manufacturer-retailer system with imperfect quality

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An integrated inventory model for suppliermanufacturer-retailer system with imperfect quality and inspection errors

Wakhid Ahmad Jauhari*) Department of Industrial Engineering, Sebelas Maret University, Jl. Ir Sutami 36 A, Surakarta 57126, Indonesia Fax: +62 271 632110 Email: [email protected]

Amanda Sofiana Production System Laboratory, Department of Industrial Engineering, Sebelas Maret University, Jl. Ir Sutami 36 A, Surakarta 57126, Indonesia Fax: +62 271 632110 Email: [email protected]

Nughthoh Arfawi Kurdhi Department of Mathematics, Faculty of Mathematics and Natural Science, Sebelas Maret University, Jl. Ir Sutami 36 A, Surakarta 57126, Indonesia Email: [email protected]

Pringgo Widyo Laksono Department of Industrial Engineering, Sebelas Maret University, Jl. Ir Sutami 36 A, Surakarta 57126, Indonesia Fax: +62 271 632110 Email: [email protected] *) Corresponding Author

Biographical Notes Wakhid Ahmad Jauhari is currently a lecturer in Sebelas Maret University. He obtained bachelor and master degrees, both in Industrial Engineering, from Sepuluh Nopember Institute of Technology (ITS) in Surabaya. His research’s interests include modelling inventory, supply chain management, simulation study and manufacturing design. Amanda Sofiana is a Researcher in Production System Laboratory, Sebelas Maret University. Her research interests are inventory management, production management and operations research.

Nughthoh Arfawi Kurdhi received his BS degree from Sebelas Maret University (2007), Indonesia, and an MS degree from Gadjah Mada University (2010), Indonesia, both in applied mathematics. He is working as a lecturer and researcher in the Mathematics Department at Sebelas Maret University, Indonesia. His research interests include operations research, optimization, supply chain management and combinatorics. He has published at international conferences, international journal papers, and books. He is currently a coordinator of Research Group Operations Reseach and Optimization at Sebelas Maret University.

Pringgo Widyo Laksono received his bachelor degree from Industrial Engineering Universitas Diponegoro (2004), Indonesia and Master of Engineering from Universitas Gadjah Mada (2012), Indonesia. Since working as lecture in Industrial Engineering Departement Universitas Sebelas Maret, he has published articles at proceeding conference, journal and books. His research interests include industrial engineering science such as production

and manufacturing systems, design

working system, optimization, and operation management. He is currently as Chief of Center of Study for Technology Development and Industrial Collaboration, Institute for Research and Community Services. Universitas Sebelas Maret.

Abstract This paper develops an integrated inventory model for a three-stage supply chain system consisting of single-supplier, single-manufacturer, and single-retailer. The demand is deterministic and the production process is imperfect and produces a certain number of defective items. The retailer conducts an inspection process to screen out the defective items from the shipment lot. The inspector may incorrectly classify non-defective item as

defective or incorrectly classify a defective item as non-defective. This model provides an optimal solution for the expected integrated total annual cost. A numerical example is presented for illustrative purpose.

Keywords: Supply chain, integrated inventory model, three-stage system, imperfect quality, inspection errors.

1. Introduction At the present time, supply chain management of a product from the producer to the final consumer becomes very important in the world of business and economics. An effective and efficient supply chain management will be able to accomplish the goals of the supply chain, which is to minimise the costs thereby increasing the efficiency of the logistics process of the product itself. The goal can be realized through the integration and communication between relevant stages. One of that influenced aspect is the management of the inventory system. A well-integrated inventory management is a key to a success of the supply chain system. If there is no proper coordination between the stages involved in the system, then the overall gain can not be achieved by all stages. Therefore, we need the integration of inventory management through better coordination by all stages of the supply chain related to decide the amount of the size of the production and delivery of products. The determination of production lot size and delivery of products to consider the benefits of all stages related to the supply chain is also called the joint economic lot-sizing problem (JELP). A major stream of the research in this area focused on developing vendor-buyer model considering defective items. This due to the fact that the vendor’s production system may be deteriorated during the production process, hence a certain number of defective items will be produced. Some scholars, including Khan and Jaber (2011), Jauhari et al. (2014), investigated the impact of defective items on JELP under the two-stage system. However, a three-stage JELP addressing imperfect quality and imperfect inspection was rarely discussed in the paper. In this paper, we intend to study the impact of incorporating defective items and inspection errors in JELP under deterministic demand. We propose an

integrated inventory model for single-supplier, single-manufacturer, single retailer system which simultaneously determines shipment size, cycle time, production batch and number of raw material’s shipment. In addition, a simple procedure is proposed to find the decision variables for minimising total cost. The remainder of the paper is organised as follows. Section 2 shows literature review on inventory model. Section 3 describes the notations and assumptions used in developing the proposed model. Section 4 presents model development and Section 5 provides a solution methodology. Finally, numerical examples and conclusions are presented in Sections 6 and 7, respectively.

2. Literature Review Most of JELP were studied in a two-stage supply chain system, namely integrated vendor-buyer model. JELP was first published by Goyal (1976), which considered a single buyer-single vendor model with deterministic demand and infinite production rate. Then, Banarjee (1986) developed a more realistic case by assuming that the production rate is limited and the delivery is made by adopting lot-for-lot policy. Subsequently, Goyal (1988) improved the lot-for-lot delivery policy assumptions by developing more general JELP that resulted a lower total cost. He showed that producing a batch which is made of equal shipments generally result lower total cost, but the whole batch must be finished before first shipment is made. In contrast to previous works that employed equal-sized shipment policy, some scholars, including Hill (1997), Hill (1999), Zhou and Wang (2007), Giri and Roy (2013) developed JELP with unequal-sized shipment policy. A comprehensive review on JELP was provided by Ben-Daya et al. (2008) and Glock (2012). In recent years, studies of JELP have grown and expanded to the supply chain system that involves multi-stages, to accommodate the wider integration of optimization and closer to the real case in the business world. Khouja (2003) was the first researcher to publish a three-stage JELP, where there are one or more companies in each stage. He examined three coordination mechanism in three-layer supply chain system that is, equal cycle time mechanism, integer multipliers of the cycle time and integer power of two multipliers mechanism.Then, Ben-Daya and Al-Nassar (2008) used the idea of Lu (1995), which uses lot streaming policy, to three-stage JELP by using a constant integer multipliers

