situation, a portion of the shortage quantitiesmay not be backordered. A particular customer may switchover to another competitive firm in the industry. However,with the advertising among other efforts, a new customercan replace the old one, at a later date. In case where theshortage costs are estimated to be a good representation of

advertising costs apportioned for unit product and loss ofprofit among other factors, an explicit computation forcontribution of the lost units of product is not necessary. Asuitable parameter for relevant cost is assumed for all theshortage quantities whether these are backlogged or not. Anannual demand needs to be adjusted in order to incorporatethe partial backordering.

From Eq. 8, the annual shortage quantity can be obtainedas,

¼Xni¼1

J 2i2TDi 1� Di=Pið Þ

A fraction bi of the shortage quantity is not backordered,and therefore, the annual manufacturing cost,

¼ cXni¼1

1

P1�ai

Di � biJ 2i2TDi 1� Di=Pið Þ

� �1�a

Equation 10 can now be adjusted as follows for thissituation,

E ¼ cXni¼1

1

P1�ai

Di � biJ 2i2TDi 1� Di=Pið Þ

� �1�a

þ 1

T

Xni¼1

Ai þ 1

2T

Xni¼1

Hi þ Kið ÞJ 2iDi 1� Di=Pið Þ þ

T

2

Xni¼1

DiHi 1� Di=Pið Þ �Xni¼1

HiJi ð17Þ

Mathematical/analytical procedure as discussed before,cannot be followed for the optimization of Eq. 17. However,conventional search process such as univariate method canbe implemented conveniently for any real data set.

7.1 Specific case

α=0 in a specific case, and the Eq. 17 can be written as,

E ¼ cXni¼1

Di=Pið Þ þ 1

T

Xni¼1

Ai þ 1

2T

Xni¼1

Hi þ Ki � cbi=Pið ÞJ 2iDi 1� Di=Pið Þ þ T

2

Xni¼1

DiHi 1� Di=Pið Þ �Xni¼1

HiJi ð18Þ

The optimal relevant cost can be obtained as,

E* ¼ cXni¼1

Di=Pið Þ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

Xni¼1

Ai

" # Xni¼1

DiHi 1� Di=Pið Þ Ki � cbi=Pið Þ= Hi þ Ki � cbi=Pið Þ" #vuut ð19Þ

Int J Adv Manuf Technol (2009) 45:382–388 387

As c/Pi is the unit production cost, and shortage costs aremuch greater than this in the real world, an optimality/feasibility condition, i.e., Ki>c/Pi, is satisfied easily.

In order to exchange the demand of products, Eq. 19 canbe used as a reference equation.

Consider the input data of Table 1.From Eq. 19, E*=$9,809.46After an exchange of the demands, the reduced relevant

cost is obtained as,

E�1 ¼ $ 9; 759:21

8 Incorporating an idle time cost

In the cyclic manufacture, a production activity usually takesplace for certain portion of the cycle time, and the remaining

portion is idle. With reference to Eq. 3,Pni¼1

Di=Pið Þ is the

annual manufacturing time. After an exchange of demand,this parameter will vary. For instance, an annual manufac-turing time has been reduced after the exchange of demandin the illustrative example of the “Illustrative example”section. This means that the idle time during the cycle hasincreased. In few cases, the problems are associated with anidle production facility such as the maintenance problems.Consistency in the quality of a product and skills of thehuman resources may also get affected up to some extent.With the occurrence of this type of problems, it seemsreasonable to introduce the idle time cost.

Consider an idle time cost per year ¼ c1 c1 < cð Þ

Idle time cost in a year ¼ c1 1�Xni¼1

Di=Pið Þ" #

Equation 3 can now be transformed as follows:

E* ¼ cXni¼1

Di=Pið Þ1�a þ c1 1�Xni¼1

Di=Pið Þ" #

þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

Xni¼1

Ai

" # Xni¼1

DiHi 1� Di=Pið Þ" #vuut ð20Þ

The above equation can be used as a reference equation forthe exchange of demand.

Similarly an idle time cost can be added in Eq. 14 withthe inclusion of shortages in a manufacturing system.

9 Concluding remarks

Almost all competitive firms in an industrial/business sectorare expected to perform in an optimal manner within the

framework of organization along with several inputparameters. However, these are continuously striving forthe cost improvement. Internal benchmarking practices arealso adopted where the standards are bound to vary withtime. A method is proposed and analyzed in which thedemand of a strategically selected item is exchanged withanother suitable item in the group. Analysis is first made forthe basic case without shortages and conditions aredeveloped for convenience in the search of another suitableitem. The process is illustrated with the help of a numericalexample. Further extensions are concerning the inclusion ofshortages that may be backlogged completely or partially.The costs are obtained at a lower level with the allowablebackorders. However, an annual shortage cost needs to beestimated with care considering the all relevant factors.

In a production cycle time, a certain period is usually idle.This idle time frequently repeats itself in case where theassociated manufacturing schedule is implemented. Idle timecost is introduced for the proposedmethod.With the inclusionof this cost, the reference equations are obtained which can beuseful for an exchange of demand. In the presence of arelevant situation, these are suitable for a trade-off concerningthe production time and idle time among other factors.

The possibilities for demand exchange can be conve-niently explored, and depending on the business strategy, theproposed approach may be implemented in a short-run/long-run. In case of the various problems being faced by the firm,an alternate schedule is available on the basis of certainmethodology. This will help in incorporating flexibility inthe industrial system and also in the decision-making processin a variety of situations.

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388 Int J Adv Manuf Technol (2009) 45:382–388