Multi-stage cleaner production process with quality improvement and lead time dependent ordering cost

Accepted Manuscript Multi-stage cleaner production process with quality improvement and lead time dependent ordering cost Min-Soo Kim, Biswajit Sarkar PII:

S0959-6526(16)31890-X

DOI:

10.1016/j.jclepro.2016.11.052

Reference:

JCLP 8441

To appear in:

Journal of Cleaner Production

Received Date: 2 June 2016 Revised Date:

3 November 2016

Accepted Date: 8 November 2016

Please cite this article as: Kim M-S, Sarkar B, Multi-stage cleaner production process with quality improvement and lead time dependent ordering cost, Journal of Cleaner Production (2016), doi: 10.1016/j.jclepro.2016.11.052. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Multi-stage cleaner production process with quality improvement and lead time dependent ordering cost Min-Soo Kima, b, Biswajit Sarkara,

Department of Industrial & Management Engineering, Hanyang University, Ansan Gyeonggi-do, 15588, South Korea. b

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a

†

Department of Applied Mathematics, Hanyang University, Ansan

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Gyeonggi-do, 15588, South Korea.

Abstract

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To maintain a multi-stage cleaner production process, the major task is eliminating all defective items during the production system and an investment is necessary to reduce the probability of in-control state to outof-control state of machinery system. In this direction, a logarithmic expression suggested by Porteus (1986) to consider in a single-stage imperfect manufacturing process for quality improvement, whereas this paper enables to consider the similar investment in a complex multi-stage imperfect manufacturing process to clean the

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production system. Using it, this paper develops a joint replenishment problem for complex multi-stage quality improvement. The study investigates a stochastic inventory model with a budget constraint for simultaneously optimizing number of shipments, replenishment interval, safety factor, backorder discounts, quality factor, and

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lead time as decision variables. Lead time is assumed as stochastic in nature, where a lead time crashing cost is used to reduce the lead time. As lead time is stochastic, a backorder price discount is allowed to save lost sells. To solve this problem, an improved algorithm is developed and two theorems are proved to obtain global

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optimal solution for this model analytically. Finally, some numerical examples and graphical illustrations are given to illustrate this model.

Keywords: Multi-stage cleaner production; Inventory; Quality improvement; Joint replenishment problem; Controllable lead time.

†

Corresponding author: Email: [email protected] (Biswajit Sarkar) Address: Department of

Industrial & Management Engineering, Hanyang University, Ansan Gyeonggi-do, 15588, South Korea. Office Phone: +82-31-400-5259, Fax: +82-31-436-8146.

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1. Introduction Traditional inventory models didn’t consider quality factor, assuming that the quality of production process is always perfect. However, in practice, the production process usually produces defective products due to some

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problems such as equipment breakdowns, labor problems, and long-run of machinery systems. Thus, Porteus (1986) suggested a logarithmic investment function within the imperfect production process to improve the quality of products and many researchers have studied on single-stage imperfect production process using it

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since last three decades. Nowadays, as the production process, becomes more complex, it becomes necessary to consider complex multi-stage imperfect production process, i.e., quality improvement of products in a complex multi-stage production system.

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This paper suggests a way to clean the complex multi-stage production system from defective items, which enables to consider a complex multi-stage system with quality improvement in a joint replenishment problem as an example that can be used in high-level industries such as electronic and automotive industries. This study has some important aspects as this paper (1) suggests a concept of complex multi-stage with quality improvement, (2) takes a first step towards single-stage quality improvement to a complex multi-stage system with quality

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improvement, (3) obtains quantity of produced defective products in periodic review inventory model for complex multi-stage problem, and (4) develops an inventory model with an imperfect complex multi-stage process and some significant features.

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In this direction, Wee et al. (2007) developed an inventory model for product’s quality and backorder. They used rework for cleaning of their production system. Sana (2010) introduced the optimum products reliability in an Sana

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imperfect manufacturing system with confirming and non-confirming products by no-resumption policy.

(2011) extended an integrated-production inventory model with the presence of imperfect and perfect items in a single-stage production system by highlighting the impact of business strategies like textile or chemical. Cárdenas-Barrón (2012) wrote a note on the simplification of another imperfect production model with some basic formulas with integer value of discrete variables. Taleizadeh et al. (2013) minimized the joint total cost for a multi-item production system with rework cost. They solved the model analytically. Wee and Widyadana (2013) developed a production model for deteriorating products with rework and stochastic preventive maintenance time. A production process (i.e., a machine or any work) often produces many sorts of products. For this reason, if

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ACCEPTED MANUSCRIPT decision maker invests to reduce process drift rate for a product i at stage j (i.e., machine j or any work j), then defective rate of other products which shares stage j also reduces. This concept is distinguished from quality improvement in a multi-stage production process and has important meaning in terms of the production process produces a lot of sorts of products. The proposed model suggest to improve this concept for a complex multi-

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stage production system with quality improvement. Porteus (1986) considered quality improvement by decreasing the probability when process moves to out-of-control state. Hong et al. (1993) and Hwang et al. (1993) developed inventory models with both quality improvement and setup cost reduction. Gunasekaran et al. (1995) discussed an inventory model for multi-stage system that manufactures multiple products to calculate an

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economic production quantity (EPQ). Ouyang and Chang (2000) investigated an inventory model with reorder point, lot size, lead time, and process quality as decision variables. Based on Moon and Choi’s (1998) model,

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Ouyang et al. (2002) investigated the inventory model with controllable lead time and partial backorder to calculate the optimal reorder point and optimal lot size. Lee (2008) extended a single-stage production model to a multi-level multi-stage inventory model, however he did not consider multi-product, but he considered assembly and components of single-product in a multi-level production system. Chakraborty et al. (2009) investigated the effects of process deterioration, inspections, and machine breakdown on optimal lot sizing

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decisions. Sarkar et al. (2010a) introduced a variable safety stock in a production system, which always produces perfect products by considering system reliability. Sarkar et al. (2010b) extended their own model (2010a) for the imperfect production system, where the system produces defective items and the defective items

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are reworked to make them perfect. Corrective and preventive both maintenance costs were used to restore the system in its original state.

Diaby et al. (2013) considered an inventory model to reduce defective rates and

setup times in a just-in-time (JIT) environment. Shah et al. (2013) developed an inventory model with non-

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instantaneous deteriorating products to maximize the total profit. Dey and Giri (2014) investigated an integrated-inventory model with imperfect production process to determine order quantity, reorder point, number of shipment per lot, and quality level. Pal et al. (2014) developed a production-inventory model with two cycles though they did not consider shortages. Sarkar and Moon (2014) extended Ouyang et al.’s (2002) model with variable backorder rate. They used Porteus’s (1986) logarithmic investment function for quality improvement and setup cost reduction. Pasandideh et al. (2015) developed an inventory model for multi-product single-stage lot size problem in an imperfect production system. Sarkar et al. (2015a) developed an inventory model with service level constraint. They considered setup cost, order quantity, reorder point, and quality factor as decision variables in a distribution free environment with known mean and standard deviation. Sarkar et al.

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ACCEPTED MANUSCRIPT (2015b) developed two continuous review inventory models with backorder discount, safety factor, controllable lead time and quality factor as decision variables, when lead time demand follows a normal distribution and doesn’t follow any distribution, but with known mean and standard deviation. In inventory model with multiple products, one of the main aim is to reduce total cost by jointly orders of

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several products at arranged replenishment interval. The joint replenishment problem (JRP) model is basically an inventory model, which considers multi-product using above concept to decrease the total cost. Some researchers often assume the JRP model without any resource constraints. However, in the real world, there are lots of restrictions in production-inventory systems (for instance budget, transportation, storage capacity, etc.).

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Therefore, in order to develop model with these practical issues, researchers should consider restrictions such as limited budget and limited space and to solve this problem, researchers should develop some efficient algorithm

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to solve the model. In this direction, Goyal (1975) introduced a JRP model with resource restriction and made a heuristic algorithm by using Lagrange multiplier method. Haksever and Moussourakis (2005) developed a mixed-integer programming model with multi-product and multi-resource constraints to calculate optimal order quantity and optimal replenishment interval. Moon and Cha (2006) developed two algorithms for JRP model with resource constraints. First algorithm was developed by modifying RAND algorithm and second algorithm

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was developed by using genetic algorithm. Hoque (2006) discussed about a JRP model with storage capacity, transport capacity, and budget constraint to calculate optimal replenishment interval of products. Recently, Amaya et al. (2013) developed a heuristic algorithm to solve a joint replenishment problem with resource

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constraints and they compared with C-RAND algorithm developed by Moon and Cha (2006). Some researchers often assume that stockouts are either totally backlogged or totally lost. However, in real market, because of many factors (for instance supplier’s reputation, customer’s loyalty, etc.), some customers

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may wait until replenishment when shortage occurs, whereas others may not wait until replenishment order arrives. That implies supplier has missed the opportunity to make more profit. If the supplier offers backorder price-discount to customer, the supplier made fewer losses than shortage losses. It is obvious that the supplier has a chance to earn more from a cost minimize perspective. In this direction, based on Ouyang et al.’s (1996) model, Pan and Hsiao (2001) discussed both backorder price-discount and order quantity as decision variables. Pan et al. (2004) considered a continuous review inventory model with order quantity, backorder discount, safety factor, and lead time as decision variables. Later, Pan and Hsiao (2005) proposed two inventory models, where the first model’s demand follows a normal distribution and second model’s demand doesn’t follow any specific distribution, but it has with known mean and variance. Ouyang et al. (2007) investigated a periodic

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ACCEPTED MANUSCRIPT review model to find out a relationship between ordering cost reduction and lead time. Taleizadeh et al. (2010), Taleizadeh et al. (2010), and Taleizadeh et al. (2010) developed three multi-product production model with limited production quantity, multi-chance constraints and stochastic replenishment rate, respectively. Lin (2010) developed integrated model to determine review period, lead time, backorder discount, and number of

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transportation. Taleizadeh et al. (2012), Taleizadeh et al. (2013), Taleizadeh et al. (2013), and Taleizadeh et al. (2013) developed several inventory models in this directions. Cárdenas-Barrón et al. (2014) wrote the contributions of the inventory models during one century of basic economic order quantity model. Sarkar et al. (2015b) discussed a continuous review model with reorder point, order quantity, backorder price- discount, lead

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time and process quality as decision variables. Considering variable backorder, Sarkar and Majumder (2013), and Sarkar (2016) developed two supply chain models with multi-stage inspections, discount policies and set up

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cost reduction of vendor respectively, where the defective items plays an important role for backorder. Sarkar and Saren (2016) developed a product inspection policy for imperfect production system, where the uninspected items are sold through warranty policy with the assumptions that they did not consider backorder. Considering random defective products through deterioration or production, Wee et al. (2013), Sarkar (2013, 2014, 2016), Sarkar (2016), and Kang et al. (2016) developed several production models with single-stage production system,

production system.

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whereas Tayyab and Sarkar (2016) relaxed the assumptions of variable backorder within multi-stage cleaner

In the competitive global market, one of the way to draw customer’s attention is quick service for the ordered

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products. If a supplier can deliver goods within a short period of time comparing to other suppliers, it may gain another opportunity to take future order. For this reason, lead time reduction is an important factor in the modern

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industry. By reducing lead time, supplier can increase customer’s satisfaction rate and can decrease the loss due to inventory shortages. In this direction, many researchers studied continuous review inventory models with controllable lead time [see for instance Ben-Daya and Raouf (1994), Ouyang et al. (1996), Pan et al. (2004), Pan and Hsiao (2005), Sarkar et al. (2015b)]. However, there are quite few models considered periodic review inventory system. Chuang et al. (2004) considered a periodic review inventory model with lost sales and backorders, and they reduced the total inventory cost by simultaneously controlling setup cost and lead time. Later, Jaggi and Arneja (2010) considered a periodic review inventory model with backorder discount. They developed their model to calculate optimal replenishment interval, optimal lead time, and optimal backorderdiscount simultaneously. Jaggi et al. (2014) developed two models with backorder-discount and controllable

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ACCEPTED MANUSCRIPT lead time. First model’s demand follows a normal distribution and second model’s demand doesn’t follow any specific distribution, but with known mean and standard deviation. Sarkar and Mahapatra (2015) developed an inventory model with reorder point, replenishment interval, and lead time as decision variables. They solved the

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model with fuzzy concept. See Table 1 for the contribution of different authors.

