Abstract. This paper derives the optimum refilling batch size and cargo policy for an economic production measure ( EPQ ) theoretical account with multiple bringings and rework of random faulty points. The authoritative EPQ theoretical account assumes a uninterrupted stock list issue policy for fulfilling demand and perfect quality for all points produced. However, in a existent life vendor-buyer incorporate system, multi-shipment policy is practically used in stead of uninterrupted issue policy and coevals of faulty points is inevitable. It is assumed that the imperfect quality points fall into two groups: the bit and the rework-able points. Failure in fix exists, therefore extra bit points generated. The finished points can merely be delivered to clients if the whole batch is choice assured at the terminal of rework. Mathematical mold is used in this survey and the long-term mean production-inventory-delivery cost map is derived. Convexity of the cost map is proved by utilizing the Hessian matrix equations. The closed-form optimum refilling batch size and optimum figure of cargos that minimize the long-term norm costs for such an EPQ theoretical account are derived. Particular instance is examined, and a numerical illustration is provided to demo its practical use.

Keywords: EPQ theoretical account ; Replenishment batch size ; Multiple bringings ; Random faulty points ; Production ; Rework ; Scrap points.

## Introduction

In fabrication houses, the economic production measure ( EPQ ) theoretical account is normally used for finding optimum refilling batch size that minimizes entire production-inventory costs for points produced in-house [ 1-2 ] . The classical EPQ theoretical account assumes that all points manufactured are of perfect quality. However, in real-life production systems, owing to assorted governable and/or unmanageable factors, coevals of faulty points during production tally seems to be inevitable

## _____________________________

Department of Business Administration, Chaoyang University of Technology, Taichung 413, Taiwan

Department of Industrial Engineering & A ; Management ; Chaoyang University of Technology, Taichung 413, Taiwan

College of Business, Ph.D. Program & A ; Department of Industrial Engineering and Systems Management, Feng Chia University, Taichung 407, Taiwan

* . Matching writer. Electronic mail: jcyang @ cyut.edu.tw

[ 3-4 ] . Surveies have been carried out to heighten EPQ theoretical account by turn toing the issues of imperfect quality points produced every bit good as quality confidence in the production. [ 5-17 ] . Examples of such research are surveyed below. Rosenblatt and Lee [ 5 ] examined an EPQ theoretical account that deals with imperfect quality. They assumed that at some random point in clip the procedure might switch from an in-control to an out-of-control province, and a fixed per centum of faulty points are produced. Approximate solutions for obtaining an optimum batch size were developed in their paper. Yum and McDowell [ 6 ] formulated allotment of review attempt job for consecutive system as a 0-1 assorted whole number additive scheduling ( MILP ) job. Their preparation permitted any combination of bit, rework, or fix at each station. Their proposed job was solved by utilizing the standard MILP package bundles. Groenevelt et Al. [ 7 ] proposed two production control policies to cover with machine dislocations. The first policy assumes that production will non restart ( called the NR policy ) after a dislocation. The 2nd policy, production is instantly resumed after a dislocation if the on-hand stock list is below a certain threshold degree ( called the AR policy ) . Both policies assume the fix clip is negligible and they studied the effects of machine dislocations and disciplinary care on the economic batch sizing determinations. Grosfeld-Nir and Gerchak [ 11 ] considered multistage production systems where faulty units can be reworked repeatedly at every phase. The output of each phase is unsure, so several production tallies may necessitate to be attempted until the measure of finished merchandises is sufficient. The tradeoff at each phase is between utilizing little tonss, perchance asking repeated rework set-ups and big tonss, which may ensue in dearly-won overrun. They showed that a multistage system where merely one of the phases requires a set-up can be reduced to a single-stage system. They besides proved that it is best to do the “ bottle-neck ” the first phase of the system and they besides developed recursive algorithms for work outing two- and three-stage systems, where all phases require set-ups, optimally. Chiu et Al. [ 14 ] examined an EMQ theoretical account with imperfect rework and random breakdown under abort/resume policy. They employed mathematical mold and derived the integrated long-term mean production-inventory cost per unit clip. Boundaries for the optimum production run times are proposed and proved in their survey. A recursive searching algorithm was developed for turn uping the optimum tally clip within the bounds that minimizes the expected production-inventory costs.