of cycle time mechanism. Further, Ben-Daya et al. (2013) expanded the model of BenDaya and Al-Nassar (2008) by adding the idea of Lee (2005), which considers raw material procurement, to the three-stage JELP. They proposed an integrated inventory model for supply chain system consisting of single-supplier, single-manufacturer and multi-buyers under deterministic demand. Studies of JELP have been done by many researchers, but those researches are limited by the assumption that each product produced by the manufacturer is on a perfect quality. In other hand, the real case can be ensured that there are defective products in the production process. Porteus (1986) probably the first to introduce an inventory model that shows a significant relationship between the quality of the product and lot size. Schwaller (1988) developed an Economic Order Quantity (EOQ) with the assumption that there is a known defect in the proportion of the size of each lot, and there is a fixed cost of inspection for sorting and removing the defective items. Salameh and Jaber (2000) developed an EOQ model in which each shipment contains some defective items. He proposed an inspection process to screen out the defective items from the lot and assumed that the defective items founded by the inspector will be sold at discounted price at the end of the inspection time. The basic model of Salameh and Jaber (2000) was then developed by many scholars into various assumptions. Wee et al. (2007) added the case of shortage backordering. Bera et al. (2007) investigated the impact of incorporating learning effect on the fuzzy inventory model. Bazan et al. (2014) considered vendor managed inventory (VMI) with consignment stock (CS) agreement for vendor-buyer system with defective items. They introduced various managerial decisions pertaining defective items and applying minor setups for restoration. Rad et al. (2014) developed an integrated vendorbuyer model with defective items and shortages. They used the assumption that the market demand is sensitive to buyer’s selling price. They showed that the coordinated model and backordering improve the total profit of the supply chain system. Paul et al. (2014) studied a joint replenishment model for multiple items in which each item has a certain percentage of defective units in each shipment. Further, Alkhedher et al. (2013) and Jauhari (2014) investigated the impact of imperfect production process in inventory decisions where the demand is modelled by a normal probability distribution.

More recently, an inventory model with considering human error factor has also received a significant research attention. Raouf et al. (1983) was the first researcher to study human error factor in the inspection process with different characteristics misclassification. Duffuaa and Khan (2005) then extended the model of Raouf et al. (1983) by proposing inspection plans for critical components in which the inspector can commit a number of misclassifications. Yoo et al. (2009) studied the impact of imperfect production process and inspection errors in EOQ model. They proposed two types of inspection errors: type 1 inspection error for incorrectly classifying a non-defective item as defective and the type 2 inspection error for incorrectly classifying a defective item as non-defective. Hsu and Hsu (2012) developed the model of a two-stage JELP taking into consideration of defective items and inspection errors with refund system. Hsu and Hsu (2013) then developed two Economic Production Quantity (EPQ) models with imperfect production processes, inspection errors, planned backorders and sales returns. The objective of the models are to determine the optimal production batch and the maximum backorder level. Khan et al. (2014) proposed a mathematical model for determining optimal inventory decisions for vendor-buyer system by considering inspection errors and learning in production. As mentioned in above description, a numerous papers have been written in response to the global competition. However, most of the JELP models deal with two-stage inventory system composed of a vendor and a buyer. Even when a three-stage JELP models are considered, most of the models are based on restrictive assumptions such as of the perfect production and inspection processes. Therefore, there is a need to analyse model that relax those assumptions to allow for a more realistic production-inventory system in supply chain. This paper proposes a mathematical model for a three-stage supply chain system consisting of single-supplier, single-manufacturer and single-retailer under deterministic demand, defective items and inspection errors.

3. Notations and Assumptions 3.1 Notations To develop the mathematical model, we use the following notations: D

annual demand

Ps

production rate of the supplier

Pm

production rate of the manufacturer

Q

shipment size from manufacturer to retailer

h0

holding cost per unit per year for supplier’s raw materials

hs

holding cost per unit per year for the supplier’s finished product and manufacturer’s raw materials per unit time

hm

holding cost per unit per year for the manufacturer’s finished products

hB

holding cost per unit per year for retailer’s finished goods

Os

supplier’s raw material ordering cost

Om

manufacturer’s raw material ordering cost

OB

retailer’s finished product ordering cost

Sm

manufacturer's setup cost

Ss

supplier's setup cost

ci

retailer’s inspection costs

cw

cost of producing defective items to the supplier

caB

retailer's cost of a post-sale defective item

caV

manufacturer's cost of a post-sale defective item

caS

supplier's cost of a post-sale defective item

cr

cost of rejecting a non-defective item

B1

number of items that are classified as defective in each delivery of Q units

B2

number of items that are returned from the market in each delivery of Q units

e1

probability of type 1 error

e2

probability of type 2 error

γ

probability that an item produced is defective

P

proportion of B2 are caused by defects of raw materials from the supplier

x

inspection rate

Ts

supplier’s cycle time

Tm

manufacturer’s cycle time

T

retailer’s cycle time

K1

integer multiplier of the manufacturer’s cycle time, Ts = K1 Tm

K2

integer multiplier of the retailer’s cycle time, Tm = K2 T (T s = K 1 K 2 T)

m1

number of raw material shipments received by the supplier within a cycle

m2

number of raw material shipments received by the manufacturer within a cycle

TCr

retailer’s total cost

TCm

manufacturer’s total cost

TCs

supplier’s total cost

EJTC expected joint total cost

3.2 Assumptions The assumptions used in this study are: 1. The demand rate is known and constant over time. 2. The shortage is not allowed. 3. The production rate is greater than the demand rate (P> D). 4. Each lot shipped from manufacturer to retailer contains a defective product with a defect rate of γ. 5. The process occurs with imperfect inspections, where there is a probability of classifying a non-defective item as defective (e1) with a probability density function f(e1), and the probability of classifying a defective item as non-defective (e2) with a probability density function f(e2 ). 6. The retailer returns all items classified as defective and those returned from the customers to the manufacturer at the end of inspection time. 7. Consumers who purchase defective product will notice the quality problems and return them to the retailer. The supplier, manufacturer and retailer will suffer the cost of postsale failure of the returned items from the market (e.g., loss of good will).

3.3 Problem Description In this study, we develop an inventory model for a three-stage supply chain system (supplier-manufacturer-retailer) for a single product by considering defective items and inspection

errors.

Supplier,

as

the

first-stage,

receives

raw

materials

from

its supplier in batch size of K1K2Q and then processes them into semi-finished goods with a certain production rate of Ps. The manufacturer, as the second-stage, then receives goods from a supplier in the production batch size of K2Q and processes them into finished goods

with a production rate of Pm. Finished goods are then shipped to retailer as a third-stage to meet the customer demand which is deterministic by D per year. Delivery of goods is allowed during the production time. Retailer orders Q units of finished goods to the manufacturer and incurs a fixed ordering cost OB. Manufacturer incurs setup cost Sm for producing finished goods in a production batch of K2Q units. The manufacturer also orders raw materials in the form of semi-finished goods to the supplier and any delivery will be charged an ordering cost Om. Supplier incurs production setup cost Ss for producing a number of semi-finished goods in a production batch K1K2Q. The supplier also orders raw materials to its supplier and any delivery will be charged an ordering cost Os. For each delivery made from manufacturer to retailer, there will be an inspection process performed by the retailer within the inspection rate of x, the inspection cost of ci per unit, and defect rate γ. When the inspection takes place, the inspector may incorrectly classify an item that is non-defective as a defective item, as referred to a type 1 error, with a probability of e1 , and incorrectly classify a defective item as nondefective item, as referred to a type 2 error, with probability of e2.