Insert Table 1 here

2. Problem definition, notation, and assumptions This section considers problem definition, notation, and assumption.

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This paper uses notation which is similar to Ouyang et al. (2007) and Sarkar et al. (2015b) for readers not to be confused.

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2a. Problem definition

The aim of this study is to clean the multi-stage production system from defective items. The defective items are produced during out-of-control state of the production system. An investment is used to improve the quality of the system such that defective production would be reduced to clean the production system. As this is a multistage production system, a budget constraint and two constraints for setup cost reduction and quality

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improvement are introduced. Lead time demand is stochastic in nature and a crashing cost is used to reduce the lead time, where a backorder price-discount is offered to reduce the lost sell cost for reducing total cost of the system. In Figure 1 and Figure 2, the graphical representation of multi-stage and complex-multi-stage are

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presented.

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Insert Figure 1 and Figure 2 here

2b. Notation

Decision variables T L


  

replenishment interval (weeks) length of lead time (weeks)

backorder price-discount of item i per unit suggested by supplier ($/unit), safety factor of item i,

i = 1, 2, …, I

i = 1, 2, …, I

probability of production stage j, which may move from in-control condition to out-of-control condition,  = 1, 2, … ,

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positive integer, which decides replenishment interval of item i ,

Parameters set of items which are made through jth work,



set of works using when ith item produces,



 = 1, 2, … ,

i = 1, 2, …, I

average demand of ith item per year (units/year),



i = 1, 2, …, I

standard deviation of demand of ith item per year (units/year),

A

initial major ordering cost per order (before lead time reduction) ($/order)



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minor ordering cost of ith item per order ($/order), i = 1, 2, …, I

C(L)

crashing cost per cycle (function of lead time) ($/cycle) backorder ratio of ith item 0 ≤  < 1,



i = 1, 2, …, I

upper bound of backorder ratio of ith item i = 1, 2, …, I

ℎ

holding cost of ith item per unit per year ($/unit/year),



purchasing cost of raw material to produce ith item ($/unit),

  

 δ "

# $% (%

)%

 = 1, 2, … ,

i = 1, 2, …, I

i = 1, 2, …, I

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defective cost of ith item per unit ($/unit),

i = 1, 2, …, I

maximum budget for investing to stage and purchasing raw material ($) marginal profit of ith item per unit ($/unit),

i = 1, 2, …, I

initial probability of production of jth stage, which may move from in-control condition to out-of = 1, 2, … ,

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investment cost required in order to reduce  to  ($),

control condition,

fractional annual cost of investment to reduce drift rate of jth stage ($/year),

percentage decreases in θ per dollar increases in   ,

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B

i = 1, 2, …, I

major ordering cost per order (function of lead time) ($/order)





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i = 1, 2, …, I

a reciprocal number of δ ,

 = 1, 2, … ,

number of item which are produced through jth stage,

 = 1, 2, … ,

 = 1, 2, … ,

pth component of lead time with $% as minimum duration (days), pth component of lead time with (% as normal duration (days),

pth component of lead time with )% as crashing cost per day,

Assumptions 7

 = 1, 2, … ,

p = 1, 2, …, &'

p = 1, 2, … , &'

p = 1, 2, …, &'

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Products are periodically ordered. Replenishment of each item is jointly made by multiple replenishment intervals.

2.

The lead time L is composed of &' mutually independent components. The pth component consists of a

minimum period $% , a normal period (% , and a crashing cost per day )% . )% is arranged in such a way

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that )* ≤ )+ ≤ ), , … , ≤ )-' . Then, it is obvious that lead time reduction should first happen on

component 1 because it has minimum unit crashing cost, and then consecutively component 2, and so on [see for instance Ben-Daya and Raouf (1994), Chuang et al. (2004)].

3.

Components of lead time L are crashed one at a time beginning with the least )% component sequentially

' (0 and .% be the duration of lead time having components 1, 2, … , p crashed to the Let . = ∑-01*

minimum duration, then .% can be represented as [see for instance Ouyang et al. (1996) and Sarkar et al.

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4.

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and so on.

(2015b)].

%

-'

.% = 2 (0 − 2(0 − $0 , 01*

01*

where p = 1, 2, … , &' and the crashing cost per cycle C(L) is given by %6*

5.

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4. = )% 5.%6* − .7 + 2 )0 (0 − $0 . 01*

For ith item ( i = 1, 2, …, I ), the backorder rate  is considered as variable and is proportional to

backorder price-discount
 suggested by the supplier. As backorder price-discount is considered in this

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paper, backorder price discount of ith item is assumed that  = 
 /
 , 0 ≤  < 1 and 0 ≤
 ≤

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 , where
 the backorder price-discount is during the maximum of backorder  and
 is the

marginal profit per unit. As the supplier would like to make more profits Thus supplier may decide not to offer the backorder discount [see for instance Pan et al. (2004) and Sarkar et al. (2015b)], if backorder

6.

discount
 is greater than marginal profit
 .

This paper assumes that quality of products and lot size are related. While producing a lot, the process may move to out-of-control state and then it begins to produce defective items and continues to do until the

predetermined lot size all is produced [see for instance Porteus (1986)]. 7.

To decrease imperfect productions, production system should be controlled during out-of-control condition. Therefore, for this purpose, decision maker needs to invest additional investment. It helps to reduce the

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ACCEPTED MANUSCRIPT production during out-of-control state. According to Porteus (1986), this paper presumes capital investment    for jth work to increase the process quality as  5 7 = " ln ?

 @ , for 0 <  ≤  . 

If investment function  5 7 = 0, it implies no investment for quality improvement. If there is an

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investment, then value of  will begin to decrease, which indicates improvement of products’ quality. In

the modern industry, the value of  is very small. Therefore, this model is considered for quite small

The reduction of lead time L involves a decreasing value of major ordering cost A, which is a pseudo

concave function of L i.e., D . > 0 and DD . ≤ 0 [see for instance Porteus (1986) and Sarkar et al.

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8.

value of  .

9.

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(2015a)].

An inventory replenishment interval T is larger than the length of lead time L.

10. The holding cost of item i during replenishment interval does not exceed the marginal profit of ith item per

3. Mathematical model

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unit (i.e.,
 ≥ ℎ  G).

This joint replenishment problem (JRP) model is developed with a complex multi-stage production system to improve quality of products to clean defective items from the whole system. The inventory of each item is reviewed at an interval of time. Because this paper assumes that the demand follows a normal distribution, buyer

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can’t exactly predict demand. Therefore, the buyer should have safety stock   H G + . to provide against

unexpected demand and to reduce cost from stockout. As the expected demand of ith item during replenishment

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interval is  G , the average inventory of ith item is  G /2 +   H G + .. Thus, the expected shortage

quantity of ith item per cycle is I

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NO PO QRSRTO UO HPO QRS

5J −   G + . −   H G + .7KJLJ =  V H G + ..

where V  = ф  −  X1 − Y Z, ф  denotes the standard normal probability density function and

Y  represents the cumulative density function. Therefore, the expected backorder quantity of ith item is

  V H G + .. Similarly, the expected lost sales quantity of ith item is 1 −   V H G + . .

Therefore, the expected annual lost sales cost of ith item is X
  +
 1 −  Z V H G + ./ G and 9

ACCEPTED MANUSCRIPT the expected annual holding cost of ith item is ℎ [ G /2 +   H G + . + 1 −   V H G + .\. Hence, the expected annual cost (EAC) is

.   G +2^ + ℎ _ +   H G + . + 1 −   V H G + .` G  G 2 a

+

1*

1 4. X
  +
 1 −  Z V H G + . + b.  G  G

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]4G,  ,  ,  , .  =

(1)

Equation (1) doesn’t consider relationship between lot size and quality. In other words, cost related to quality is ignored. Therefore, Equation (1) should be rebuild to consider a relationship between lot size and quality. Let 

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(in generally, the value of  is very small as close to zero) be the probability of production stage j (for instance

one machine, or one work, etc.) which may move from in-control state to out-of-control state during the

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production. If the process moves to out-of-control condition, defective items are continuously produced until all predetermined lot sizes are produced. Porteus (1986) found out that the expected quantity of defective items is

approximately c+ /2 during lot size Q is produced in EOQ model with single-stage [see for instance Porteus (1986), Sarkar et al. (2015b)]. However, in modern industries, complex products no longer are produced through

only single-stage manufacturing process. Thus, the expected quantity of defective items needs to find out for applying to a complex multi-stage inventory model. This paper obtains that the expected quantity of defective ith

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item can be approximated by  + G51 − ∏∈f51 −  77/2. Therefore, the annual expected defective cost is

  + G51 − ∏∈f51 −  77/2 (see Appendix A).

Quality of jth stage can be improved by allocating a capital investment. Thus, quality factor is no longer a

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fixed parameter. By considering a capital investment of jth stage   , as in Assumption 7, Equation (1) is

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rebuild which considers both defective cost and the investment cost. Hence, the total expected annual cost (TEAC) can be described as

a

G]45G,  ,  ,  ,  , . 7 = 2 g 2  " h ? 1* ∈f

  + G51 − ∏∈f51 −  77  @+ i + ]4G,  ,  ,  , . ,  2

(2)

subject to

0 <  ≤  , ∀ = 1, 2, … , .

Furthermore, by Assumption 5,  = 
 /
 . Therefore, (2) becomes G]45G,  ,
 ,  ,  , .7 =

 .    G + 2 g 2  " h ? @ + + ℎ _ +   H G + .` G   G 2 a

1* ∈f

10

(3)

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 4.   + G51 − ∏∈f51 −  77 `+ l  V H G + . + + m,
  G  G 2

0 ≤
 ≤
 , ∀ = 1, 2, … , ,

(5)

0 <  ≤  , ∀ = 1, 2, … , , 2 g  G + 2  " h ? 1*

∈f

(6)

 @i ≤ n, ∀ = 1, 2, … , , ∀ = 1, 2, … , 

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a

(4)

(7)

where
 =
 − 
 +  π+pq /
  and TEAC5G, .,  ,
 ,  ,  7 is the total expected annual cost.

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Equation (4) is a non-linear program and to solve this equation, constraints (5), (6), and (7) are ignored initially.

To solve this equation, taking the first partial derivatives of TEAC5G,  ,
 ,  ,  , .7 with respect to

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G,  ,
 ,  ,  , and L, respectively. (Note that this model only uses integer value of  , however,  is considered as continuous variable to use convex function property). One can obtain

vG]4 .  4. ℎ   ℎ   
  V H G + . = − + + 2 w− − + + − + + vG G  G  G 2  G + 2H   G + . a

+

  V 

1*

2H   G + .

kℎ _1 −

  + 51 − ∏∈f51 −  77 

 `+ l+ m,
  G 2

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vG]4 

 = ℎ  H G + . − wℎ _1 − `+ m 51 − Y 7 H G + ., v
  G vG]4 1 2
 ℎ  =^ _ −  ` − b  H G + . V , v
  G



(8)

(9)

(10)

(11)


  V H G + . 4.  + G51 − ∏∈f51 −  77 − + + , 2 + G  G

(12)

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  + G ∏ x ∈f6yz51 −  x 7 #  " vG]4 =− + 2 , v  2 ∈a

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vG]4  ℎ  G ℎ   G 

  V G =− + + + + kℎ _1 − `+ l v 2
   G 2H   G + .  G 2H   G + . −

)% vG]4 D . ℎ   

  V  = +2w + kℎ _1 − `+ l − m. v. G
   G 2H   G + .   G 2H   G + . a

1*

It is obvious that for fixed 5G,  ,
 ,  ,  7, TEAC5G,  ,
 ,  ,  , .7 is concave in L because of

11

(13)

ACCEPTED MANUSCRIPT ,

,

v + G]4 DD . ℎ    G + .6+ 

  V  G + .6+ = − 2 g + kℎ _1 − ` + l i < 0.  v.+ G 4
  G 4 a

1*

(14)

Therefore, for fixed 5G,  ,
 ,  ,  7 the minimum value of TEAC5G,  ,
 ,  ,  , .7 exists at the end point

of the interval[.% , .%6* \. Similarly, it is also clear that for fixed 5G,  ,
 ,  , .7, TEAC5G,  ,
 ,  ,  , .7 is

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(see Appendix B). Hence, for fixed 5G,  ,
 ,  , .7 , the minimum value of

convex in 

TEAC5G,  ,
 ,  ,  , .7 exists either at critical point or at the end point of the interval [.% , .%6* \. For fixed . ∈ [.% , .%6* \, by setting 8 − 11 equal to zero, one can obtain

.  4. ℎ   ℎ      V  

 = 2 w− − + + + kℎ _1 − `+ l + + + G  G  G 2
  G 2 H   G + . 2H   G + . 1*
  V H G + .   + 51 − ∏∈f51 −  77 + m,  G + 2

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−

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a

Y  = 1 −

(15)

ℎ , 

 ℎ _1 − `+
  G


 =


 + ℎ  G , 2

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2#  "  = g 2   + } G ∈a

~

(16)

(17) 6*

51 −  x 7i .

