Capacity Planning to Optimize the Proﬁt of an Apparel ...

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Capacity Planning to Optimize the Profit of an Apparel

Manufacturer in Developing Country∗

Kader Md. MOHAFIQUL∗∗, Hirohisa NARITA∗∗∗,Lian-Yi CHEN∗∗∗∗ and Hideo FUJIMOTO∗∗∗∗

The export oriented apparel manufacturers of developing countries produce apparel onthe make-to-order basis instead of make-to-sell. As a result, there is no chance of stockout ofgoods that could be sold or overstock that has to be liquidated at deep discount at the end ofthe period. But an analogy can be drawn between stock of goods and capacity of productionas idle capacity or shortage of capacity exists frequently while matching capacity with ordersobtained. Considering capacity as a resource in inventory to be used in the future this paperapplies newsboy problem to decide its use in production.

Key Words: Optimization, Newsboy Problem, Capacity Planning, Apparel Manufacturer

1. Introduction

Traditional production systems produce products andstock them as inventory until they are sold (make-to-stock). In order to reduce inventory and increase the levelof customization, some firms have designed their produc-tion systems to produce a product only after it is ordered.Such systems are referred to as make-to-order. The ready-made garments (RMG) or apparel manufacturers of devel-oping countries like Bangladesh adopt one or both of thesystems mentioned above on purpose. Some of the fac-tories serve the domestic demand and they use the make-to-stock systems. Some serve the export demand and usemake-to-order system. This paper is concerned with theexport oriented firms capacity planning.

World’s apparel industry has a very complex supplychain. It is not uncommon for design, production of fab-ric, cutting and sewing of garments, and sales in retail out-lets to take place in different countries spread all over theglobe(1). The apparel manufacturers of Bangladesh do thecutting and sewing of garments according to the ordersand specification of foreign buyers. As a result, there is

∗ Received 30th July, 2004 (No. 04-4178)∗∗ Department of Computer Science and Engineering, Grad-

uate School of Engineering, Nagoya Institute of Technol-ogy, Gokiso-cho, Showa-ku, Nagoya 466–8555, Japan.E-mail: [email protected]

∗∗∗ Tsukuri College, Graduate School of Engineering, NagoyaInstitute of Technology

∗∗∗∗ Omohi College, Graduate School of Engineering, NagoyaInstitute of Technology

no chance of stockout of goods that could be sold or over-stock that has to be liquidated at deep discount at the endof the period or to carry the unsold goods as inventory tothe next period. Usually this happens to the outlet of theforeign buyers.

Export oriented production entirely depends on theorders received from the foreign buyers. There are sev-eral steps starting from the negotiation of the orders withforeign buyers to the shipment of the products, but the im-portant ones are shown in Fig. 1. In this whole process,capacity planning plays a significant role, because a firmmust ensure that the committed orders are delivered timelyto the customer.

Depending on a firm’s size, marketing skill and lengthof experience, it may lease its production facility to othersor rent from others. This paper deals with the decisionmaking of export-oriented firms regarding their capacityutilization during production. This paper considers pro-duction facility as an asset to utilize in the future and usesthe newsboy technique to make decision.

The rest of the paper is followed by a literaturereview in section 2, apparel manufacturing scenario of

Fig. 1 Steps in the apparel manufacturing

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Bangladesh in section 3, mathematical model and the so-lution in section 4, a numerical example in section 5. Fi-nally, conclusion and direction of future research are insection 6.

2. Literature Review

Several researches are carried out about the problemsof apparel manufacturing. The impact of working capitaland inventory holding costs on a firm’s stockout and mark-down costs is discussed by Raman and Kim(2). Demanduncertainty and overstocking and understocking cost offirms whose production of garments occurs in variouscountries like Hong Kong, China, Jamaica, Bangladesh,is dealt by Fisher and Raman(1). Prospects, marketing andstatus of textile along with readymade garments (RMG) ofBangladesh is discussed in Bhuiyan and Hossain(3), Haqueand Kader(4), and Chowdhury(5). The line balancing ofRMG industry in Bangladesh is studied by Mohiuddin andKader(6). None of these papers deals with capacity plan-ning like that in this paper.

On the other hand, most papers on newsboy problemdiscuss the pricing, ordering and marketing policies andseldom involve the policy of capacity planning. In pric-ing policy, the case of progressive discount to sell the ex-cess inventory is studied by Khouja(7). An extension ofthe problem, multi-product resource constrained situationis dealt by Khouja and Mehrez(8). In ordering policy, a sit-uation with two orders in a period, is considered by Lauand Lau(9), (10). A newsboy problem with centralized ware-house that outperforms one with multi-location is dealt byCherikh(11). Considering the above situations, this paperpresents the application of newsboy problem in differentfield, like capacity planning.