4. Model Formulation As mentioned in above section, we attempt to develop a three-stage JELP under defective items, inspection errors and deterministic demand. Our aim is to find a lot size that minimises total cost for three parties in supply chain. Inventory costs incurred by retailer, manufacturer, and supplier, as well as the joint total cost are calculated by following the cycle time of the supplier (K1K2T) with the aim to facilitate the formulation of the model calculations per cycle period is same for all stages. 4.1 Retailer’s cost formulation The total cost for the retailer consists of retailers’s holding cost (HCr), ordering cost, inspection cost, and post-sale failure cost. Since the formulation of the model is calculated over a period of the longest cycle time which is the cycle time of supplier (K1K2T), then the equation of retailer’s cycle time is multiplied by K1K2 later on. Fig. 1 shows the retailer’s inventory level for each ordering cycle. By definition, B1, B2, t1 are calculated as follows:

𝐵1 = 𝑄(1 − 𝛾 )𝑒1 + 𝑄𝛾(1 − 𝑒2 )

(1)

𝐵2 = 𝑄𝛾𝑒2

(2)

𝑡1 = 𝑄⁄𝑥

(3)

We assume that for each item returned from the market, a good quality item is replaced to the customer. Based on this assumption, there are two kinds of demand, the regular demand and the demand to replace the returned items. Let D' be the effective demand, then we have 𝐷 ′ = 𝐷 + 𝐵2 ⁄𝑇. By definition, the cycle length of each delivery of size Q is 𝑇 = (𝑄 − 𝐵1 )⁄𝐷′ . Substituting 𝐷 ′ = 𝐷 + 𝐵2 ⁄𝑇 and solving the equation, we have: 𝑇=

(𝑄−𝐵1 −𝐵2 ) 𝐷

=

𝑄(1−𝛾)(1−𝑒1 )

(4)

𝐷

Inventory level

Q

Q - B1

L

B1

L

A

B

B2 time

t1 T T

Fig. 1. Retailer’s inventory profile The holding cost per retailer’s delivery cycle is : (𝑄−𝐵1 )𝑇

𝐻𝐶𝑟 = ℎ𝐵 { 𝐵1 𝑡1 +

2

+

𝐵2 𝑇 2

}

(5)

Substituting B1 into equation (1), B2 into equation (2), and t1 into equation (3), then the holding cost per supplier’s cycle time is given by: (1−𝛾)𝑒1 +𝛾(1−𝑒2 )

𝐻𝐶𝑟 = ℎ𝐵 𝑄 2 𝐾1 𝐾2 {

𝑥

1

+ 2𝐷 (1 − 𝛾 )(1 − 𝑒1 ){(1 − 𝛾 )(1 − 𝑒1 ) + 2𝑒2 𝛾 } (6)

After adding ordering cost, inspection cost, and post-sale failure cost for each supplier’s cycle time, the retailer’s total cost per supplier’s cycle time is given by:

(1−𝛾)𝑒1 +𝛾(1−𝑒2 )

𝑇𝐶𝑟 = ℎ𝐵 𝑄 2 𝐾1 𝐾2 {

𝑥

1

+ 2𝐷 (1 − 𝛾 )(1 − 𝑒1 ){(1 − 𝛾 )(1 − 𝑒1 ) + 2𝑒2 𝛾 }} +

𝑂𝐵 𝐾1 𝐾2 + 𝑐𝑖 Q 𝐾1 𝐾2 + 𝑐𝑎𝐵 𝑄𝛾𝑒2 𝐾1 𝐾2

(7)

4.2 Manufacturer’s cost formulation The total cost for the manufacturer consists of manufacturer’s raw material holding cost (HCRm), manufacturer’s finished product holding cost (HCFm), production setup cost, ordering cost, cost for producing defective items, type 1 inspection error cost, and post-sale failure cost. Since the formulations of the model are calculated over a period of the longest cycle time which is the cycle time of supplier (K1K2T), then the equation of manufacturer’s cycle is multiplied by K1 later on. Inventory level

K2Q/m2 Pm

Time K2T

Fig. 2. Manufacturer’s raw material inventory profile Inventory level

Q

T K2T K2Q/Pm

Q

T K2Q

(K2-1) Q

Q

time (K2-1) T K2T

Fig. 3. The combined inventory profile for the manufacturer and the retailer

Fig. 2 shows the average inventory of the manufacturer’s raw material in the area under the triangles divided by K2T. The holding cost per supplier’s cycle time for the manufacturer’s raw material is given as follows: 𝐾 𝐾 2 𝑄2

𝐻𝐶𝑅𝑚 = ( 21𝑚2 𝑃 )ℎ𝑠

(8)

2 𝑚

From Fig. 3 we can find the holding cost of manufacturer’s finished product by calculating the inventory level of manufacturer’s finished product per manufacturer's production cycle time. The inventory level of manufacturer’s finished product is obtained by reducing the area of a trapezoid which represents the total cumulative production per manufacturer's production cycle time with shading area representing the cumulative demand of retailer per cycle. The formulation is given as follows: 𝑄

𝐻𝐶𝐹𝑚 = ℎ𝑚 {[𝐾2 𝑄 (𝑃 + (𝐾2 − 1)𝑇) −

𝐾2 𝑄(𝐾2 𝑄/𝑃𝑚 ) 2

𝑚

] − 𝑇[𝑄 + 2𝑄 + ⋯ + (𝐾2 − 1)𝑄 ]} (9)

Then from the above equation, T is substituted with T from equation (4), thus holding cost of manufacturer’s finished product per supplier’s cycle time becomes: 1

(𝐾2 −1)(1−𝛾)(1−𝑒1)

𝐾

𝐻𝐶𝐹𝑚 = ℎ𝑚 𝐾1 𝐾2 𝑄 2 {𝑃 − 2 𝑃2 + ( 𝑚

2𝐷

𝑚

)}

(10)

After adding production setup cost, ordering, cost, the cost for producing defective items, type 1 inspection error cost, and post-sale failure cost, the manufacturer’s total cost per supplier’s cycle time is given by: 𝐾 𝐾 2 𝑄2

1

(𝐾2 −1)(1−𝛾)(1−𝑒1)

𝐾

𝑇𝐶𝑚 = ( 21𝑚2 𝑃 ) ℎ𝑠 + ℎ𝑚 𝐾1 𝐾2 𝑄 2 {𝑃 − 2 𝑃2 + ( 2 𝑚

𝑚

𝑚

2𝐷

𝑂𝑚 𝑚2 𝐾1 + 𝑐𝑤 𝐾1 𝐾2 𝑄 𝛾 + 𝑐𝑟 𝐾1 𝐾2 𝑄(1 − 𝛾 )𝑒1 + 𝑐𝑎𝑉 𝐾1 𝐾2 𝑄𝛾𝑒2

)} + 𝑆𝑚 𝐾1 + (11)

Inventory level

K1K2Q/m1 Ps

K1K2T

Fig. 4. Supplier’s raw material inventory profile

B1

B2 B1 B2

B3

B3

B2

B1

Fig. 5. The combined inventory profile for the supplier and the manufacturer 4.3 Supplier’s cost formulation The total cost for the supplier consists of supplier’s raw material holding cost (HCRs), supplier’s finished product holding cost (HCFs), production setup cost, ordering cost, and post-sale failure cost. Fig. 4 shows the average inventory of the supplier’s raw material in the area under the triangles divided by K1K2T. The holding cost per supplier’s cycle time for the supplier’s raw material is:

𝐻𝐶𝑅𝑠 = (

𝐾1 2 𝐾2 2𝑄2 2 𝑚1 𝑃𝑠

)ℎ0

(12)