The “ uninterrupted ” stock list issue policy for fulfilling merchandise demand is another unrealistic premise of EPQ theoretical account. In existent life vendor-buyer integrated production-inventory- bringing system, multiple or periodic cargos of finished merchandises are frequently used. Goyal [ 18 ] foremost studied the incorporate stock list theoretical account for a individual supplier-single client job. He proposed a method that is typically applicable to those stock list jobs where a merchandise is procured by a individual client from a individual provider. He gave illustration to exemplify his proposed method. Many surveies have since been carried out to turn to assorted facets of vendor-buyer supply concatenation optimisation issues [ 19-27 ] . Golhar and Sarker [ 20 ] developed a simple algorithm to calculate the optimum batch size for a system where a just-in-time ( JIT ) purchaser demands frequent bringings of little tonss of certain merchandises. They found that the generalised sum cost map for the proposed theoretical account is a piecewise convex map. When production uptime and rhythm clip are each equal to an whole number multiple of the cargo interval, a perfect matching of shipment size occurs, and for such a state of affairs, the generalised theoretical account specializes to more traditional stock list theoretical accounts. Economic impact of telling and apparatus costs decrease is besides investigated. Hill [ 22 ] studied a theoretical account in which a fabrication company purchases a natural stuff, manufactures a merchandise and ships a fixed measure of the merchandise to a individual client at fixed and regular intervals of clip. His aim is to find a buying and production agenda which minimizes entire cost of buying, fabrication and stockholding. Viswanathan [ 23 ] examined the integrated vendor-buyer stock list theoretical accounts with two different schemes: ( 1 ) each refilling measure delivered to the purchaser is indistinguishable and ( 2 ) at each bringing all the stock list available with the seller is supplied to the purchaser. As a consequence, there is no 1 scheme that obtains the best solution for all possible job parametric quantities. He besides provided consequences of a elaborate numerical probe that analyzed the comparative public presentation of the two schemes for assorted job parametric quantities. Diponegoro and Sarker [ 26 ] determined an ordination policy for natural stuffs every bit good as an economic batch size for finished merchandises that are delivered to clients often at a fixed interval of clip for a finite planning skyline. The job was so extended to counterbalance for the lost gross revenues of finished merchandises. A closed-form solution to the job was obtained for the minimum sum cost. They besides developed a lower edge on the optimum solution for job with lost sale. Because small attending has been paid to the probe of joint consequence of multiple bringings and quality confidence issues on the optimum refilling batch size and cargo policy of the EPQ theoretical account, this paper is intended to bridge the spread.

## MODEL DESCRIPTION AND MATHEMATICAL Modeling

This paper examines an economic production measure theoretical account with multiple cargos and quality confidence issues. See a fabrication system which has an one-year production rate P, and its procedure may randomly bring forth ten part of faulty points at a production rate d. All points produced are screened and review cost per point is included in the unit production cost C. The imperfect quality points fall into two groups, a I? part of them is the bit and the other ( 1-I? ) part of them is considered to be rework-able. The rework procedure starts instantly after the regular production, at a rate of P1 in each rhythm. It is non a perfect procedure either, a I?1 part ( where 0 & lt ; = I?1 & lt ; =1 ) of reworked points fails during the rework procedure and becomes bit. The one-year production rate P is assumed to be larger than the amount of one-year demand rate I» and the production rate of faulty points d. That is ( P-d-I» ) & gt ; 0, where the production rate of faulty points d can be expressed as d=Px. Let d1 denote production rate of bit points during the rework procedure, so d1 can be expressed as d1=P1I?1. Unlike authoritative EPQ theoretical account presuming a uninterrupted stock list issue policy, this survey paper considers a multi-delivery policy. It is assumed that the finished points can merely be delivered to clients if the whole batch is choice assured at the terminal of rework. Fixed measure N installments of the finished batch are delivered by petition to clients at a fixed interval of clip during production downtime t3 ( see Figure 1 ) . Extra notation used in this paper is given as follows.