3. RMG and Its Growth in Bangladesh

Bangladesh — the country of world famous muslinfabric and the Great Royal Bengal Tiger has now emergedas a child labor free, apparel giant in the world textileand apparel market. The country exports its apparel prod-ucts worth nearly 5 billion US$ per year to the USA,EU, Canada and other countries of the world. This isabout 75% of its annual national exports earning(4). Atpresent the country is the 6th largest apparel supplier tothe USA and EU countries. The major products are Knitand Woven Shirts and Blouses, Trousers, Skirts, Shorts,Jackets, Sweaters, Sportswear and many more casual andfashion apparels.

Starting in late 70 s as a small nontraditional sec-tor of export, Ready-made Garment (RMG) emerged asa promising export-earning sector of the country by theyear 1983. At that time there were only 46 industries asmembers of Bangladesh Garments Manufacturers and Ex-porters Association (BGMEA), which increased to 1 650only in 10 years in the year 1993(6). Today BGMEA is

Fig. 2 RMG exports of Bangladesh

Fig. 3 Order collection and production planning

proud to have about 4 000 garment manufacturers and ex-porters as its members. Figure 2 provides the RMG exportdata of Bangladesh.

4. Mathematical Model

Apparels and fashion change according to two mainseasons, summer and winter. Accordingly the productionin the industry changes in its types, and the manufacturerconsiders two periods for production, summer and winter.In the mathematical model, a firm is considered that pro-duces a single product in each period. All the orders itcollects in a period and before, is planned to produce inthe next period. But the planning of capacity is done priorto the period of order collection as shown in Fig. 3. Thisadvanced decision about capacity planning helps the firmin many ways, namely (i) the decision maker gets a num-ber of firms to be chosen for its capacity leasing, (ii) getsthe opportunity to bargain for the contract, (iii) less risk ofbeing remain idle, etc.

Considering a single product for a single period, letX – random variable representing the sum of orders re-ceived in a period, follows N(µ,σ2),k0 – upper limit of the in-house capacity,

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Fig. 4 Leasing model of capacity

K – capacity decided to use in the period for own order,thus when K < k0, k0−K is the leased capacity and whenK > k0, K−k0 is the rented capacity,h(x) – density function of X,p – profit from unit capacity employed for own production,l – profit from unit leased capacity,cr – unit rental cost of capacity,cp – shortage capacity penalty cost (e.g. incurring from theloss of goodwill).

As mentioned earlier, a firm may lease out its capac-ity to avoid being remained idle or rent capacity to avoidshortage of capacity required to produce goods ordered.Both of the cases will be discussed in this paper separately.

4. 1 Leasing modelCapacity is leased when either (i) a firm is newly es-

tablished or (ii) having weak marketing force, or (iii) ca-pacity is relatively large. Whatever the firm receives asorders for a period does not exceed its capacity. If the firmdoes not lease out its capacity to other firms its part of thecapacity may remain idle losing the revenue from lease.By leasing, the decision maker commits to produce part orwhole of the order of its client factory and reduces its riskof remaining idle. The revenue thus received producinga unit, l, is lower than that of its own order, p. Thus ifthe firm leases much capacity and misses the opportunityto produce own orders, it loses (p− l) per unit capacity.Since the amount of total order is not known at the be-ginning of the period it may be lower or higher than Kasshown in Fig. 4.

The profit function∗1 is

P(K)=xp+ (k0−K)l, x<K

=K p+ (k0−K)l− (x−K)cp, x≥K (1)

The expected profit is

E{P(K)}=(k0−K)l+ p∫ K

0xh(x)dx

+K p∫ ∞

Kh(x)dx−cp

∫ ∞

K(x−K)h(x)dx

(2)

∗1 The cost for maintaining the facility will be same irrespec-tive of the order size.

It can be shown that expected profit is concave downwardon K and thus from first derivative we get∫ K

0h(x)dx=P(x≤K∗)=

p+cp− l

p+cp(3)

4. 1. 1 Analysis of solution As long as p+cp > l,(3) yields the optimal capacity to use in the period underconsideration. Let Φ(.) be the standard normal distribu-tion, φ(.) be its density function, and Φ−1(.) be its inversefunction.∫ K

0h(x)dx=P(x≤K∗)=

p+cp− l

p+cp=Φ

(K∗ −µσ

)

=

∫ t1

0φ(x)dx; t1 =

K∗ −µσ

K∗=max

{µ+σΦ−1

(p+cp− l

p+cp

),o

}(4)

It can be shown by taking appropriate derivatives thatwhen the unit revenue of own order, p and unit shortagepenalty cost, cp increase, it is better to use more capacityto increase profit or to avoid capacity shortage. On theother hand if unit revenue of leased capacity, l, increasesmore capacity is leased to avoid decrease of profit.