Fig. 5 shows the inventory profile of the supplier and the manufacturer. The trapezoid area represents the total cumulative production of supplier per supplier's production cycle time and the coloured area in the bottom portion gives the cumulative demand of the manufacturer. The inventory of the supplier’s finished goods cannot be calculated by subtracting the inventory of the manufacturer from the system’s inventory as done for the manufacturer. Since the manufacturer does not receive continuous shipments of incoming goods as in the case of the retailer, the on-hand inventory at the beginning of the supplier’s cycle depend on the length of manufacturer’s production downtime. Thus, in order to find the average inventory of supplier’s finished goods, we subtract the cumulative demand of the manufacturer (coloured area) from the supplier’s cumulative production (area under the trapezoid dashed line). Let A be the area corresponding to the cumulative supplier’s production. The cumulative supplier’s production is represented by the area of the triangle till the end of production plus the area of the rectangle from the end of production until the end of the cycle. Thus, we have: 1

𝐴 = (𝐾1 𝐾2 𝑄 × 𝐾1 𝐾2 𝑇) − (2 × 𝐾1 𝐾2 𝑄 ×

𝐾1 𝐾2𝑄 𝑃𝑠

)

(13)

Substituting T in equation (13) with T in equation (4), then A becomes: 2 𝐾1 (1−𝛾)(1−𝑒1)

1

𝐴 = 2 𝐾1 𝐾2 2 𝑄 2 (

𝐷



𝐾1 𝑃𝑠

)

(14)

Then we have B which is the area corresponding to the cumulative demand of the manufacturer. The area B is divided into 3 parts, namely the B1, B2, and B3. 1 𝐾1 𝐾2 2𝑄2 (𝑚2 +1)

𝐵1 = 2

𝑚2 𝑃𝑚

1

𝐵2 = 2 ((𝐾1 (𝐾1 + 1)𝐾2 2 𝑄𝑇 −

𝐾1 (𝐾1 +1)𝐾22 𝑄2 𝑃𝑚

1

𝑄 (1−𝛾)(1−𝑒1 )

2

𝐷

= (𝐾1 (𝐾1 + 1)𝐾2 2 𝑄 1

= 2 𝐾1 𝐾2 2 𝑄 2 ((𝐾1 + 1)

(1−𝛾)(1−𝑒1 ) 𝐷

− −



2 𝐾1 𝐾22 𝑄2 𝑚2 𝑃𝑠

𝐾1 (𝐾1 +1)𝐾2 2 𝑄2 𝑃𝑚 (𝐾1 +1) 𝑃𝑚

2

−𝑚

2 𝑃𝑠

)



2 𝐾1 𝐾2 2 𝑄2 𝑚2 𝑃𝑠

)

)

1 𝐾1 (𝐾1 −1)𝐾22 𝑄2

𝐵3 = 2

𝑃𝑚

The area B is simply the sum of the three areas B1, B2, and B3 which is given by :

(𝑚 +1)

1

𝐵 = 2 𝐾1 𝐾2 2 𝑄 2 ( 𝑚 2𝑃

2 𝑚

(𝐾1 +1)(1−𝛾)(1−𝑒1)

+

𝐷



(𝐾1 +1) 𝑃𝑚

2

−𝑚

+

2 𝑃𝑠

(𝐾1 −1) 𝑃𝑚

)

(15)

The formulation to get the total holding cost of supplier’s finished products per supplier’s production cycle time is obtained by reducing the area A with area B (1−𝛾)(1−𝑒1 )(𝐾1 −1)

1

𝐻𝐶𝐹𝑠 = 2 𝐾1 𝐾2 2 𝑄 2 ℎ𝑠 (

𝐷

1

2

1

1

− 𝑃 (𝐾1 − 𝑚 ) − 𝑃 (𝑚 −1)) 𝑠

2

𝑚

(16)

2

After adding production setup cost, ordering cost, and post-sale failure cost, the supplier’s total cost is given by: 𝐾1 2 𝐾2 2𝑄2

𝑇𝐶𝑠 = (

2 𝑚1 𝑃𝑠

1

) ℎ0 + 2 𝐾1 𝐾2 2 𝑄 2ℎ𝑠 (

(1−𝛾)(1−𝑒1 )(𝐾1 −1) 𝐷

1

2

1

1

− 𝑃 (𝐾1 − 𝑚 ) − 𝑃 (𝑚 −1)) + 𝑠

2

𝑚

2

𝑆𝑠 + 𝑂𝑠 𝑚1 +𝑐𝑎𝑆 𝑃𝐾1 𝐾2 𝑄𝛾𝑒2

(17)

4.4 The integrated supplier-manufacturer-retailer inventory model The formulation of the total supplier-manufacturer-retailer integrated cost per supplier’s production cycle time is the sum of equation (7), equation (11), and equation (17), which is given by: (1−𝛾)𝑒1 +𝛾(1−𝑒2 )

𝐽𝑇𝐶 = {

𝑥

1

+ 2𝐷 (1 − 𝛾 )(1 − 𝑒1 ){(1 − 𝛾 )(1 − 𝑒1 ) + 2𝑒2 𝛾 }} 𝑄 2 ℎ𝐵 𝐾1 𝐾2 + 𝐾 𝐾 2𝑄2

1

𝐾

𝑂𝐵 𝐾1 𝐾2 + 𝑐𝑖 Q 𝐾1 𝐾2 + 𝑐𝑎𝐵 𝑄𝛾𝑒2 𝐾1 𝐾2 + ( 21𝑚2 𝑃 ) ℎ𝑠 + ℎ𝑚 𝐾1 𝐾2 𝑄 2 {𝑃 − 2 𝑃2 + 2 𝑚

(𝐾2 −1)(1−𝛾)(1−𝑒1)

(

2𝐷

𝐾1 2 𝐾2 2 𝑄2

1

(

1

2 𝑚1 𝑃𝑠

(1−𝛾)(1−𝑒1 )(𝐾1 −1)

1

) ℎ0 + 2 𝐾1 𝐾2 2 𝑄 2 ℎ𝑠 (

𝐷

1

Since the supplier’s cycle time is 𝑇𝑐 = 𝐾1 𝐾2 𝑄 (1−𝐸[𝛾]) (1−𝐸[𝑒1 ]) 𝐷

2

− 𝑃 (𝐾1 − 𝑚 ) − 𝑠

−1)) + 𝑆𝑠 + 𝑂𝑠 𝑚1 + 𝑐𝑎𝑆 𝑃𝐾1 𝐾2 𝑄𝛾𝑒2

𝐸[𝑇𝑐 ] =

𝑚

)} + 𝑆𝑚 𝐾1 + 𝑂𝑚 𝑚2 𝐾1 + 𝑐𝑤 𝐾1 𝐾2 𝑄 𝛾 + 𝑐𝑟 𝐾1 𝐾2 𝑄(1 − 𝛾 )𝑒1 +

𝑐𝑎𝑉 𝐾1 𝐾2 𝑄𝛾𝑒2 + (

𝑃𝑚 𝑚2

𝑚

2

(18)

𝐾1 𝐾2 𝑄 (1−𝛾)(1−𝑒1) 𝐷

, then we have : (19)

Using the renewal reward theorem, the expected total annual cost of the suppliermanufacturer-retailer is given by: 𝐸𝐽𝑇𝐶(𝑄, 𝐾1 , 𝐾2 , 𝑚1 , 𝑚2 ) =