(18)

 x ∈f6yz

Therefore, for fixed . ∈ [.% , .%6* \, from 15 − 18, the optimal values of G,  ,
 ,  can be calculated

3.1. Proposition

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∗ when constraints are ignored. (These values are denoted by G ∗ , ∗ ,
 , ∗ ∀ = 1, 2, … , , ∀ = 1, 2, … , )

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For fixed . ∈ [.% , .%6* \ and fixed  , Hessian Matrix of the TEAC5G,  ,
 ,  ,  , .7 is always positive at

∗ the point 5G ∗ , ∗ ,
 , ∗ 7. ∀ = 1, 2, … , , ∀ = 1, 2, … , .

Proof. See Appendix C.

Now, it is considered that 5 − 7 constraints 0 ≤
 ≤
 , 0 <  ≤  , and ∑a1* ^  G +

∑∈f  " h _

ƒ„ „

`b ≤ n .

Third

G ≤ ^n − ∑a1* ∑∈f  " h _

ƒ„ „

term

restricts

a

range

of

T,

because

this

term

equals

to

`b / ∑a1*    . This constraint will use to determine starting value in the

iterative method in KS algorithm by using Newton-Raphson procedure. KS algorithm method is developed by 12

ACCEPTED MANUSCRIPT ∗ using Newton-Raphson method, as iterative procedure, and convex property. If ∗ >  and
 >
 , then

those inequalities imply that there will be no capital investment for quality improvement and there will be no ∗ =
 for every backorder discount offered by supplier, respectively. For this reason, setting ∗ =  and


i, j.

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3.2. KS algorithm

As this mathematical model is for a complex multi-stage production system in a JRP model thus, T can’t be well defined. Therefore, the traditional algorithm for JRP model such as RAND algorithm can’t be used to solve this problem. In order to solve this problem, KS algorithm for this paper is developed by using convex property, method

by KS

algorithm

using

Newton-Raphson

method.

For

fixed

SC

iterative

5G,  ,
 ,  , .7,

time is also significantly reduced.

(Initialization and Loop) Step 1.1

Input all parameters.

Step 1.2

Set  = 1,  = 1, 2, … ,  and perform Step 2 through Step 7.

Step 1.3

∗ ∗ ∗ ∗ If ∀  , TEAC5G…∗ , … ,
… , …∗ , ∀ = 1, .∗%… 7 ≤ TEAC5G…∗ , … ,
… , …∗ ,  , .∗%… 7, then set

TE D

Step 1

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TEAC5G,  ,
 ,  ,  , .7 is convex in  (see Appendix B). Therefore, by using this property, computation

∗∗ ∗∗ ∗∗ ∗∗ ∗ ∗ 5G…∗∗ , … ,
… , …∗∗ , … , .%… 7 = 5G…∗ , … ,
… , …∗ , ∀ = 1, .∗%… 7. Then go to Step 8.

EP

Otherwise, let  ≠ 1 be  x . Increase  x one by one and perform Step 2 through Step

(Find solutions)

AC C

Step 2

∗ ∗ ∗ ∗ 7 until TEAC5G…∗ , … ,
… , …∗ ,  , .∗%… 7 > TEAC5G…∗ , … ,
… , …∗ ,  − 1, .∗%… 7.

For each .% , ‡ = 0, 1, … , &', perform Step 2.1 through Step 2.6.

Step 2.1

Step 2.2

Set
 =
 ,  %*

%*

=  , and  = 0 (indicates V5 7 = 0.39894). %*

%*

Substituting
 ,  , and V5 7 into (15), using Newton-Raphson method, evaluate %*

%*

%*

G %* . By (6), maximum value of T is B/∑a1*    . Thus, this value is set as a starting

Step 2.3

value to evaluate G %* .

Use G %* to obtain Y5 7 from (16), hence find  %+

normal distribution table and calculate V5 7. %+

13

%+

by identifying the standard

ACCEPTED MANUSCRIPT Step 2.4 Step 2.5 Step 2.6

Use G %* and 

%*

%+

to obtain 

%+

from (18).

Repeat Step 2.2 to Step 2.5 until the values of G %0 ,
 ,  , and  %0

Denote these optimal solutions by GŠ,
‹ , Š , and Š .

 Ž and ŒŽ , and  (Compare Œ ‘ ’“ and “ ) Step 3.1

%0

no longer change.

If ∀ , ,
‹ <
 and Š <  , then the values found in Step 2 are the optimal

∗ solutions for the given .% . Denote these solutions by 5G ∗ , ∗ ,
 , ∗ 7. Then go to Step 4.

If there exists an  D ∈ y1, 2, … , z such that
‹ x ≥
 x and ∀ , Š <  , then set


‹ x =
 x and use Step 2 to calculate the new 5GŠ, Š , Š 7 from (15), (16), and (18). If

SC

Step 3.2

%0

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Step 3

Use G %* to determine
 from (17).

∀ , Š <  , then the optimal solutions for given .% is

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Step 3.3

∗ 5G ∗ , ∗ ,
 , ∗ 7 = 5GŠ, Š ,
‹ x , Š 7 and then go to Step 4. Otherwise, go to Step 3.4.

If there exists a  DD ∈ y1, 2, … , z such that Š xx ≥  xx and ∀ ,
‹ <
 , then set

Š xx =  xx and use Step 2 to calculate the new 5GŠ , Š ,
‹ 7 from (15), (16), and (17). If ∀ ,
‹ <
 , then the optimal solutions for given .% is

TE D

Step 3.4

∗ 5G ∗ , ∗ ,
 , ∗ 7 = 5GŠ, Š ,
‹ , Š xx 7 and then go to Step 4. Otherwise, go to Step 3.4.

If there exists an  D ∈ y1, 2, … , z such that
‹ x ≥
 x and there exists a  DD ∈

y1, 2, … , z such that Š xx ≥  xx , then set
‹ x =
 x and Š xx =  xx , and use Step 2

EP

∗ to calculate the new 5GŠ , Š 7 from (15) and (16). Then, denote 5G ∗ , ∗ ,
 , ∗ 7 =

5GŠ , Š ,
‹ x , Š xx 7 and go to Step 4.

(Compare œ , ^ž − ∑¤“1¥ ∑Ž∈£“ ŸŽ  Ž ¡¢ _

AC C

Step 4

Step 4.1

Step 4.2

ŒŽ ŒŽ

`b / ∑¤“1¥ ¦“ ¢“ §“ and ¨∗ )

If .% ≤ G ∗ ≤ ^n − ∑a1* ∑∈f  " h _

ƒ„ „

`b / ∑a1*    , then the value of T found in

Step 2 is the optimal solution for the given .% . Then go to Step 6.

If G ∗ < .% , then set G ∗ = .% , and use Step 2 to calculate the new 5Š ,
‹ , Š 7 from (16), (17), and (18). Then go to Step 5.

Step 4.3

If ^n − ∑a1* ∑∈f  " h _

ƒ„ „

`b / ∑a1*    < G ∗ , then set

14

ACCEPTED MANUSCRIPT G ∗ = ^n − ∑a1* ∑∈f  " h _

„

`b / ∑a1*    , and use Step 2 to calculate the new

5Š ,
‹ , Š 7 from (16), (17), and (18). Then go to Step 5.

 Ž and ŒŽ , and  (Compare Œ ‘ ’“ and “ )

Step 5.1

Step 5.2

If ∀ , ,
‹ <
 and Š <  , then the values found in Step 4 are the optimal

∗ solutions for the given .% . Denote these solutions by 5G ∗ , ∗ ,
 , ∗ 7. Then go to Step 6.

If

RI PT

Step 5

ƒ„

there exists an  D ∈ y1, 2, … , z such that
‹ x ≥
 x and ∀ , Š <  , then set


‹ x =
 x and use Step 2 to calculate the new 5Š , Š 7 from (16) and (18). If

∗ 5G ∗ , ∗ ,
 , ∗ 7 = 5G ∗ , Š ,
‹ x , Š 7 and then go to Step 6. Otherwise, go to Step 5.4.

If

there exists a  DD ∈ y1, 2, … , z such that Š xx ≥  xx and ∀ ,
‹ <
 , then set

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Step 5.3

SC

∀ , Š <  , then the optimal solutions for given .% is

Š xx =  xx and use Step 2 to calculate the new 5Š ,
‹ 7 from (16) and (17). If ∀ ,
‹ <
 , then the optimal solutions for given .% is

If

there exists an  D ∈ y1, 2, … , z such that
‹ x ≥
 x and

there exists a  DD ∈

y1, 2, … , z such that Š xx ≥  xx , then set
‹ x =
 x and Š xx =  xx , and use Step 2

TE D

Step 5.4

∗ 5G ∗ , ∗ ,
 , ∗ 7 = 5G ∗ , Š ,
‹ , Š xx 7 and then go to Step 6. Otherwise, go to Step 5.4.

∗ to calculate the new Š from (16). Then, denote 5G ∗ , ∗ ,
 , ∗ 7 = 5G ∗ , Š ,
‹ x , Š xx 7

EP

and go to Step 6. Step 6

(Calculate total expected annual cost)

Step 7

ª) (Choose minimum cost for each œ ,  = , ¥, … , ©

Step 8

AC C

∗ Utilize (4) to calculate the total expected annual cost G]45G ∗ , ∗ ,
 , ∗ ,  , .% 7. ∗ ∗ ∗ G]45G…∗ , … ,
… , …∗ ,  , .∗%… 7 = min%1,*,…,-' G]45G ∗ , ∗ ,
 , ∗ ,  , .% 7 gives the optimal solution

in fixed  ,  = 1,2, … , .

(Choose minimum cost for each n)

∗∗ ∗∗ ∗∗ ∗∗ ∗ ∗ TEAC5G…∗∗ , … ,
… , …∗∗ , … , .%… 7 = min1*,+,…,a TEAC5G…∗ , … ,
… , …∗ ,  , .∗%… 7 gives the minimal

∗∗ ∗∗ ∗∗ ∗∗ total cost of the problem and 5G…∗∗ , … ,
… , …∗∗ , … , .%… 7 are optimal values in this problem.

Step 1 sets parameter for initialization and fixes the number of shipments to find solution. This step makes

15

ACCEPTED MANUSCRIPT the loop and condition for stopping algorithm using convex property (note Appendix B). For fixed  , Step 2 is

performed to obtain the optimal solution ignored constraints (5)-(7) by using Newton-Raphson iterative method.