## [ Insert Figure 1 about here ]

Q = fabrication batch size, to be determined for each rhythm,

n = figure of fixed measure installments of the finished batch to be delivered by petition to clients, to be determined for each rhythm,

T = rhythm length,

H1 = maximal degree of on-hand stock list in units when regular production procedure ends,

H = the maximal degree of on-hand stock list in units when rework procedure coatings,

t1 = the production uptime for the proposed EMQ theoretical account,

t2 = clip required for reworking of faulty points,

t3 = clip required for presenting all quality assured finished merchandises,

Tennessee = a fixed interval of clip between each installment of finished merchandises delivered during production downtime t3,

I† = overall bit rate per rhythm ( amount of bit rates in t1 and t2 ) ,

C = unit production cost,

K = apparatus cost,

H = unit keeping cost,

CR = unit rework cost,

CS = disposal cost per bit point,

h1 = keeping cost for each reworked point,

K1 = fixed bringing cost per cargo,

CT = bringing cost per point shipped to clients,

h2 = keeping cost for each point kept by client

Is ( T ) = on-hand stock list of bit points at clip T,

I ( T ) = on-hand stock list of perfect quality points at maker ‘s terminal at clip T,

Id ( T ) = on-hand stock list of faulty points at clip T,

Ic ( T ) = on-hand stock list of perfect quality points at client ‘s terminal at clip T,

TC ( Q, n ) = entire production-inventory-delivery costs per rhythm for the proposed theoretical account,

TC1 ( Q, n ) = entire production-inventory-delivery per rhythm when no faulty points produced ( the particular instance: the authoritative EPQ theoretical account with a multi-delivery policy ) ,

E [ TCU ( Q, n ) ] = the long-term norm costs per unit clip for the proposed theoretical account,

E [ TCU1 ( Q, n ) ] = the long-term norm costs per unit clip for the particular instance.

The undermentioned simple equations can be obtained straight from Figure 1:

( 1 )

( 2 )

( 3 )

( 4 )

( 5 )

( 6 )

The on-hand stock list of faulty points produced during the production uptime t1 are as follows ( see Figure 2 ) . Among them a I? part is scrap and the other ( 1-I? ) part of faulty points is considered to be rework-able.

( 7 )

## [ Insert Figure 2 about here ]

During the rework procedure, a part I?1 of reworked points fails and becomes bit. Figure 3 depicts the on-hand stock list of scrap points during t1 and t2. One notes that maximal degree of scrap points I†xQ is

( 8 )

## [ Insert Figure 3 about here ]

During bringing clip t3, n fixed-quantity installments of the finished batch are delivered to clients at a fixed interval of clip. Cost for each bringing is:

( 9 )

And entire bringing costs for n cargos in a rhythm are:

( 10 )

Entire keeping costs of finished merchandises during t3 at maker ‘s terminal can be obtained as follows ( mention to Appendix-A ) .

( 11 )

Entire retention costs for points kept at client ‘s terminal are as follows ( see Figure 4 and besides refer to Appendix-B ) .

. ( 12 )

## [ Insert Figure 4 about here ]

Entire production-inventory-delivery cost per rhythm TC ( Q, n ) consists of variable production cost, apparatus cost, variable rework cost, disposal cost, fixed and variable bringing cost, keeping cost at the maker ‘s terminal during production uptime t1, make overing clip t2, and bringing clip t3, variable keeping cost for points reworked, and keeping cost at the client ‘s terminal for finished goods during the bringing clip t3. Therefore, the overall production-inventory-delivery cost per rhythm TC ( Q, N ) is

( 13 )

Because the proportion ten of faulty points is assumed to be a random variable with a known chance denseness map, in order to take the entropy of faulty rate into history, the expected values of ten can be used in the cost analyses. Substituting all related variables from Eqs. ( 1 ) to ( 12 ) in Eq. ( 13 ) and using the reclamation wages theorem, the expected production-inventory-delivery cost per unit clip E [ TCU ( Q ) ] can be obtained ( see Appendix-C for inside informations ) :

( 14 )

## DERIVATIONS OF OPTIMAL REPLENISHMENT LOT SIZE & A ; SHIPMENT POLICY

For cogent evidence of convexness of E [ TCU ( Q, n ) ] one could use Hessian matrix equations ( Rardin [ 28 ] ) and obtains the undermentioned derived functions:

( 15 )

( 16 )

( 17 )

( 18 )

( 19 )

Substituting Eq. ( 15 ) through ( 19 ) in the undermentioned Hessian matrix equations [ 28 ] and with farther derivations, one can obtain

( 20 )

Because K, I» , Q, and ( 1-I†E [ x ] ) are all positive, so Eq. ( 20 ) is positive. Hence, E [ TCU ( Q, n ) ] is a purely bulging map for all Q and n different from nothing. It follows that for the optimum refilling batch size Q* and optimum figure of bringing n* , one can distinguish E [ TCU ( Q, n ) ] with regard to Q and with regard to n, and work out the additive system of the aforesaid Eqs. ( 15 ) and ( 17 ) by first puting these partial derived functions equal to zero.