𝐸[𝐽𝑇𝐶] 𝐸[𝑇𝑐 ]

𝐸𝐽𝑇𝐶(𝑄, 𝐾1 , 𝐾2 , 𝑚1 , 𝑚2 ) = {

(1−𝐸[𝛾])𝐸[𝑒1 ]+𝐸[𝛾](1−𝐸[𝑒2 ]) 𝑥

(1−𝐸[𝛾])(1−𝐸[𝑒1 ]){(1−𝐸[𝛾])(1−𝐸[𝑒1 ])+2𝐸[𝑒2 ]𝐸[𝛾]} 2𝐷 𝐷𝐶𝑖

𝐷𝐶

(1−𝐸[𝛾])(1−𝐸[𝑒1 ])

𝐷𝐶𝑟 (1−𝐸[𝛾])𝐸[𝑒1 ] (1−𝐸[𝛾])(1−𝐸[𝑒1 ]) 𝐷

1 𝐾 𝑄ℎ

𝑄𝐷ℎ

𝐸[𝛾]𝐸[𝑒 ]

aV 2 + (1−𝐸[𝛾])(1−𝐸[𝑒

1 ])

𝐷

𝐷𝑆

1 )𝐾2

1 ])𝐾2

1

1

𝐾

𝑚

𝑚

𝑤 + (1−𝐸[𝛾])(1−𝐸[𝑒

1 ])

+2

𝑚1 𝑃𝑠

1

𝐷𝑆𝑠 𝑄(1−𝐸[𝛾])(1−𝐸[𝑒1 ])𝐾1 𝐾2

+

𝑠

2

𝐷𝑂𝑠 𝑚1 𝑄(1−𝐸[𝛾])(1−𝐸[𝑒1 ])𝐾1𝐾2

+

+

𝐷 (1−𝐸[𝛾])(1−𝐸[𝑒1 ])

ℎ 𝐷𝑄𝐾

𝑠 2 (𝐾1 − 1) − (𝐾1 − 𝑚 ) 𝑃 − (𝑚 − 1) 𝑃 } 2(1−𝐸[𝛾])(1−𝐸[𝑒 2

+

+ {𝑃 − 2𝑃2 +

1 𝐾1 𝐾2 𝑄ℎ0

𝑚 2 + 𝑄(1−𝐸[𝛾])(1−𝐸[𝑒

2

1 ])

𝐷𝐶 𝐸[𝛾]

𝑚 + 𝑄(1−𝐸[𝛾])(1−𝐸[𝑒

𝐷𝑂 𝑚

𝐵 + 𝑄(1−𝐸[𝛾])(1−𝐸[𝑒

1

1 ])

2 𝑚

𝑚 } (1−𝐸[𝛾])(1−𝐸[𝑒

𝐷𝐶

𝐷𝑂

+ 2 𝑚2 𝑃 𝑠 (1−𝐸[𝛾])(1−𝐸[𝑒 1 ])

(1−𝐸[𝛾])(1−𝐸[𝑒1 ])

{

𝐸[𝛾]𝐸[𝑒 ] 1 ])

2𝐷

𝑄𝐷ℎ

𝐵 } (1−𝐸[𝛾])(1−𝐸[𝑒

1 ])

aB 2 + (1−𝐸[𝛾])(1−𝐸[𝑒

(𝐾2 −1)(1−𝐸[𝛾])(1−𝐸[𝑒1 ])

+

1 ])

𝑚

+

+

𝑐aS 𝑃𝐷𝐸[𝛾]𝐸[𝑒2 ]

(20)

(1−𝐸[𝛾])(1−𝐸[𝑒1 ])

5. Solution Procedure We develop a solution procedure to find the optimal solution of the proposed model. Taking the first partial derivative of 𝐸𝐽𝑇𝐶(𝑄, 𝐾1 , 𝐾2 , 𝑚1 , 𝑚2 ) with respect to the decision variables 𝑄, 𝐾1 , 𝐾2 , 𝑚1 , 𝑚2 , we obtain : 𝜕𝐸𝐽𝑇𝐶(𝑄,𝐾1 ,𝐾2 ,𝑚1,𝑚2) 𝜕𝑄 𝐷𝑂𝐵

(1−𝐸[𝛾])(1−𝐸[𝑒1 ]) 𝐷𝑚 𝑂

𝑄2 (1−𝐸[𝛾])(1−𝐸[𝑒1 ]) 𝐷ℎ𝑚 (

=

(1−𝐸[𝛾])𝐸[𝑒1 ]+𝐸[𝛾](1−E[𝑒2 ]) (1−𝐸[𝛾])(1−𝐸[𝑒1 ])((1−𝐸[𝛾])(1−𝐸[𝑒1 ])+2𝐸[𝛾]𝐸[𝑒2 ]) + ) 𝑥 2𝐷

𝐷 ℎ𝐵 (

1 ])𝐾2

(1−𝐸[𝛾])(1−𝐸[𝑒1 ])(−1+𝐾2 ) 1 𝐾 + − 2 ) 2𝐷 𝑃𝑚 2𝑃𝑚

(1−𝐸[𝛾])(1−𝐸[𝑒1 ]) 𝐷ℎ𝑠 𝐾2 (

𝐷𝑚 𝑂

2 𝑚 − 𝑄2 (1−𝐸[𝛾])(1−𝐸[𝑒

1 𝑠 − 𝑄2 (1−𝐸[𝛾])(1−𝐸[𝑒

1 ])𝐾1 𝐾2

𝐷ℎ 𝐾

𝑠 2 + 2(1−𝐸[𝛾])(1−𝐸[𝑒

1 ])𝑚2 𝑃𝑚

1 2 (1−𝐸[𝛾])(1−𝐸[𝑒1 ])(−1+𝐾1) −1+𝑚2 𝐾1 −𝑚2 − − ) 𝐷 𝑃𝑚 𝑃𝑠

2(1−𝐸[𝛾])(1−𝐸[𝑒1 ])

𝐷ℎ 𝐾 𝐾

𝜕𝐸𝐽𝑇𝐶(𝑄,𝐾1 ,𝐾2 ,𝑚1,𝑚2) 𝜕𝐾2 𝐷𝑄ℎ𝑚 (

(1−𝐸[𝛾])(1−𝐸[𝑒1 ])

𝑚 − 𝑄2 (1−𝐸[𝛾])(1−𝐸[𝑒

1 ])𝐾2

(21)

𝐷𝑚 𝑂

𝐷𝑚 𝑂

2 𝑚 = − 𝑄(1−𝐸[𝛾])(1−𝐸[𝑒

(1−𝐸[𝛾])(1−𝐸[𝑒1 ]) 1 − ) 2𝐷 2𝑃𝑚

𝐷𝑆

1 ])𝑚1 𝑃𝑠

𝐷𝑆𝑠

+

+

0 1 2 + 2(1−𝐸[𝛾])(1−𝐸[𝑒

𝑄2 (1−𝐸[𝛾])(1−𝐸[𝑒1 ])𝐾1𝐾2

2 1 ])𝐾2

1 𝑠 − 𝑄(1−𝐸[𝛾])(1−𝐸[𝑒

𝐷𝑄ℎ

𝑠 + 2(1−𝐸[𝛾])(1−𝐸[𝑒

1 ])𝑚2 𝑃𝑚

2 1 ])𝐾1 𝐾2

+



+



𝐷𝑄ℎ𝑠 (

1 2 (1−𝐸[𝛾])(1−𝐸[𝑒1 ])(−1+𝐾1 ) −1+𝑚2 𝐾1 −𝑚2 − − ) 𝐷 𝑃𝑚 𝑃𝑠

2(1−𝐸[𝛾])(1−𝐸[𝑒1 ])