Step 3 compares the solution obtained in Step 2 with initial value of the backorder discounts and quality factors to consider constraints (5) and (6). If the solution obtained in Step 2 does not satisfy the constraints (5) and/or

RI PT

(6), then the solution is modified by initial values of them and is recalculated as to satisfy those constraints. In Step 4, replenishment interval obtained in Step 2 is compared with the lead time (as in Assumption 9) and upper bound of replenishment interval made by budget constraint. If the replenishment interval does not have value between lead time and the upper bound, then the value is changed into lead time or the upper bound, and new

SC

solution is recalculated. In Step 5, the solution is compared with initial values of the backorder discounts and quality factors to consider constraints (5) and (6). Step 6 calculates the total expected annual cost by using the solution obtained in Step 2~5. Step 7 determines the minimum total cost for each lead time. Finally, Step 8

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generates the minimum total cost for each number of shipments  and the optimal solutions such as the

replenishment interval, number of shipments, safety factors, quality factors, backorder discounts, and lead time.

∗∗ ∗∗ ∗∗ ∗∗ This algorithm always provides optimal solution because of the fact that the TEAC5G…∗∗ , … ,
… , …∗∗ , … , .%… 7

4. Numerical examples

TE D

is a convex function (see for reference Proposition 3.1).

In order to consider JRP model, it is used that minor ordering cost data and purchasing cost data from

EP

Moon and Cha (2006). It is used that the parametric values, which are used by Ouyang et al. (2002) to illustrate above problem and algorithm (refer item 3 and work 3 in Table 2). However, because they considered single

AC C

item and non-complex single-stage, below additional parameters ought to be used (Table 2 and Table 4).

Example 1

Insert Table 2, 3, and 4 here

It is assumed the relationship between ordering cost and lead time (see for instance Lin (2008)) as

 − / = ­ h. − ./. ,

which

implies

. =  + ".,

where

 = 1 − 1/­

and

"=

 /­.  and ­> 0 is constant parameter, which represents a linear relationship between ordering cost and

lead time reduction. The total annual expected cost is calculated when ­ = 0.75, 1.00, 1.25, 2.50, 5.00, and it is compared when ­ → ∞ i.e., fixed ordering cost unaffected by lead time to verify relationship between ordering

cost and lead time reduction in Table 5 and Table 6. The result is illustrated in Figure 3 that ­ is in inverse 16

ACCEPTED MANUSCRIPT proportion to the reduction of the total annual expected cost. Example 2 Using same data used in Example 1, it is assumed the relationship between ordering cost and lead

time (see for instance Lin (2008)) as  − / = ° h./. , which implies . = K + ± ln ., where Insert Table 5, 6, 7, and 8 K =  1 + ° ln . , ± = −° > 0, and °< 0 is constant parameter, which represents a linear relationship here

between ordering cost and lead time reduction. It is calculated that total annual expected cost when ° =

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0, −0.2, −0.5, −0.8, −1.0. Those calculated values are compared to verify relationship between ordering cost and lead time reduction in Table 7 and Table 8. The result is illustrated in Figure 4 that ° is in inverse

SC

proportion to the reduction of the total annual expected cost.

Insert Figure 3 and Figure 4 here

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4.1. Sensitivity analysis

The sensitivity analysis of the key parameters of Example 1 is performed with changes

−50%, −25%, +25%, +50%. The results are shown in Table 9. Insert Table 9 here

1.

TE D

The sensitivity analysis results represent the following significant features. The increasing value of initial major ordering cost causes a reduced value of safety stock and an increasing value of replenishment interval. In addition, the increasing initial major ordering cost indicates the

EP

increasing value of backorder price-discount and a decreasing value of quality factor due to increasing cost resulted from increasing ordering cost. The increasing value of holding cost gives a decreasing value of safety stock, because many inventories

AC C

2.

increase the expected total cost of system. Therefore, a manager would rather order more frequently than store inventory. In addition, the increasing value of holding cost results an increasing value of backorder price-discount and quality factor. 3.

The defective cost does not influence the replenishment interval, safety stock level, backorder pricediscount, and major ordering cost. The increasing defective cost causes a decreasing value in quality factor, which implies that more quality improvement is required to reduce the expected total cost when the defective cost is high.

4.

If fractional annual cost of investment increases, safety stock also increases, because high investment cost hinders quality improvement. The defective items are produced more frequently. As a result, more safety 17

ACCEPTED MANUSCRIPT stock is required. In addition, the increasing value of fractional annual cost of investment implies decreasing value of backorder price discount.

4.2. Managerial insights

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This sub-section provides significant managerial insights for a complex multi-stage production system with quality improvement. These insights enables manager to decide which process is preferentially invested.

All parameter values of items and processes change to the value of item 3 and process 3 in Table 2. The

parameter values are modified by the values with changes −50%, −25%, +25%, +50% and process quality

SC

is improved 0.00005 from the initial value to obtain which process is preferentially improved. Table 10 represents the following significant insights into quality improvement.

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Insight 1 They are able to reduce more expenses by improving process, which has the worst initial quality. Insight 2 Within the ordering cost, holding cost, and marginal cost of item, which influences more. The managers would have ideas over this to control quality factor to clean the production system. Insight 3 They are able to reduce more expenses by preferentially investing in process that produces item, which incurs the highest defective cost, keeping all other parameters are fixed.

5. Conclusions

TE D

Insert Table 10 here

EP

For cleaning the multi-stage production system, an effort was done by cleaning the defective items within the production system. Based on the quality improvement for multi-stage, it could be assured that the throughput of This study suggested considering a complex multi-stage

AC C

the manufacturing system would be perfect always.

production system with quality improvement by a continuous investment, it was found that the multi-stage production system become more stable regarding the production of perfect products. This research initialized a first step towards the complex multi-stage production system with quality improvement. A backorder pricediscount was successfully established based on the several suggestions from the literature. Specifically, this paper proposed a novel representation for a complex multi-stage production system with quality improvement to clean the production system by reducing the defective items at almost zero level within the JRP framework. The model was solved analytically with global minimum solutions. Two theorems were derived to prove the global optimality of decision variables. An efficient new and improved computational algorithm was provided to obtain

18

ACCEPTED MANUSCRIPT optimal solution of this model. Furthermore, two numerical examples and graphical representations were given to illustrate this model.

The sensitivity analysis results represented significant features based on cleaning the

multi-stage production system from defective items by using quality improvement and lead time dependent ordering cost, which would enable the managers to decide about the investment to clean the defective items

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from the production system. Using this concept and improved algorithm, researchers can investigate the quality related issues for any complex multi-stage production system to obtain more accurate decision making [see for references Sarkar et al., 2015; Sarkar et al., 2015; Sarkar and Moon, 2014]. A major extension of this paper is possible if the number of shipment with unequal lot size is considered, where the fixed and variable

SC

transportation cost and carbon emission cost can be considered and the aim would be cleaned the multi-stage production system [see for references Wang et al. 2015; Sarkar et al. 2015; Sarkar et al. 2016]. This research can

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be further extended by considering random breakdown during production as this model considered only the production system moves from in-control state to out-of-control state. Another interesting idea is to consider the rate of defective item will depend on the random failure rate of the production system, where corrective maintenance and preventive maintenance should be considered to control the system with perfect production [see for references Sarkar et al., 2010; Sarkar et al., 2010; Sett et al., 2016]. The demand may be considered as

TE D

fuzzy [see for reference Sarkar and Mahapatra, 2015] or single-period fuzzy [see for references Taleizadeh et al., 2011; Taleizadeh et al., 2013]. Using utilization and allocation decisions, this model can extended in another

EP

direction [see for reference Nobil et al. (2016)].

Appendix A

AC C

Referring Porteus (1986), the expected defective quantity of ith product in demand during replenishment interval of product i is   G −

'51 − 'NOPOQ 7 

where  = 1 − ~ 51 −  7 ∈f

As ' = ∏∈f51 − θ 7 is approximated to one using a Taylor series expansion method of ' NOPOQ and obtain ª 'NOPOQ = µ NOPOQ¶P ≅ 1 +   G5h' 7 +

+ [  Gh'\ 2

Therefore, it can be estimated that the quantity of defective items of ith item per cycle 19

ACCEPTED MANUSCRIPT '51 − 'NOPOQ 7 

=   G −

+ + G + h' + 1 − 1 −   G 5h'7 −   2 =   G − 

=

=

+ + G + 2

 + + + G + 2 

+ + G + 51 − ∏∈f51 − θ 77 2

= s  − = s  −

s  '51 − 'NOPOQ 7   G

s  ?1 − 1 −   G5h'7 −   G

M AN U

Therefore, the expected defective cost of ith item per year is

+ + G + h' + @ 2

= s  −

  G + − ' + + + G + ` ' 2    G

=

  G ` 2

AC C

=

 + + + G + ` 2   G

s  _  G −

= s  − s  _1 −

+ [  Gh'\ 2

EP

= s  −

TE D

ª As ' = 1 − θ ≅ 1 and 'NOPOQ = µ NOPOQ¶P ≅ 1 +   G5h' 7 +

s  _

SC

=   G −

  G −

RI PT

 + + + +   G −   G   ' 2' +   =   G −  

s +  G 2

s +  G51 − ∏∈f51 − θ 77 2

Appendix B

For fixed 5G,  ,
 ,  , .7, TEAC5G,  ,
 ,  ,  , .7 is convex in  . In other words,

20

ACCEPTED MANUSCRIPT ,

,

v + G]4 2 ℎ   G +  G + .6+  V G +  G + .6+ 

 = , − − kℎ _1 − `+ l + 4 4
  G v  G

− Proof.


  V 

+ H G + .

+

24. 2
  V H G + . + > 0. , G , G

,

,

RI PT

ℎ   G +  G + .6+  V G +  G + .6+ 


  V  Let η = + kℎ _1 − `+ l− + . 4 4
  G  H G + . ,

,

v + G]4 2 ℎ   G +  G + .6+
  V  

  V G +  G + .6+ = − − − kℎ _1 − ` + l  4
  G 4 v+ , G + H G + .

= º − η, )ℎµ»µ º =

2 24. 2
  V H G + . 2
  V  + , + − +  G , G , G  H G + .

By Theorem 1 (see Appendix D),

 + G + 51 − ∏∈f51 −  77 2
  V  − + 4 G + .  H G + .

 + 4.3 G + 4. 2, G G + . −

2
  V 

+ H G + .

 + 4.3 G + 4. 2, G G + .

AC C

=

+

+

 + G + 51 − ∏∈f51 −  77 2
  V H G + . ℎ  G + + + 4 G + . 4 G + . , G

EP

=

2 24. 2
  V H G + .  + 4. ℎ  G + + + − + , G 2+  G + . 4 G + . , G , G

TE D

>

SC

24. 2
  V H G + . + , G , G

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+

Appendix C

+

2
  V .

, G H G + .

+

Proof of proposition 3.1.

 + G + 51 − ∏∈f51 −  77 ℎ  G + + >0 4 G + . 4 G + .