After simplification with the method proposed by Sil-ver and Peterson(12), the expression for expected profit, (2)becomes

E{P(K)}= (k0−K)l+ pµ− (p+cp)σGu(t) (5)

4. 2 Rental modelThis is the opposite situation of leasing. Capacity is

rented when (i) the firm is relatively old in the industry, (ii)having a strong marketing force, (iii) capacity is relativelysmall. Since it cannot produce all the orders by its owncapacity, it contracts others to produce for it. The firmreceives a revenue as it receives from own capacity, but ithas to pay the rent, cr, to the firm it rents capacity from.As the firm rents other firms to make the excess ordersthat it cannot produce by its own capacity during a certainperiod, it should not rent too much or too less. In the firstcase, its rented capacity remains idle and incur loss payingthe rent. In the second case it may lose revenue, as itscapacity does not permit to produce potential orders.

According to the two situations shown in Fig. 5 theprofit function is

P(K)=xp− (K−k0)cr, x≤K

=K p− (K−k0)cr− (x−K)cp, x>K (6)

Expected profit function is

E{P(K)}=(k0−K)l+ p∫ K

0xh(x)dx

+p∫ ∞

KKh(x)dx−cp

∫ ∞

K(x−K)h(x)dx

(7)


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Fig. 5 Rental model of capacity

Table 1 Comparison of the model parameters and results

To find the maximizing value we setdE{P(K)}

dx= 0

and get∫ K

0h(x)dx=

p+cp−cr

p+cp.

As shown in the leasing model, using thestandard normal distribution, we can show, thevalue of K maximizing expected profit is K∗ =

max

{µ+σΦ−1

(p+cp−cr

p+cp

),o

}, and the expected profit

function is

E{P(K)}=−(K−k0)cr+ pµ− (p+cp)σGu(t) (8)

The expressions for capacity decided to use, K, ex-pected profit for that capacity, E{P(K)}, and variation ofcapacity K with p, cp, l and cr are shown in Table 1.

5. Numerical Example

The examples of the two situations are provided sep-arately. Leasing is considered first. From a certain ca-pacity when a firm leases out, its decided capacity to usedecreases. In this case its maximum capacity may be theits own capacity when it decides not to lease at all. On theother hand, when a firm rents capacity its decided capac-ity to use for production increases. Thus the lower limit ofcapacity used is its own capacity when renting no capacity.

5. 1 Leasing model (L-M)Suppose a firm, with a capacity of producing 5 000

units of shirts in summer period, wants to decide about its

Table 2 Capacity used and expected profit (L-M)

Fig. 6 Variation of expected profit (L-M)

capacity leasing. The past record shows that total ordersize follows N(4 000,1 0002). It earns 5$ when it producea shirt from its own order and earns 4$ from its leasing.The shortage penalty cost is ignored.

From calculation the optimum capacity to use for ownorder production is 3 155 units. So it can go for a contractof making 1 845 units maximum to optimize the profit.The tabular calculation and graphical variation of the ex-pected profit with capacity used for own order productionare shown in Table 2 and Fig. 6 respectively.

This value of K, and the expected profit will changeif the value of p or l changes. For example, when the rev-enue from using unit capacity for own order, p increasesthe firm loses the probable revenue p− l per unit capacity.Thus when all others remain constant, and p increases,the capacity decided to use, K also increases, to avoid theshortage of capacity when order is available. Similarly,when revenue from the leased unit capacity, l increases,the value of p− l decreases. That permits the decisionmaker to lease more, and as a result K decreases. These


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(a) As l changes

(b) As p changes

Fig. 7 Changes of K and E{P(K)} with p and l (L-M)

facts are shown in the following graphs (Fig. 7).5. 2 Rental model (R-M)

Suppose a firm, with a capacity of producing 5 000units of shirts in summer period, wants to decide about itscapacity leasing. The past record shows that total ordersize follows N(7 000,1 0002). It earns 5$ when it producea shirt from its own order and gives 3$ rent for each unit.The shortage penalty cost is ignored.