𝐷𝑄ℎ 𝐾

𝐷𝑆

0 1 + 2(1−𝐸[𝛾])(1−𝐸[𝑒

1 ])𝑚1 𝑃𝑠

𝑚 − 𝑄(1−𝐸[𝛾])(1−𝐸[𝑒

2 1 ])𝐾2

𝐷𝑆𝑠

(22)

𝑄(1−𝐸[𝛾])(1−𝐸[𝑒1 ])𝐾1 𝐾22

𝜕𝐸𝐽𝑇𝐶(𝑄,𝐾1 ,𝐾2 ,𝑚1,𝑚2) 𝜕𝐾1

=

𝐷𝑄ℎ0 𝐾2

𝜕𝐸𝐽𝑇𝐶(𝑄,𝐾1 ,𝐾2 ,𝑚1,𝑚2)

𝐷𝑚1 𝑂𝑠 − 𝑄(1−𝐸[𝛾])(1−𝐸 [𝑒1 ])𝐾12 𝐾2

2(1−𝐸[𝛾])(1−𝐸[𝑒1 ])

+

𝑠 − 𝑄(1−𝐸[𝛾])(1−𝐸 [𝑒

(23)

2 1 ])𝐾1 𝐾2

𝐷𝑂

𝐷𝑄ℎ 𝐾 𝐾

𝑠 = 𝑄(1−𝐸[𝛾])(1−𝐸 [𝑒

1 ])𝐾1 𝐾2

𝜕𝑚1

+

(1−𝐸[𝛾])(1−𝐸[𝑒1 ]) 1 − ) 𝐷 𝑃𝑠

𝐷𝑄ℎ𝑠 𝐾2 (

𝐷𝑆

2(1−𝐸[𝛾])(1−𝐸[𝑒1 ])𝑚1 𝑃𝑠

0 1 2 − 2(1−𝐸[𝛾])(1−𝐸 [𝑒 ])𝑚2 𝑃 1

(24)

1 𝑠

1

𝜕𝐸𝐽𝑇𝐶(𝑄,𝐾1 ,𝐾2 ,𝑚1,𝑚2) 𝜕𝑚2



=

𝐷𝑂𝑚 𝑄(1−𝐸[𝛾])(1−𝐸[𝑒1 ])𝐾2



𝐷𝑄ℎ𝑠 𝐾2 2(1−𝐸[𝛾])(1−𝐸[𝑒1 ])𝑚22 𝑃𝑚

+

2

𝐷𝑄ℎ𝑠 𝐾2 ( 2 − 2 ) 𝑚2 𝑃𝑚 𝑚2 𝑃𝑠 (25) 2(1−𝐸[𝛾])(1−𝐸[𝑒1 ])

Thus, by setting expressions (21) to (25) equal to zero, rearranging and simplifying leads to the following equations: 𝑄∗ = 𝑂𝐵 𝐾1 𝐾2 +𝑚2 𝑂𝑚 𝐾1 +𝑚1𝑂𝑠 +𝑆𝑚 𝐾1 +𝑆𝑠 (1−𝐸[𝛾])𝐸[𝑒1 ]+𝐸[𝛾](1−𝐸[𝑒2 ]) (1−𝐸[𝛾])(1−𝐸[𝑒1 ])((1−𝐸[𝛾])(1−𝐸[𝑒1 ])+2𝐸[𝛾] 𝐸[𝑒2 ]) 𝐾1 𝐾2[( + )ℎ𝐵 + 𝑥 2𝐷 1 2 −1+ 𝐾1 − (1−𝐸[𝛾])(1−𝐸[𝑒1 ])(−1+𝐾2 ) 1 𝐾 ℎ 𝐾 ℎ 𝐾 (1−𝐸[𝛾])(1−𝐸[𝑒1 ])(−1+𝐾1 ) 𝑚2 𝑚2 ℎ0 𝐾1 𝐾2 ℎ𝑚 ( + − 2 )+ 𝑠 2 + 𝑠 2 ( − − )+ ] 2𝐷 𝑃𝑚 2𝑃𝑚 2𝑚2 𝑃𝑚 2 𝐷 𝑃𝑚 𝑃𝑠 2𝑚1 𝑃𝑠



(26) 𝐾2 ∗ = (1−𝐸[𝛾])𝐸[𝑒1 ]+𝐸[𝛾](1−𝐸[𝑒2 ]) (1−𝐸[𝛾])(1−𝐸[𝑒1 ])((1−𝐸[𝛾])(1−𝐸[𝑒1 ])+2𝛾𝑒2 ) + )ℎ𝐵 + 𝑥 2𝐷 (1−𝐸[𝛾])(1−𝐸[𝑒 ]) 1 1 )] ℎ𝑚 ( − 𝑃𝑚 2𝐷 1 2 −1+ 𝐾1 − 𝑚2 𝑚2 ℎ 𝐾 𝑂𝐵 𝐾1 (ℎ ((1−𝐸[𝛾])(1−𝐸[𝑒1])− 1 )+ ℎ𝑠 +1ℎ ((1−𝐸[𝛾])(1−𝐸[𝑒1])(−1+𝐾1)− − )+ 0 1 ) 𝑚 𝑠 2𝐷 2𝑃𝑚 2𝑚2 𝑃𝑚 2 𝐷 𝑃𝑚 𝑃𝑠 2𝑚1 𝑃𝑠

(𝑚2 𝑂𝑚 𝐾1 +𝑚1 𝑂𝑠 +𝑆𝑚 𝐾1 +𝑆𝑠 )([



(27) ∗

𝐾1 = √

(ℎ𝑚 (

(1−𝐸[𝛾])(1−𝐸[𝑒1 ]) 1 ℎ (1−𝐸[𝛾])(1−𝐸[𝑒1 ]) ℎ𝑠 2ℎ − )− 𝑠 + + 𝑠 )(𝑚1 𝑂𝑠 +𝑆𝑠 ) 𝐷 𝑃𝑚 𝐷 𝑃𝑚 𝑃𝑠 𝑚2 (1−𝐸[𝛾])(1−𝐸[𝑒1 ]) 1 ℎ (𝑚2 𝑂𝑚 +𝑆𝑚 )(ℎ𝑠 ( − )+ 0 ) 𝐷 𝑃𝑠 𝑚1 𝑃𝑠

(28)

ℎ 𝑆

0 𝑠 𝑚1 ∗ = √ (1−𝐸[𝛾])(1−𝐸[𝑒1 ]) 1 𝑃𝑠 𝑂𝑠 ℎ𝑠 ( − ) 𝐷

(29)

𝑃𝑠

2ℎ 𝑆

𝑠 𝑚 𝑚2 ∗ = √ (1−𝐸[𝛾])(1−𝐸[𝑒1 ]) 𝑃𝑠 𝑂𝑚 (ℎ𝑚 −ℎ𝑠 )( − 𝐷

1 ) 𝑃𝑚

(30)

Based on the above calculations, we adopt the algorithm proposed by Ben-Daya et al. (2013) to obtain solutions of the proposed model. In this algorithm, the notions ⌊𝑎⌋ and ⌈𝑎⌉ denote the largest integer less than or equal to a and the smallest integer greater than or equal to a, respectively. First, the procedure is started by calculating integer variables m1 and m2 by employing equation (29) and equation (30). Then we set the values of m1 and m2 in this step as the upper bond for search range on optimal m1 and m2. Thus, by considering the values of upper bond, we can determine the possible combinations of m1 and m2.