For fixed . ∈ [.% , .%6* \ and fixed  , Hessian matrix is as given below:

21

v + G]4 ¾ ∗+ ½ vG ½v + G]4 ½ v ∗ vG ∗ ½  + ½v G]4 ∗ ½v
 vG ∗ ½ + v G]4 ½ ∗ ¼ v vG ∗

v + G]4 vG ∗ v∗

v + G]4 ∗ vG ∗ v
 + v G]4 ∗ v∗ v


v + G]4 v∗+ v + G]4 ∗ v
 v∗

v + G]4 ∗+ v
 + v G]4 ∗ v∗ v


v + G]4 v∗ v∗

∗ where G]4 = G]45G ∗ , ∗ ,
 , ∗ ,  , .7

v + G]4 Á vG ∗ v∗ À v + G]4 À v∗ v∗ À À v + G]4 À ∗ v
 v∗ À À + v G]4 À v∗+ ¿

,

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ACCEPTED MANUSCRIPT

∗ v + G]4 2. 2 +  V∗  G ∗ + .6+ 

 = + 2 g − kℎ _1 − `+ l  + ∗, ∗, vG G  G 4
  G ∗ a

1*

SC

,

v + G]4 #  " = ∗+ v ∗+

v + G]4 2  V∗  = H G ∗ + . v
 ∗+
  G ∗

M AN U

ℎ ∗  +  G ∗ + .6+
  V∗  24. 2
  V∗ H G ∗ + . − − + + i 4  G ∗, G ∗+ H G ∗ + .  G ∗,

∗ v + G]4 

 = wℎ _1 − `+ m  ф∗ H G ∗ + .  ∗+
  G ∗  v

TE D

∗ v + G]4 v + G]4  V∗ H G ∗ + . 2
 ∗ = = − _ −  ` ∗ v
 vG ∗ vG ∗ v
  G ∗+


v + G]4 v + G]4 ∗ = ∗ = 0 v
 v∗ v∗ v


EP

∗ v + G]4 v + G]4 1 2
 ℎ  ∗ _ −  ` − b  Y∗  − 1H G ∗ + . = 0 ∗ = ∗ ∗ = ^ ∗ v
 v v v
  G



AC C

v + G]4 v + G]4 ∗ ∗ = =0 v v v∗ v∗

v + G]4 v + G]4
  51 − Y∗ 7H G ∗ + . ∗ ∗ = = v vG vG ∗ v∗  G ∗+ and

  + ∏ x ∈f6yz Â1 − ∗x à v + G]4 v + G]4 ∗ ∗ = = 2 v vG vG ∗ v∗ 2

detÄ**  =

v + G]4 vG ∗+

∈a

22

ACCEPTED MANUSCRIPT ,

,

∗ 2. 2 ℎ ∗  +  G ∗ + .6+ +  V∗  G ∗ + .6+ 

 = + 2 g − − kℎ _1 − `+ l  ∗, ∗, G  G 4 4
  G ∗ a

−

1*

 V∗ 


G ∗+ H G ∗ + .

+

24. 2 V∗ 
 H G ∗ + . + m.  G ∗,  G ∗,

a

1*

,

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2. 2 + 4.  V∗ 
 2 V∗ 
 H G ∗ + . = + 2 ^ − + m − Å , G ∗,  G ∗,  G ∗, G ∗+ H G ∗ + . ,

∗ ℎ ∗  +  G ∗ + .6+ +  V∗  G ∗ + .6+ 

 where Å = 2 g + kℎ _1 − `+ li. 4 4
  G ∗ a

1*

SC

By Theorem 2 (see Appendix D)

2. 2 + 4.  V∗ 
 2 V∗ 
 H G ∗ + . > ∗, + 2 ^ − + m ∗,  G  G ∗, G G ∗+ H G ∗ + . 1*

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a

 + + 51 − ∏∈f51 − ∗ 77  + 4.  V∗ 
 + ℎ  . + − − mÇ. − Æ ∗, + 2 w ∗+ 2G  G ∗ + . 2G ∗+ H G ∗ + . 4 G ∗ + . G 4 G ∗ + . 1* a

. 5 + 4.73 G ∗ + 4.
  V∗  G ∗ + 4. + ℎ  = ∗, + 2 w + + G 2 G ∗,  G ∗ + . 4 G ∗ + . 2 G ∗, H G ∗ + . a

v + G]4 vG ∗+ detÄ++  = ÈÈ + v G]4 v∗ vG ∗

TE D

> 0.

 + + 51 − ∏∈f51 − ∗ 77 m. 4 G ∗ + .

v + G]4 + vG ∗ v∗ È v + G]4 v + G]4  = detÄ − ? @ ** v + G]4 È vG ∗ v∗ v ∗+ 

EP

+

1*

v∗+

AC C

.#  " #  " 5 + 4.73 G ∗ + 4.
  V∗  G ∗ + 4. + ℎ  > + 2 w + + 4 G ∗ + . 2 G ∗,  G ∗ + . G ∗, ∗+ ∗+ 2 G ∗, H G ∗ + . a

1*

+

  + ∏ x ∈f6yz Â1 − ∗x à  + + 51 − ∏∈f51 − ∗ 77 + m−g 2 i . 4 G ∗ + . 2 ∈a

By setting 10 equals to 0, one can obtain 2

∈a

  + ∏ x ∈f6yz Â1 − ∗x à 2

=

#  " ∗ G ∗

#  " . 1 5 + 4.73 G ∗ + 4.
  V∗  G ∗ + 4. + ℎ  = Æ + 2 w + + G ∗, ∗ ∗ ∗ 2 G ∗,  G ∗ + . 4 G ∗ + . 2 G ∗, H G ∗ + . a

1*

23

ACCEPTED MANUSCRIPT +

  + ∏ x ∈f6yz Â1 − ∗x à  + + 51 − ∏∈f51 − ∗ 77 1 m − 2 i. 4 G ∗ + . G∗ 2 ∈a

As ∗ ≤ 1 − ∏∈f51 − ∗ 7, ∏ x ∈f6yz Â1 − ∗x à < 1, and the value of ∗ is very less by Assumption 7

#  " . 5 + 4.73 G ∗ + 4.
  V∗  G ∗ + 4. > ∗+ Æ ∗, + 2 w + mÇ. G 2 G ∗,  G ∗ + .  2 G ∗, H G ∗ + . a

v + G]4 È vG ∗+ v + G]4  detÄ,, = v ∗ vG ∗  Èv + G]4 ∗ v
 vG ∗

v + G]4 vG ∗ v∗ v + G]4 v∗+ v + G]4 ∗ v
 v∗

v + G]4 ∗ vG ∗ v
 È

v + G]4 v + G]4 v + G]4 v + G]4 detÄ++  − ? ∗ ∗ @ ∗ ∗ = ∗+ v v
 vG v
 v
 v∗+ v + G]4 È ∗+ v


+

SC

> 0.

RI PT

1*

2#  "   V∗ H G ∗ + . . 5 + 4.73 G ∗ + 4.
  V∗  G ∗ + 4. > Æ + 2 k + lÇ 2 G ∗,  G ∗ + . G ∗,
  G ∗ ∗+ 2 G ∗, H G ∗ + . 1*

+ ∗ #  " + V∗ +  G ∗ + . 2
 _ −  `  .
 + G ∗Î ∗+

+ + + ∗ #  " 
 + V∗ +  G ∗ + 4.   V∗ +  G ∗ + . 2
 wυ + − _ − 1` m
 ∗+ + G ∗Î
 + G ∗Î

TE D

>

M AN U

−

a

5 + 4.73 G ∗ + 4. 2  V∗ H G ∗ + . . mÇ . where υ = Æ ∗, + 2 w 2 G ∗,  G ∗ + . G
  G ∗ a

By 16,

EP

>

#  "  + V∗ +
  G ∗ + 4. ℎ+ + G ∗+ ∗  Æυ + w −  G + . mÇ.   +
 ∗+ + G ∗Î


#  "  + V∗ + X

  G ∗ + 4. −  ℎ+ + G ∗+  G ∗ + 4.ZÇ. Æυ + + ∗+ + G ∗Î


AC C

=

1*

+ By Assumption (9),
 ≥ ℎ+ + G + .

>

#  "  + V∗ + X
 − 
 ZÇ. Æυ + ∗+ + G ∗Î


2#  "   V∗ H G ∗ + . . 5 + 4.73 G ∗ + 4. ≥ Æ + 2 w mÇ. G ∗, 2 G ∗,  G ∗ + .
  G ∗ ∗+

> 0.

a

1*

24

v + G]4 ∗+ È vG

v + G]4 vG ∗ v∗

v∗ vG ∗

v + G]4 v∗ v∗

v + G]4 v∗ vG ∗ detÄÎÎ  = È + v G]4 ∗ v
 vG ∗ + Èv G]4 =

v + G]4 v ∗+

v + G]4 v∗+ v + G]4 ∗ v
 v∗

v + G]4 ∗ vG ∗ v
 + v G]4 ∗ v∗ v
 v + G]4 ∗+ v
 + v G]4 ∗ v∗ v


v + G]4 vG ∗ v∗ È v + G]4 v∗ v∗ È v + G]4 ∗ v
 v∗ + v G]4 È v∗+

v + G]4 v + G]4 v + G]4 detÄ,,  − ? ∗ ∗ @ ? @? @ v
 ∗+ vG v v ∗+ +

RI PT

ACCEPTED MANUSCRIPT

#  " 2  V∗ H G ∗ + . . 5 + 4.73 G ∗ + 4. > ∗+ Æ Æ + 2 w mÇ
  G ∗ G ∗, 2 G ∗,  G ∗ + . 

SC

1*

∗  + V∗ + 

 
m − 
kℎ _1 − `+ l  ф∗ H G ∗ + .     + ∗Î
  G ∗   G


M AN U

+

a

,

2#  "  ,
 V∗ 51 − Y∗ 7  G ∗ + .+ . − ∗+
 , G ∗Ñ +

∗ 2#  "  + V∗ H G ∗ + . . 5 + 4.73 G ∗ + 4. 
 gÆ ∗, + 2 w mÇ kℎ _1 − ` ∗+ ∗, ∗ ∗ 2 G  G + .
 G 
  G a

+

1*

+
 
51 − Y∗ 7  G ∗ + . ∗ ∗+.−    l ф  G Ç. H    G ∗ + G ∗Î

TE D

>

+

+

2#  "  + V∗ H G ∗ + . . 5 + 4.73 G ∗ + 4. gÆ + 2 w mÇ ℎ _1 > G ∗, 2 G ∗,  G ∗ + . ∗+
  G ∗

EP

∗ 
 ` ф∗ H G ∗ + ..


1*

AC C

> 0.

−

a

As the principal minors of the Hessian Matrix are all positive, the Hessian matrix is positive definite at the point ∗ 5G ∗ , ∗ ,
 , ∗ 7. Hence, the total expected annual cost (TEAC) function has a global minimum at that point.

Appendix D

Proof of Theorem 1 and Theorem 2 Theorem 1

 + G + 51 − ∏∈f51 −  77  + 4. ℎ  G + − − 4 G + . 2+  G + . 4 G + .

25

ACCEPTED MANUSCRIPT ,

,

*

ℎ   G +  G + .6+  V G +  G + .6+ 

  V 
  G + .6+ > + kℎ _1 − `+ l− . 4 4
  G +

Proof.

By equating 11 to 0, one can obtain

−

or equivalently,

RI PT

 + G51 − ∏∈f51 −  77  + 4. ℎ  G ℎ   G  V G 

 = + + kℎ _1 − ` + l +  2
  G 2 + G 2H   G + . 2H   G + .
  V H G + . . + G

,

,

SC

 + 4. ℎ  G + ℎ   G +  G + .6+  V G +  G + .6+ 

 = + + kℎ _1 − `+ l + 4 4
  G 2  G + . 4 G + . *

M AN U

 + G + 51 − ∏∈f51 −  77
  V  G + .6+ − . + 4 G + . 2+

 + G + 51 − ∏∈f51 −  77 ℎ  G +  + 4. − − 4 G + . 2+  G + . 4 G + . ,

,

*

TE D

ℎ   G +  G + .6+  V G +  G + .6+ 


  V  G + .6+ = + kℎ _1 − `+ l− . 4 4
  G 2+

Therefore,

EP

 + G + 51 − ∏∈f51 −  77  + 4. ℎ  G + − − 4 G + . 2+  G + . 4 G + . ,

,

*

AC C

ℎ   G +  G + .6+  V G +  G + .6+ 

  V 
  G + .6+ > + kℎ _1 − `+ l− . 4 4
  G +

Theorem 2.