With farther derivations, one obtains the optimum refilling batch size Q* and the optimum figure of bringing n* as shown in Eqs. ( 21 ) and ( 22 ) below.

( 21 )

( 22 )

## SPECIAL CASE

Suppose all points produced are of perfect quality ( i.e. x=0 ) , the proposed EPQ theoretical account becomes the same as the authoritative EPQ theoretical account with a multi-delivery policy. On-hand stock list of perfect quality point is depicted in Figure 5.

## [ Insert Figure 5 about here ]

Entire production-inventory-delivery cost per rhythm TC1 ( Q, N ) is

( 23 )

The expected production-inventory-delivery cost E [ TCU1 ( Q, n ) ] for this particular theoretical account can be derived as follows.

( 24 )

Convexity of E [ TCU1 ( Q, n ) ] can be proved as shown in Eq. ( 25 ) , and optimum batch size Q* and optimum figure of bringing n* can besides be derived consequently as shown in Eqs. ( 26 ) to ( 27 ) .

( 25 )

and

( 26 )

. ( 27 )

## Numeric EXAMPLE

Assume a merchandise can be manufactured at a rate of 60,000 units per twelvemonth and it has experienced a level demand rate of 3,400 units per twelvemonth. During production uptime, the random faulty rate ten is assumed and it follows a unvarying distribution over the interval [ 0, 0.3 ] . Among faulty points, a part I? =0.1 is considered to be scrap and the other part is considered to be rework-able with a fix rate P1=2,100 units per twelvemonth. During the rework procedure, a part I?1=0.1 of reworked points fails and becomes bit.

It is besides assumed that finished points can merely be delivered to clients if the whole batch is choice assured at the terminal of rework. Fixed measure N installments of the finished batch are delivered by petition to clients at a fixed interval of clip during the bringing clip t3 as depicted in Figure 1. Extra values of parametric quantities used in the illustration are given as follows.

K1 = $ 2,000 per cargo, a fixed cost,

CT = $ 0.1 per point delivered,

C = $ 100 per point,

CS = $ 20, disposal cost for each bit point,

CR = $ 60, repaired cost for each point reworked,

K = $ 20,000 per production tally,

H = $ 20 per point per twelvemonth,

h1 = $ 40 per point reworked per unit clip ( twelvemonth ) ,

h2 = $ 80 per point kept at the client ‘s terminal per unit clip.

The optimum figure of cargos n*=3 can be obtained from Eq. ( 22 ) , so by utilizing Eq. ( 21 ) one has the optimum refilling batch size Q*=1,735. The long-term norm cost map E [ TCU ( Q* , n* ) ] = $ 485,541 can besides be obtained from Eq. ( 14 ) . The optimum refilling batch size and shipment policies for the particular instance Q*=2,018 and n*=3 can besides be computed by utilizing Eqs. ( 27 ) and ( 26 ) . Using Eq. ( 24 ) , one has the long-term norm cost for the particular instance E [ TCU1 ( Q* , n* ) ] = $ 427,938.

## CONCLUDING Remark

Authoritative EPQ theoretical account assumes a uninterrupted stock list issue policy for fulfilling merchandise demand and a perfect quality production for all points produced during the production procedure. However, in real-life vendor-buyer integrated production-inventory-delivery system, discontinuous issue policy is frequently used and coevals of nonconforming points during a production tally is inevitable. This paper investigates the aforesaid issues by integrating a multiple bringing policy and the quality confidence into EPQ with rework and failure in fix. Mathematical mold is employed here, and the long-term mean production-inventory-delivery cost map is derived and proved to be convex. The closed-form solutions in footings of optimum refilling batch size and optimum figure of cargos to the job are obtained. It may be noted that without an in-depth probe and robust analysis of such a realistic system, the optimum production-shipment policies can non be revealed. For future survey, one interesting subject among others will be to analyze the consequence on the same determinations when deficit with backlogging is permitted.

## Recognitions

The writers greatly appreciate the support of the National Science Council ( NSC ) of Taiwan under grant No. NSC-99-2410-H-324-007-MY3.