Second, the

values of integer multipliers K1 and K2 can be computed by

employing equation (27) and equation (28). Third, by investigating each combination of m1, m2 and K1, K2, the values of Q and EJTC can be determined by using equation (26) and equation (20), respectively. Further, the optimal solution of the proposed model can be found by searching the combination of the decision variables that provides a minimum EJTC. However, the algorithm proposed here is only guaranteed to converge to local optimal solutions due to the difficulties associated with establishing the convexity of the expected total cost function. The algorithm for solving the above problem is as follows:

1. Determine a continous value of variables m1 and m2 using equation (29) and equation (30). 2. Let 𝑚1𝑈 = ⌈𝑚1 ⌉ dan 𝑚2𝑈 = ⌈𝑚2 ⌉ denote the upper bound for the search range on the optimal values of m1 and m2. 3. For each combination in the search range 𝑚1 𝜖 [1, 𝑚1𝑈 ] and 𝑚2 𝜖 [1, 𝑚2𝑈 ], determine the values of the integer multiplier K1 and K2 using equation (27) and equation (28) 3.1.

Let K1= ⌊𝐾1 ⌋ and K2= ⌊𝐾2 ⌋ and calculate Q using equation (26) and the corresponding value of EJTC using equation (20).

3.2.

Let K1= ⌊𝐾1 ⌋ and K2=⌈K2⌉ and calculate Q using equation (26) and the corresponding value of EJTC using equation (20).

3.3.

Let K1=⌈K1⌉ and K2= ⌊𝐾2 ⌋ and calculate Q using equation (26) and the corresponding value of EJTC using equation (20).

3.4.

Let K1=⌈K1⌉ and K2=⌈K2⌉ and calculate Q using equation (26) and the corresponding value of EJTC using equation (20).

4. Set (m1 *, m2 * , K1 * , K2 * , and Q* ) be the values leading to the minimum EJTC found in Step 3, then (m1 *, m2 * , K1 * , K2 * , and Q* ) is the optimal solution.

6. Numerical Examples In this paper, we consider an example given by Hsu and Hsu (2012) and Ben Daya et al. (2013), where : D

= 133,000 units / year

Ps

= 399,000 units / year

Pm

= 140,000 units / year

Os

= $ 600 / delivery

Om

= $ 300 / delivery

OB

= $ 50 / shipment

Sm

= $ 200 / batch production

Ss

= $ 800 / batch production

P

= 0.10

x

= 155,000 units / year

ci

= $ 0.5 / unit / year

cw

= $ 50 / unit / year

caB

= $ 200 / unit / year

cav

= $ 300 / unit / year

caS

= $ 350 / unit / year

cr

= $ 100 / unit / year

h0

= $ 0.08 / unit / year

hs

= $ 0.8 / unit / year

hm

= $ 2 / unit / year

hB

= $ 5 / unit / year The probability density function of uniform distribution of the defect rate and

inspections errors are provided as follows: 1

f(𝛾 ) =

{𝛽

,0 ≤ 𝛾 ≤ 𝛽 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

𝛽

𝛽𝛾

∫0 𝛾 𝑓 (𝛾 ) 𝑑𝛾 = ∫0 𝜆

𝛽

1

𝑓 (𝑒1 ) = {𝜆

𝐸 [𝑒2 ] =

0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

1

𝑓 (𝑒2 ) =

{𝜂

, 0 ≤ 𝑒2 ≤ 𝜂 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

𝐸 [𝛾 ] =

𝛽

𝑑𝛾 = 2

𝜆 𝑒1

𝐸 [𝑒1 ] = ∫0 𝑒1 𝑓 (𝑒1 ) 𝑑𝑒1 = ∫0 𝜂 ∫0 𝑒2

, 0 ≤ 𝑒1 ≤ 𝜆

𝜆 𝜂 𝑒2 𝑓 (𝑒2 ) 𝑑𝑒2 = ∫0 𝜂

𝜆

𝑑𝑒1 = 2

𝜂

𝑑𝑒2 = 2

𝛽 = 𝜆 = 𝜂 = 0.04 , then we have : 𝐸 [𝛾 ]

= 0.02

𝐸 [𝑒1 ] = 0.02 𝐸 [𝑒2 ] = 0.02 In this numerical example, we define that the cost of false acceptance is greater than the cost of false rejection. For a critical component such as spare parts, it is known that accepting the defective items will increase the cost of maintenance. Therefore the values of caB, cav and caS are given to be higher than that of cr. Applying the proposed algorithm to the numerical example, the optimal solutions are m1 = 1, m2 = 2, K1 = 1, K2 = 18, Q = 1,427.53 units and the joint total cost is $531,125.83. Table 1 shows the solutions for different values of defect rate. One can see that if the defect rate is gradually increased, retailer cost, manufacturer cost, supplier cost and joint total cost increase as well. However, the percentage increase in manufacturer cost is relatively higher than retailer cost and supplier cost. The percentage increases in retailer cost, supplier cost and manufacturer cost are 1.04%, 6.83% and 13.92%, respectively. This is happened because the greater the defect rate, then the greater the number of mistakes made by the inspector during the inspection process which gives an impact on the rising of type 1 and type 2 costs. The greater the number of defective goods sold to the end consumer, the greater the number of goods returns to the manufacturer. Therefore, it can be seen from the table that the increase in defect rate will consequently increase the cost of producing defective items (cw), the cost of rejecting a non-defective item (cr) and the post-sale failure cost to the

manufacturer. Further, it is also obtained from the table that the shipment size is not sensitive to the changes in defect rate. To gain further insights into the behavior of the model, we examine the effect of the changes in the probability of inspection errors on the optimal solutions. Table 2 presents the influence of the probability of type 1 inspection error on the proposed model. Clearly, the results are similar to the defect rate does on model’s solutions. It shows that increasing the probability of type 1 inspection error, will also increase retailer cost, manufacturer cost, supplier cost and joint total cost. We may see that the percentage increase in manufacturer cost is 37.37%, which is higher than the retailer cost (2.14%) and supplier cost (1.87%). We may also observe from the table that the higher probability of type 1 inspection error, the higher shipment size and production batch. If we compare the results in table 1 and table 2, one can see that the percentage increases of manufacturer cost and retailer cost in table 2 are higher than in table 1. While the percentage increase of supplier cost in table 2 is smaller than in table 2. These results indicate that the proposed model is more sensitive to the change in probability of type 1 inspection error than the change in defect rate. Table 1 The impact of defect rate on model β