 + + 51 − ∏∈f51 − ∗ 77  + 4.  V∗ 
 ℎ +  . + 2 w + − − m G ∗, 2G ∗+  G ∗ + . 2G ∗+ H G ∗ + . 4 G ∗ + . 4 G ∗ + . 1* a

,

,

∗ ℎ ∗  +  G ∗ + .6+ +  V∗  G ∗ + .6+ 

 > 2 g + kℎ _1 − `+ li. 4 4
  G ∗ a

1*

Proof. By equating (7) to 0, one can obtain

26

ACCEPTED MANUSCRIPT ∗ .  4. ℎ   ℎ ∗     V∗  

 = 2 w− − + + + kℎ _1 − `+ l  ∗+ ∗+ ∗+ ∗ ∗ G  G  G 2
  G ∗ 2H   G + . 2H   G + . a

−


  V∗ H G ∗ + .   + 51 − ∏∈f51 − ∗ 77 + m.  G ∗+ 2

Multiply  G ∗ /2 G ∗ + 2. on only right term. Since

PO Q ∗

+PO Q ∗ R+S

=

*

5ÒÓO Ô∗ ÕÒÖ7 ÓO Ô∗

,

.  + 4. ℎ +  ℎ ∗  +  G ∗ + .6+ > 2 g− + + 2G ∗+  G ∗ + . 4 G ∗ + . G ∗, 4 a

1*

,

=

+R

*

ÒÖ ÓO Ô∗

< 1,

RI PT

1*

SC

∗  + + 51 − ∏∈f51 − ∗ 77 +  V∗  G ∗ + .6+ 


  V∗  + kℎ _1 − `+ l − + i. 4
  G ∗ 4 G ∗ + . 2G ∗+ H G ∗ + .

or equivalently, a

1*

,

M AN U

 + + 51 − ∏∈f51 − ∗ 77 .  + 4.
  V∗  ℎ +  + 2 w + − − m G ∗, 2G ∗+  G ∗ + . 2G ∗+ H G ∗ + . 4 G ∗ + . 4 G ∗ + . ,

∗ ℎ ∗  +  G ∗ + .6+ +  V∗  G ∗ + .6+ 

 > 2 g + kℎ _1 − `+ li. 4 4
  G ∗ a

EP

References

TE D

1*

Amaya, C.A., Carvajal, J., Castano, F., 2013. A heuristic framework based on linear programming to solve the

AC C

constrained joint replenishment problem (C-JRP). Int. J. Prod. Econ. 144 (1), 243-247. Ben-Daya, M., Raouf, A., 1994. Inventory models involving lead time as decision variable. J. Oper. Res. Soc. 45 (5), 579-582.

Cárdenas-Barrón, L. E., 2012. A complement to “A comprehensive note on: An economic order quantity with imperfect quality and quantity discounts”, App. Math. Model., 36 (12), 6338–6340. Cárdenas-Barrón, L.E., Chung, K.J., Treviño-Garza, G., (2014). Celebrating a century of the economic order quantity model in honor of Ford Whitman Harris. Int. J. Prod. Econ., 155, 1-7. Chakraborty, T., Giri, B.C., Chaudhuri, K.S., 2009. Production lot sizing with process deterioration and machine breakdown under inspection schedule. Omega. 37 (2), 257-271.

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ACCEPTED MANUSCRIPT Chuang, B.R., Ouyang, L.Y., Chuang, K.W., 2004. A note on periodic review inventory model with controllable setup cost and lead time. Comput. Oper. Res. 31 (4), 549-561. Dey, O., Giri, B.C., 2014. Optimal vendor investment for reducing defect rate in a vendor-buyer integrated system with imperfect production process. Int. J. Prod. Econ. 155, 222-228.

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Diaby, M., Cruz, J.M., Nsakanda, A.L., 2013. Shortening cycle times in multi-product, capacitated production environments through quality level improvements and setup reduction. Eur. J. Oper. Res. 228 (3), 526-535. Goyal, S.K., 1975. Analysis of joint replenishment inventory systems with resource restriction. J. Oper. Res. Soc. 26 (1), 197-203.

SC

Gunasekaran, A., Korukonda, A.R., Virtanen, I., Yli-olli, P., 1995. Optimal investment and lot-sizing policies for improved productivity and quality. Int. J. Prod. Res. 33 (1), 261–278.

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Haksever, C., Moussourakis, J., 2005. A model for optimizing multi-product inventory systems with multiple constraints. Int. J. Prod. Econ. 97 (1), 18-30.

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Hoque, M.A., 2006. An optimal solution technique for the joint replenishment problem with storage and

TE D

transport capacities and budget constraints. Eur. J. Oper. Res. 175 (2), 1033-1042. Hwang, H., Kim, D.B., Kim, Y.D., 1993. Multiproduct economic lot size models with investment costs for setup reduction and quality improvement. Int. J. Prod. Res. 31 (3), 691–703.

EP

Jaggi, C.K., Arneja, N., 2010. Periodic inventory model with unstable lead-time and setup cost with backorder discount. Int. J. App. Dec. Sc. 3 (1), 53-73. Jaggi, C.K., Ali, H., Arneja, N., 2014. Periodic inventory model with controllable lead time where backorder

AC C

rate depends on protection interval. Int. J. Ind. Engg. Comput. 5 (2), 235-248. Kang, C.W., Ullah, M., Sarkar, B., Iftikhar, H., Rehman, A., 2016. Impact of random defective rate on lot size focusing work-in-process inventory in manufacturing system. Int. J. Prod. Res. Lee, H.H., 2008. The investment model in preventive maintenance in multi-level production systems. Int. J. Prod. Econ. 112 (2), 816-828. Lin, Y.J., 2010. A stochastic periodic review integrated inventory model involving defective items, backorder price discount, and variable lead time. 4OR. 8 (3), 281-297. Moon, I.K., Choi, S.J., 1998. A note on lead time and distributional assumptions in continuous review inventory models. Comput. Oper. Res. 25, 1007-1012.

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ACCEPTED MANUSCRIPT Moon, I.K., Cha, B.C., 2006. The joint replenishment problem with resource restriction. Eur. J. Oper. Res. 173 (1), 190-198. Moon, I., Shin, E., Sarkar, B., 2014. Min–max distribution free continuous-review model with a service level constraint and variable lead time. App. Math. Comput. 229, 310-315.

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SC

lead time. J. Oper. Res. Soc. 47 (6), 829-832.

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TE D

Pal, B., Sana, S.S., Chaudhuri, K., 2014. Joint pricing and ordering policy for two echelon imperfect production inventory model with two cycles. Int. J. Prod. Econ. 155, 229-238. Pan, C.H., Hsiao, Y.C., 2001. Inventory models with back-order discounts and variable lead time. Int. J. Syst.

EP

Sci. 32 (7), 925-929.

Pan, C.H., Lo, M.C., Hsiao, Y.C., 2004. Optimal reorder point inventory models with variable lead time and backorder discount considerations. Eur. J. Oper. Res. 158 (2), 488-505.

AC C

Pan, C.H., Hsiao, Y.C., 2005. Integrated inventory models with controllable lead time and backorder discount considerations. Int. J. Prod. Econ. 93-94, 387-397. Pasandideh, S.H.R., Niaki, S.T.A., Nobil, A.H., Cárdenas-Barrón, L.E., 2015. A multiproduct single machine economic production quantity model for an imperfect production system under warehouse construction cost. Int. J. Prod. Econ. 169, 203-214. Porteus, E.L., 1986. Optimal lot sizing, process quality improvement and setup cost reduction. Oper. Res. 34, 137–144. Sana, S. 2010. A production–inventory model in an imperfect production process. Euro. J. Oper. Res., 200 (2), 451–464.

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ACCEPTED MANUSCRIPT Sana, S. 2011. A production-inventory model of imperfect quality products in a three-layer supply chain. Dec. Sup. Syst. 50 (2), 539–547. Sarkar, B., Sana, S. S., Chaudhuri, K., 2010a. Optimal reliability, production lotsize and safety stock: an economic manufacturing quantity model. Int. J. Manag. Sc. Eng. Manag. 5 (3), 192-202.

imperfect production process. Int. J. Prod. Econ. 155, 204-213.

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Sarkar, B., Moon, I.K., 2014. Improved quality, setup cost reduction, and variable backorder costs in an

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SC

74-82.

Sarkar, B., Mahapatra, A.S., 2015. Pericodic review fuzzy inventory model with variable lead time and fuzzy

M AN U

demand. Int. Trans. Oper. Res. 00, 1-31.

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TE D

management. App. Math. Model. 37 (5), 3138-3151. Sarkar, B., 2016. Supply chain coordination with variable backorder, inspections, and discount policy for fixed lifetime products. Math. Prob. in Engg. 2016.

EP

Sarkar, B., Cárdenas-Barrón, L.E., Sarkar, M., Singgih, M.L., 2014. An economic production quantity model with random defective rate, rework process and backorders for a single stage production system. J. Manuf.

AC C

Sys. 33 (3), 423-435.

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ACCEPTED MANUSCRIPT App. Maths. Comput. 224, 362-371. Sarkar, B., Mandal, B., Sarkar, S., 2015b. Quality improvement and backorder price discount under controllable lead time in an inventory model. J. Manuf. Sys. 35, 26-36. Sarkar, B., Moon, I., 2011. An EPQ model with inflation in an imperfect production system. App. Maths.

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SC

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EP

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Taleizadeh, A.A., Niaki, S.T.A., Meibodi, R.G., 2013c. Replenish-up-to multi-chance-constraint inventory control system under fuzzy random lost-sale and backordered quantities. Knowledge-Based Sys. 53, 147-

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random defective rate. J. Cleaner Prod.

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Wee, H.M., Jonas, Y., Chen, M.C., 2007. Optimal inventory model for items with imperfect quality and shortage backordering. Omega, 35 (1), 7-11. Wee, H.M., Widyadana, G.A., 2013. A production model for deteriorating items with stochastic preventive maintenance time and rework process with FIFO rule. Omega, 41 (6), 941-954. Wee, H. M., Wang, W. T., Cárdenas-Barrón, L.E., 2013. An alternative analysis and solution procedure for the EPQ model with rework process at a single-stage manufacturing system with planned backorders, Comput. Ind. Engg. J. 64(2), 748-755.

Figure captions 32

ACCEPTED MANUSCRIPT Figure 1. Multi-stage production process. Figure 2. Complex multi-stage production process. Figure 3. Graphical illustration of scaling parameter (λ) versus the TEAC of Example 1.

TE D

M AN U

SC

RI PT

Figure 4. Graphical illustration of scaling parameter ( ) versus the TEAC of Example 2.

Multi-stage production process

AC C

EP

Figure 1.

33

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

1500

1391.02 1098.22

AC C

1000

1879.46

EP

TEAC

2000

TE D

Figure 2. Complex multi-stage production process

535.56

500

264.62

0

0

0.75

1

1.25

2.5

5

∞

Scaling parameter λ

Figure 3. Graphical illustration of scaling parameter (λ) versus TEAC of Example 1.

34

ACCEPTED MANUSCRIPT

2500 1948.31

1554.18

1500

RI PT

TEAC

2000

945.03

1000

368.51

SC

500

0 -0.8

-0.5 Scaling parameter γ

-0.2

M AN U

-1

0

0

Figure 4. Graphical illustration of scaling parameter ( ) versus the TEAC of Example 2.

TE D

Table captions Table 1 Contribution of different authors. Table 2 Input parameters of Example 1.

EP

Table 3 Components of lead time.

Table 4 Set of work for production of item i.

AC C

Table 5 The results of Example 1.

Table 6 Summary of optimal solutions for each

in Example 1.

Table 7 The results of Example 2. Table 8 Summary of optimal solutions of each gamma in Example 2. Table 9 Sensitivity analysis of key parameters in Example 1. Table 10 Computational results for managerial insights regarding quality improvement.

35

M AN U

SC

RI PT

ACCEPTED MANUSCRIPT

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Table 1 Contribution of different authors.

Quality improvement

Porteus (1986)

Single stage

Multistage

Multiitem

Backorder Price discount

Controllable lead time

Budget constraint

√ √

√

Hong and Hayya (1995)

√

√

Gunasekaran et al.