As the average value of orders received is greater thanthe capacity of the firm it must rent capacity to satisfythe demand. By doing this it earns more revenue andapproaches its expected optimum. But it should not rentmuch that its capacity become greater than its total ordersobtained in the period, thus causing the capacity remainedidle. Thus with the normally distributed demand, whenthe firm rents 6 800 unit capacity, it maximizes its profit to27,066$ (curve with cr =3.0 in Fig. 8).

Fig. 8 Changes of E{P(K)} with K at different cr (R-M)

Fig. 9 Profit difference when K is set at average order size

6. Conclusion

In the apparel manufacturing industry it is a verycommon practice to lease out and rent capacity to matchcapacity with the orders received in a period. This alsohappens in many other industries when capacity of a firmis not sufficient to manufacture all the orders obtained.This paper models the situation of the renting and leas-ing of capacity of a firm producing a single product. Thisstudy shows that mere depending on the averages in ca-pacity planning decision in apparel manufacturing firmsproducing make-to-order basis is misleading, as it doesnot optimize the expected profit (Fig. 9). Following arethe findings of the study:

( 1 ) The profit maximizing capacity can be found outsystematically, which is not necessarily the average of theorders received in the period.

( 2 ) When p and cp increase K should be increase toearn p− l (during leasing) or p−cr (during renting) moreper unit of capacity and to avoid shortage penalty cost andvice versa.

( 3 ) When l or cr increases K should be decreasedwith leasing more capacity or renting less capacity. Con-sequently the risk of capacity being idle will decrease.

Though the study enhances capacity planning using


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newsboy problem in a very simple fashion, it seems thatthere is a very good chance to extend the research far-ther in future considering multiple products with capac-ity constraints. Another approach of importance is that ofdecision-making whether to lease or rent for a single firm.

References

( 1 ) Fisher, M. and Raman, A., Reducing the Cost ofDemand Uncertainty through Accurate Response toEarly Sales, Operations Research, Vol.44, No.1 (1996),pp.87–99.

( 2 ) Raman, A. and Kim, B., Quantifying the Impact ofInventory Holding Cost and Reactive Capacity on anApparel Manufacturer’s Profitability, Production andOperations Management, Vol.11, No.3 (2002), pp.358–373.

( 3 ) Bhuiyan, M.A. and Hossain, M.T.Z., Prospects of Man-Made Fiber Production in Bangladesh in View of Southand Southeast Asian Countries, Presented in the Pro-ceedings of International Conference on Manufactur-ing (ICM) 2002, Dhaka, Vol.2 (2002), pp.629–637.

( 4 ) Haque, A.S.M.S. and Kader, M.M., Marketing ofExport Quality Yarn in Bangladesh: Problems andProspects, Presented in the Proceedings of Interna-tional Conference on Manufacturing (ICM) 2002,Dhaka, Vol.2 (2002), pp.593–601.

( 5 ) Chowdhury, A.H., Present Status of Textile Sector inBangladesh and Its Future Prospect, Presented in the

Proceedings of International Conference on Manufac-turing (ICM) 2002, Dhaka, Vol.2 (2002), pp.654–660.

( 6 ) Mohiuddin, G. and Kader, M.M., Line Balancing andthe Garments Industries of Bangladesh, Presented inthe 38th Annual Convention of IEB, Dhaka, 18-22 Jan-uary 1994, Published in the Journal of AOTS, BAAS,(1994), pp.119–132.

( 7 ) Khouja, M., The Newsboy Problem under ProgressiveMultiple Discounts, European Journal of OperationalResearch, Vol.84 (1995), pp.458–466.

( 8 ) Khouja, M. and Mehrez, A., A Multi-Product Con-strained Newsboy Problem with Progressive Multi-ple Discounts, Computers and Industry Engineering,Vol.30 (1996), pp.95–101.

( 9 ) Lau, H.-S. and Lau, A.H.-L., Reordering Strategies fora Newsboy-Type Product, European Journal of Opera-tions Research, Vol.103 (1997). pp.557–572.

(10) Lau, H.-S. and Lau, A.H.-L., A Semi-Analytical So-lution for a Newsboy Problem with Mid-Period Re-plenishment, Journal of Operational Research Society,Vol.48 (1997), pp.1245–1253.

(11) Cherikh, M., On the Effect of Centralization on Ex-pected Profits in a Multi-Location Newsboy Prob-lem, Journal of Operational Research Society, Vol.51(2000), pp.755–761.

(12) Silver, E.A. and Peterson, R., Decision Systems forInventory Management and Production Planning, 2ndEd., (1985), pp.398–422, John Wiley & Sons.


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