√

√

√

Hong (1997)

√ √

Lee et al. (1997) Ouyang and Chang

Continuous review system

√

Hong et al. (1993)

(1995)

Periodic review system

√

AC C

Hwang et al. (1993)

Stochastic demand

EP

Author (year)

Inventory policy

√

√

√(fixed)

√

√

Ouyang et al. (2002)

√

√

√(fixed)

√

√

Lee (2005)

√

(2000)

Lee (2008)

√ √

36

ACCEPTED MANUSCRIPT √

Dey and Giri (2014)

√

√

Sarkar and Moon (2014)

√

√

√

√ √

√

Pasandideh et al. (2015) Sarkar et al. (2015a)

√

√

Sarkar et al. (2015b)

√

√

This model

√

√

√

√

√ √

ï

√

√

√

√

√

√

RI PT

Diaby et al. (2013)

√

√(C-JRP)

√Ù indicates complex multi-stage quality improvement, C-JRP indicates constrained joint replenishment

M AN U

SC

problem, and fixed indicates fixed backorder price-discount

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Table 2 Input parameters for Example 1.

item 2

item 3

item 4

3000

1500

600

200

25

12

7

1

47

50

45

43

6.25

6.25

6.25

6.25


 ($/unit)

25

22

20

24

140

143

150

147



70

80

75

75

0.90

0.90

0.95

0.85

work 1

work 2

work 3

work 4

0.18

0.15

0.1

0.2

 (units/year) 

 ($/order)

AC C

 ($/unit)

EP

item 1

ℎ ($/unit/year)

 ($/defective unit)

 ($/year)

37

ACCEPTED MANUSCRIPT "



410

450

400

430

0.00018

0.00023

0.0002

0.00025

 : $200/order, B: $8,000 (in Example 1), $6,000 (in Example 2)

Components of lead time.

RI PT

Table 3 normal duration (% (days)

minimum duration $% (days)

crashing cost per unit )% ($/day)

2

20

6

1.2

3

16

9

6

0.4

SC

20

5.0

Table 4

TE D

M AN U

1

Set of work for production of ith item. Stage 1

1

4

3 4

Stage 2

1

2

2

3

AC C

2

EP

Item

1

3

38

Stage 3

4

Table 5

4

3

L

5.6

22.4

57.4

4.

200

133

66.7

33.3

∗

8

6

4

4.652

G∗

1

1

1

1

*∗

1.815

1.940

2.107

2.046

*∗

71.92

71.44

70.96

71.12

∗
*

1

1

1

1

+∗

1.880

2.002

2.166

2.106

+∗

73.19

72.77

72.35

72.48

∗
+

Item 2

6

0

Item 1

The results of Example 1 for different values of λ.

4.

75

∗

4.154

4.867

G∗

1

1

1

*∗

1.815

1.940

2.092

2.027

*∗

71.92

71.44

71.00

71.17

∗
*

1

1

1

1

+∗

1.880

2.002

2.151

2.088

+∗

73.19

72.77

72.38

72.53

∗
+

6

TE D

57.4

100

6

1

L

57.4

4.

120

100

∗

6

4.269

4.991

G∗

1

1

1

1

*∗

1.815

1.940

2.081

2.017

*∗

71.92

71.03

71.03

71.20

∗
*

1

1

1

1

+∗

1.880

2.141

2.141

2.078

+∗

73.19

72.40

72.40

72.56

∗
+

EP

λ

8

L

22.4

150

8

Item 2

3

5.6

200

Item 1

4

0

3

22.4

160

8

8

4

5.6

200

Item 2

6

0

Item 1

8

39

0.75

1.00

1.25

AC C

M AN U

SC

RI PT ACCEPTED MANUSCRIPT

Table 5

The results of Example 1 for different values of λ.

57.4

4.

180

175

∗

4.596

5.346

G∗

1

1

*∗

2.051

1.988

*∗

71.10

71.29

∗
*

1

1

+∗

2.111

2.050

+∗

72.47

72.63

∗
+

TE D

L

EP

SC

∗
+

M AN U

+∗

Item 2

+∗

Item 1

∗
*

λ

*∗

Savings

*∗

TEAC

G∗

Process (work) quality (*106Ñ 

∗

72.61

Item 4

4.

2.059

RI PT

Item 3

L

1

2.50

71.26

Î∗

1.997

,∗

1

+∗

∗
Î

5.230

*∗

Î∗

150

Î∗

57.4

∗
,

3

,∗ 2088.1

,∗ 19683.71

1

72.77

1.80309 1.45797 5.96145 0.60744

75.89

2.002

74.57

2.157

1

2.089

1

71.44

73.19

72.45

1.940

1.880

2.121

1

1

1 6

71.92

71.08 180

1.815

2.060 5.6

1

1 6

8

4.489 577.77

200

160

20837.51

0

22.4

1.39806 1.13047 4.62231 0.47098

8

4

74.88

0

1879.46

1 1.985

23807.91

18570.84

75.77 1 1.04854 0.84785 3.46673 0.35324

2.09709 1.69570 6.93346 0.70648

2.216 76.15 75.35

74.42

1 2.054 1.862

2.149

1 1

3

22.4

72.77

Item 2

76.54

Saving

4

2.002

73.19

1

1.880

71.44

1

1.940

71.92

1

1.815

6

1

190

8

5.6

200

3

L

22.4

57.4

4.

200

200

200

∗

8

6

4.700

5.458

G∗

1

1

1

1

*∗

1.815

1.940

2.042

1.980

*∗

71.92

71.44

71.13

71.31

∗
*

1

1

1

1

+∗

1.880

2.002

2.102

2.041

+∗

73.19

72.77

72.49

72.65

∗
+

8

0

6

Item 1

1.934

TEAC

1391.02

5.00

1

Process (work) quality(*106Ñ )

∗
Î

1.72349 1.39360 5.69825 0.58062

19059.28

433.33

Item 4 Î∗ 74.62

2.01952 1.63298 6.67700 0.68035

20981.95

0

1209.2

4

5.6

200

40

Item 3 Î∗ 2.071 74.46

1.39806 1.13047 4.62231 0.47098

23807.91

Î∗

∗
,

1 2.134 74.88

1.04854 0.84785 3.46673 0.35324

,∗

,∗ 75.94 1 1.985

75.35

+∗

,∗ 2.139 75.80 1 1.862

*∗

1 2.201 76.15 1

1531.08

1 2.054 76.54

20240.73

1 1.934

20562.61

1098.22

6

0

Item 2 ∗
Î

1.68059 1.35892 5.55643 0.56617

19352.08

346.66

8

Item 1 Î∗

74.65

1.96513 1.58900 6.49719 0.66200

21068.62

0

Saving

Î∗

2.061

74.49

1.39806 1.13047 4.62231 0.47098

23807.91

TEAC

1

Process (work) quality(*106Ñ )

∗
,

1

2.124

74.88

1.04854 0.84785 3.46673 0.35324

Item 4

,∗ 75.96

1

1.985

75.35

Item 3 ,∗ 2.129

75.82

1

1.862

Inf (∞)

1 2.190

76.15

1

Î∗

1 2.054

76.54

,∗

1

1.934

+∗

1

*∗

AC C

ACCEPTED MANUSCRIPT

RI PT Process(work) quality(*106Ñ  Î∗

TEAC

Saving

M AN U

SC

0

Table 6

Summary of optimal solutions for each λ in Example 1.

∗

4

G∗

4

.∗

1

*∗

2.107

*∗

70.96

∗
*

1

+∗

2.166

+∗

72.35

∗
+

Item 2

λ

66.7

Item 1

0.75

2.50

1.25

1.00

160

120

100

4.489

4.269

4.154

4

4

4

1

1

1

2.060

2.081

2.092

71.08

71.03

71.00

1

1

1

2.121

2.141

2.151

72.45

72.40

72.38

TE D

Item 4 ,∗

Saving

264.62

TEAC

∗
Î

1.56919 1.26884 5.18812 0.52864

20185.68

86.66

21771.81

0

∞

5.00

200

180

4.700

4.596

EP

Item 3 +∗

Î∗

*∗

∗
,

∗
Î

,∗

Î∗

,∗ 590.03

173.33

21181.78

1

21241.95

0

1.60378 1.29681 5.30246 0.54029

76.01

1.39806 1.13047 4.62230 0.47098

23807.91

74.71

2.110

74.88 1.04854 0.84785 3.46673 0.35324

2.041

1 1 1.985 75.35

535.56

75.86 1 1.862

19914.74

2.171 76.15 1

1.86846 1.51083 6.17757 0.62946

1 2.054 76.54

74.54

1 1.934

2.103

1

Process(work) quality(*106Ñ)

Î∗ 74.73

1.82523 1.47588 6.03464 0.61489

21328.62

0

Item 4 Î∗ 2.032 74.56

1.39806 1.13046 4.62230 0.47098

23807.91

∗
Î

1.53680 1.24265 5.08101 0.51773

20450.3

0

4

4

1

1

2.042

2.051

71.13

71.10

1

1

2.102

2.111

72.49

72.47

41

Item 3 ∗
,

1 2.094 74.88

1.04854 0.84785 3.46673 0.35324

Î∗

,∗ 76.03 1 1.985

75.35

,∗

,∗ 2.101 75.88 1 1.862

+∗

1 2.161 76.15 1

*∗

1 2.054 76.54

291.62

1 1.934

Î∗

74.76

1.78490 1.44326 5.90129 0.60131

21415.28

0

Saving

Î∗

2.024

74.58

1.39806 1.13046 4.62230 0.47098

23807.91

TEAC

21480.19

1

Process(work) quality(*106Ñ)

∗
,

1

2.085

74.88

1.04854 0.84785 3.46673 0.35324

Item 4

,∗ 76.05

1

1.985

75.35

Item 3 ,∗ 2.093

75.90

1

1.862

Î∗

1 2.153

76.15

1

,∗

1

2.054

76.54

+∗

1

1.934

*∗

1

AC C

ACCEPTED MANUSCRIPT

SC

,∗

Î∗

TEAC

Saving

M AN U +∗

RI PT Process(work) quality(*106Ñ) *∗

TE D

∗
Î

EP

Î∗

Item 4 Î∗

Item 3 ∗
,

1098.22

,∗

1879.46

2.01952 1.63298 6.67700 0.68035

19352.08

535.56

,∗

18570.84

74.46

1.96513 1.58900 6.49719 0.66200

19914.74

264.62

2.09709 1.69570 6.93346 0.70648

2.134

74.49

1.86846 1.51083 6.17757 0.62946

20185.68

0

74.42

1

2.124

74.54

1.82523 1.47588 6.03464 0.61489

20450.3

2.149

75.80

1

2.103

74.56

1.78490 1.44326 5.90129 0.60131

1

2.201

75.82

1

2.094

74.58

75.77

1

2.190

75.86

1

2.085

2.216

1

2.171

75.88

1

Table 7

The

γ

0

– 0.2

– 0.5

42

1

1

2.161

75.90

1391.02

1

2.153

19059.28

1

AC C

ACCEPTED MANUSCRIPT

TE D

SC

Savings

M AN U

TEAC

Process (work) quality (*106Ñ  ∗
Î

Item 4 Î∗

Item 3 Î∗

Item 2 ∗
,

460.65

EP

RI PT Item 1 ,∗

Î∗

,∗

0

,∗

∗
+

21771.81

+∗

+∗

1.53680 1.24265 5.08101 0.51773

*∗

+∗

74.76

∗
*

2.024

*∗ 1

*∗ 76.05

G∗ 2.093

∗ 1

4. 72.65

L 2.041

71.31

1

1.980

0

1

20450.3

5.458

1.78490 1.44326 5.90129 0.60131

200

74.58

57.4

2.085

3

1

1

75.90

71.13

2.153

2.042

1

1

72.49

4.700

2.102

200

0

22.4

21415.28

4

1.39806 1.13046 4.62230 0.47098

2.002

74.88

1

1.985

71.44

1

1.940

76.15

1

2.054

6

1

200

72.77

5.6

0

6

23807.91

1.880

1.04854 0.84785 3.46673 0.35324

1

75.35

71.92

1.862

1.815

1

1

76.54

8

1.934

200

21311.16

368.51

Saving ∗
Î

1.58860 1.28453 5.25227 0.53518

20081.79

TEAC Î∗

74.72

1.84157 1.48909 6.08866 0.62040

Î∗

Î∗

2.037

74.55

,∗ 1

2.098

+∗

1

74.88

*∗

Process (work) quality(*106Ñ )

1

0

+∗

75.88

1.985

Item 4

73.19

8

+∗

2.165

Item 3

∗
*

1

Item 2

*∗

2.055

72.46

1

*∗

+∗

,∗

Î∗

43

Item 1 *∗ 1

2.115

∗
,

G∗ 71.27

1

,∗

∗ 1.994 71.09

,∗

4. 1 2.055

∗
+

L 5.280 1

76.15

76.02

160.8 4.555

2.054

2.106

57.4 172.3

1

1

3 22.4

72.77

72.62

4

2.002

99.73

1

21315.55

71.44

1.39806 1.13046 4.62230 0.47098 1.940

0

1

23807.91

Saving

6

1.04854 0.84785 3.46673 0.35324

TEAC

1184.87

188.5

75.35

20586.94

945.03

5.6

1.862

∗
Î

1.67743 1.35637 5.54599 0.56510

19505.27

249.32

6

1

Î∗

74.65

1.93784 1.56693 6.40694 0.65283

21165.96

0

76.54

Î∗

2.060

74.50

1.39806 1.13047 4.62230 0.47098

23807.91

1.934

∗
,

1

2.118

74.88

1.04854 0.84785 3.46673 0.35324

1

,∗

75.96

1

1.985

75.35

73.19

,∗

2.128

75.83

1

1.862

1.880

∗
+

1

2.185

76.15

1

1

+∗

72.56

1

2.054

76.54

71.92

+∗

2.077

72.42

1

1.934

1.815

∗
*

1

2.135

72.77

1

1

*∗

71.20

1

2.002

73.19

8

*∗

2.016

71.04

1

1.880

200

G∗

1

2.075

71.44

1

0

∗

5.001

1

1.940

71.92

8

4. 101.9

4.329

1

1.815

Process (work) quality (*106Ñ )

L 57.4

130.7

6

1

Item 4

3 22.4

171.2

8

Item 3

4

5.6

200

Item 2

6

0

Item 1

8

AC C

ACCEPTED MANUSCRIPT

SC

RI PT

∗ 4.703

G∗

1

1

*∗

1.940

2.098

2.041

*∗

71.44

70.98

71.13

∗
*

1

1

1

+∗

Î∗

74.59

∗
Î

2.05115 1.65856 6.78159 0.69100

1.78349 1.44212 5.89664 0.60083

18896.12

19816.28

TEAC

1554.18

1955.53

Saving

Process (work) quality (*106Ñ  Î∗

2.085

74.44

398.91

Item 4 ∗
,

1

2.141

21016.37

Item 3

,∗

75.90

1

1.39806 1.13047 4.62231 0.47098

Item 2

,∗

2.152

75.79

74.88

Î∗

∗
+

1

2.207

1.985

,∗

+∗

72.49

1

1

+∗

2.102

72.37

76.15

*∗

2.158

2.054

0

1

23807.91

Saving

1.04854 0.84785 3.46673 0.35324

TEAC

2499.74

75.35

19272.07

1948.31

+∗

,∗

Î∗

1.04854 0.84785 3.46673 0.35324

*∗

quality (*106Ñ )

1.86679 1.50948 6.17204 0.62889

18501.99

498.64

Process (work)

74.54

2.09709 1.69570 6.93346 0.70648

20916.64

∗
Î

74.42

1.39806 1.13047 4.62231 0.47098

72.77

Î∗

74.88

75.35

0

23807.91

2.002

M AN U

4. 43.07 4.090 1

1.862

Î∗

2.103

1

∗
,

1

2.149

76.54

TE D

L 57.4 89.10 6

1.934

,∗

75.86

1

1.985

1

,∗

2.170

75.77

1

1.862

73.19

∗
+

1

2.216

76.15

1

EP

Table 7

3 22.4 154.0

Item 1

4 5.6

1.880

+∗

72.45

1

2.054

76.54

1

+∗

2.120

72.35

1

1.934

71.92

6 1.815

∗
*

1

2.166

72.77

1

1

*∗

71.08

1

2.002

73.19

8

*∗

2.060

70.96

1

1.880

200

G∗

1

2.107

71.44

1

0

∗ 4.494

1

1.940

71.92

8

4. 3.834

4

1

1.815

Item 4

L 57.4 61.37

6

1

Item 3

3 22.4

142.5

8

Item 2

4 5.6

200

Item 1

6 0

44

The results of Example 2 for different values of λ. γ – 0.8

– 1.00

8

AC C

ACCEPTED MANUSCRIPT

M AN U

SC

RI PT Table 8 Summary of optimal solutions for each γ in Example 2.

Item 3

Item 4

Process (work) quality (*106Ñ )

1

2.098

EP

75.88

TE D

Savings

Item 2

2.165

*∗

1

∗
Î

72.46

Î∗

2.115

Î∗

1

1554.18

∗
,

TEAC

71.09

18896.12

45

Item 1

2.055

74.42

1948.31

,∗

Î∗

1

2.149

18501.99

,∗

,∗

4

1

0.691 00

∗
+

+∗

4.555

75.77

6.781 59

+∗

0

172.3

2.216

1.658 56

+∗

20450.3

368.51

– 0.20

∗
*

20081.79

945.03

*∗

0.601 31

19505.27

*∗

5.901 29

0.620 40

.∗

1.443 26

6.088 66

0.652 83

G∗

1.489 09

6.406 94

∗

1.841 57

1.566 93

γ

74.55

1.937 84

1

0.706 48

74.58

74.50

72.35

6.933 46

2.085

2.118

74.44 2.166

1.695 70

1

1

2.141 1

2.097 09

75.90

75.83

1 70.96

2.153

2.185

75.79 2.107

1

1

2.207 1

72.49

72.42

1 4

2.102

2.135

72.37

4

1

1

2.158

71.13

71.04

1

2.042

2.075

70.98

1

1

2.098

4

4

1

4.700

4.329 4

200

130.7 4.090

0

– 0.50 89.10 61.37

1.784 90

– 0.80

2.051 15

– 1.00

AC C

ACCEPTED MANUSCRIPT

Item 2

Item 3

Item 4

Process (work) quality (*106Ñ 

Î∗

TEAC

TE D

M AN U

RI PT

19059.28

,∗

2.01952 1.63298 6.67700 0.68035

18808.82

+∗

74.46

2.06849 1.67257 6.83891 0.69844

18354.17

*∗ 2.134

74.44

2.09709 1.69570 6.93347 0.70648

18137.51

∗
Î

1

2.144

74.42

2.09709 1.69570 6.93347 0.70648

24029.27

Î∗ 75.80

1

2.150

74.42

2.09710 1.69571 6.93349 0.70648

21330.39

Î∗ 2.201

75.78

1

2.150

74.88

2.09709 1.69570 6.93348 0.70648

∗
,

1

2.210

75.77

1

1.985

74.65

,∗ 72.38

1

2.216

75.77

1

2.060

,∗

2.151

72.36

1

2.216

76.15

1

∗
+

1 2.161

72.35

1

2.054

75.96

+∗

71.00 1

2.166

72.35

1

2.128

+∗

2.092 70.97

1

2.166

72.77

1

∗
*

1

2.102 70.96

1

2.002

72.56

SC

Item 1

*∗

1

2.107 70.96

1

2.077

1

1

1

1

2.149

2.149

2.149

2.149

74.42

74.42

2.79613 2.26095 9.24467 0.94197

1.67764 1.35655 5.54675 0.56518

18417.04

18690.13

EP

Table 9

*∗

4.055 1

2.107

71.44

1

G∗

83.33 4 1

1.940

71.20

∗

+25% 50 4 1

2.016

Changes (in %)

Sensitivity analysis of key parameters of Example 1.

Parameters

–25% 33.33 4 1

4.154

–50% 66.67 4

100

+50% 66.67

+50%

+25%

72.23

15864.56

2.226

1.83544 1.48413 6.06840 0.61833 1

74.29

70.83

2.210

2.168

1

1

75.66

4.570

2.275

66.67

1

–25%

72.09

12875.82

2.306

1.50706 1.21860 4.98270 0.50771

1

74.14

70.67

2.289

2.249

1

1

74.14

5.566

2.353

66.67

1

–50%

72.35

75.77

2.149

18787.60

2.166

72.35

2.216

75.77

1

2.148

1.39805 1.13046 4.62228 0.47098

1

2.166

72.35

1

2.216

75.77

1

2.141

74.42

70.96

1

2.166

72.35

1

2.216

75.77

1

2.149

2.107

70.96

1

2.166

72.35

1

2.214

75.78

1

1

2.107

70.96

1

2.166

72.35

1

2.208

75.77

4

1

2.107

70.96

1

2.166

72.35

1

2.216

66.67

4

1

2.107

70.96

1

2.165

72.36

1

+50% 66.67

4

1

2.107

70.96

1

2.158

75.77

+25%

66.67

4

1

2.107

70.97

1

2.216

–25%

66.67

4

1

2.105

70.98

1

–50%

66.67

4

1

2.099

75.77

+50%

66.67

4.017

1

2.216

+25%

66.67

4.081

1

–25%

66.67

74.44

74.43

74.42

74.42

74.42

1.02773 0.83102 3.39792 0.34623

1.56606 1.26631 5.17776 0.52758

2.62137 2.11964 8.66686 0.88310

3.14566 2.54359 10.4003 1.05972

4.19425 3.39149 13.8672 0.14130

17835.78

18217.49

18897.24

19196.73

18200.28

46



ℎ





–50%

AC C

ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT

Table 10 Computational results for managerial insights regarding quality improvement.

 ℎ


 

Process (work) quality + , Î 0.00025 0.00015 0.0001 0.00025 0.00015 0.0001 0.0002 0.00015 0.0001 0.00025 0.0001 0.0001 0.00025 0.00015 0.00005 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.00015 0.0002 0.0002 0.0002 0.00015 0.0002 0.0002 0.0002 0.00015

Total cost 13461.19 13353.79 13357.05 13371.65 13394.65 13461.20 13373.74 13367.97 13356.47 13350.71

Percentage deviation 0.7978 0.7736 0.6652 0.4943 0.6497 0.6926 0.7780 0.8208

Item 1 +50% 10.5 10.5 10.5 67.5 67.5 67.5 30 30 30 225 225 225 112.5 112.5 112.5

Item 2 +25% 8.75 8.75 8.75 56.25 56.25 56.25 25 25 25 187.5 187.5 187.5 93.75 93.75 93.75

Item 3 –25% 5.25 5.25 5.25 33.75 33.75 33.75 15 15 15 112.5 112.5 112.5 56.25 56.25 56.25

Item 4 –50% 3.5 3.5 3.5 22.5 22.5 22.5 10 10 10 75 75 75 37.5 37.5 37.5

* 0.0002 0.00015 0.0002 0.0002 0.00015 0.0002 0.0002 0.00015 0.0002 0.0002 0.00015 0.0002 0.0002 0.00015 0.0002

Process (work) quality + , 0.0002 0.0002 0.0002 0.0002 0.0002 0.00015 0.0002 0.0002 0.0002 0.0002 0.0002 0.00015 0.0002 0.0002 0.0002 0.0002 0.0002 0.00015 0.0002 0.0002 0.0002 0.0002 0.0002 0.00015 0.0002 0.0002 0.0002 0.0002 0.0002 0.00015

Total cost 13461.20 13362.23 13362.23 13461.20 13362.23 13362.23 13461.20 13362.23 13362.23 13497.52 13398.55 13398.55 13217.23 13133.51 13164.00

Percentage deviation 0.7352 0.7352 0.7353 0.7353 0.7352 0.7352 0.7332 0.7332 0.6334 0.4027

RI PT

* 0.0003 0.00025 0.0003 0.0003 0.0003 0.0002 0.00015 0.0002 0.0002 0.0002

AC C



Work 4 –50% 0.0001 0.0001 0.0001 0.0001 0.0001 20 20 20 20 20

EP

Parameters

Work 3 –25% 0.00015 0.00015 0.00015 0.00015 0.00015 30 30 30 30 30

SC

 × "

Work 2 +25% 0.00025 0.00025 0.00025 0.00025 0.00025 50 50 50 50 50

M AN U



Work 1 +50% 0.0003 0.0003 0.0003 0.0003 0.0003 60 60 60 60 60

TE D

Parameters

47

Î 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002