Q(units)

m1

m2

K1

K2

Retailer Cost ($)

Manufacturer Cost ($)

Supplier Cost ($)

Total Cost ($)

0.04

1,427.53

1

2

1

18

77,772.80

438,767.77

14,585.26

531,125.83

0.06

1,427.83

1

2

1

18

78,567.27

513,664.92

15,684.51

607,916.70

0.08

1,428.09

1

2

1

18

79,379.01

588,564.42

16,786.51

684,729.94

0.1

1,428.34

1

2

1

18

80,208.58

663,466.33

17,891.36

761,566.26

0.12

1,428.55

1

2

1

18

81,056.53

738,370.73

18,999.14

838,426.40

Table 2 The impact of type 1 inspection error on model λ

Q(units)

m1

m2

K1

K2

Retailer Cost ($)

Manufacturer Cost ($)

Supplier Cost ($)

Total Cost ($)

0.04

1,427.53

1

2

1

18

77,772.80

438,767.77

14,585.26

531,125.83

0.08

1,428.11

1

2

1

18

79,378.75

710,430.33

14,847.72

804,656.81

0.12

1,428.56

1

2

1

18

81,056.13

982,102.55

15,121.56 1,078,280.24

0.16

1,428.90

1

2

1

18

82,809.57 1,253,785.04 15,407.54 1,352,002.14

0.2

1,429.11

1

2

1

18

84,644.17 1,525,478.47 15,706.45 1,625,829.09

Table 3 shows the effect of the probability of type 2 inspection error on the model. From the table we observe that the probability of type 2 inspection error gives significant impact to the model’s solutions. The retailer cost, manufacturer cost, supplier cost and joint total cost increase gradually due to the increase in probability of the type 2 inspection error. The percentage increases of retailer cost and manufacturer cost are 0.0004% and 3.59%, respectively, which are smaller than supplier cost (11.26%). This is happened because the greater the type 2 inspection error conducted by inspectors, resulting in the increase of the post-sale failure cost. When comparing both inspection errors, it is clear that the type 1 inspection error gives much more impact on joint total cost than that of the type 2 inspection error. We note that the defect rate clearly affect the proposed model. Due to the imperfect production process, the production batch and shipment lot may contain a number of defective items and thus increasing the total cost. The defective items should be reworked in order to improve its quality or may be can be sold to the secondary market with a lower price. Selling the defective items to secondary market will reduce the customer satisfaction and profit, thus reducing the supply chain capability to compete in global market. Facing this situation, the manager may consider a quality improvement project to be applied in production system. In reality, an investment on quality improvement can be done by upgrading machine and tools, investing an eqipment to detect unreliable process early and giving a training to the operator or labour to increase their skill. We also note that the inspection errors clearly affect individual cost and supply chain cost. The increase in inspection errors consequently increase the cost of misclassifications, i.e. false rejection and false acceptance, and manufacturer, retailer and supplier costs. We conclude that the type I inspection error has a much more impact on total cost than that of type II inspection error. Therefore, the manager should pay more attention to controlling type I inspections error to reduce the total cost. In addition, an effort such as helping the manufacturer to conduct a full inspection may also be considered by the manager. This strategy was first analysed by Rezae and Salimi (2012). The retailer

buyer can help the manufacturer to improve production quality by paying some more than usual purchasing price to avoid receiving defective items. When the inspection process shifts from the buyer to the manufacturer, the associated cost is believed to be decreased. This due to the fact that the manufacturer is more familiar with the product and its deficiencies. Table 4 shows the effect of hm on the proposed model. It shows that increasing on the holding cost of the manufacturer’s finished products, will also increase slightly the retailer cost, manufacturer cost, supplier cost and joint total cost. The percentage increases of the retailer cost, manufacturer cost and supplier cost are 0.05%, 0.2%, and 0.69%, respectively. Further, table 4 shows that the increase in the holding cost of the manufacturer’s finished products can make the value of Q optimal decrease with an average of 3.34%. We also observe that the change in the holding cost of the manufacturer’s finished products affects the value of the decision variables K1 and K2. Table 3 The impact of type 2 inspection error on model η

Q(units)

m1

m2

K1

K2

Retailer Cost ($)

Manufacturer Cost ($)

Supplier Cost ($)

Total Cost ($)

0.02

1,427.53

1

2

1

18

77,772.80

438,767.77

14,585.26

531,125.83

0.04

1,427.51

1

2

1

18

77,773.11

455,385.82

16,524.06

549,682.98

0.06

1,427.50

1

2

1

18

77,773.43

472,003.86

18,462.85

568,240.14

0.08

1,427.49

1

2

1

18

77,773.74

488,621.90

20,401.65

586,797.29

0.1

1,427.47

1

2

1

18

77,774.06

505,239.94

22,340.45

605,354.45

Table 4 The impact of the manufacturer holding cost on model hm

Q(units)

m1

m2

K1

K2

Retailer Cost ($)

Manufacturer Cost ($)

Supplier Cost ($)

Total Cost ($)

2

1,102.93

1

2

1

23

81,206.95

438,463.00

14,617.72

534,287.67

3

1,038.17

1

1

2

13

81,264.64

439,517.49

14,894.66

535,676.79

4

1,012.94

1

1

2

13

81,300.68

440,388.54

14,933.26

536,622.47

5

989.46

1

1

2

13

81,341.81

441,226.52

14,977.38

537,545.71

6

967.55

1

1

2

13

81,387.32

442,034.50

15,026.21

538,448.03

Table 5 presents the impact of the changes in manufacturer ordering cost and retailer ordering cost on the proposed model. One can see that if ordering cost is increased gradually, the costs incurred by the retailer, manufacturer and supplier increase with average of 1.94%, 0.17%, and 0.25%, respectively. Moreover, as the ordering cost increases, the shipment size increases as well. The increase of shipment size due to the increase of ordering cost is about 13.08%. Facing an expensive ordering cost, the system tends to increase ordering quantity, thus minimising the impact of this cost on ordering cost per unit product. There is also change in the value of the decision variables K2 which increases the production batch of the manufacturer and the supplier on average of 1,83%. Table 5 The impact of the ordering cost on model Om

OB

($)

($)

300

Q(units)

m1

m2

K1

K2

Retailer Cost ($)

Manufacturer Cost ($)

Supplier Cost ($)

Total Cost ($)

50

1,427.53

1

2

1

18

77,772.80

438,767.77

14,585.26 531,125.83

350

70

1,656.79

1

2

1

16

79,364.37

439,564.57

14,515.24 533,444.18

400

100

2,047.27

1

2

1

13

81,284.40

440,424.39

14,507.17 536,215.96

450

120

2,254.72

1

2

1

12

82,425.26

441,142.37

14,475.90 538,043.54

500

150

2,511.30

1

2

1

11

83,988.06

441,893.56

14,441.36 540,322.97

Table 6 The comparison between the integrated model with the independent model Model

Q

m1

m2

K1

K2

Retailer Cost ($)

Manufacturer Cost ($)

Supplier Cost ($)

Total Cost ($)

1,639

1

2

1

12

88,770.90

438,716.36

18,431.80

545,919.06

1,428

1

2

1

18

77,772.80

438,767.77

14,585.26

531,125.83

(unit) Independent model Integrated model Cost Reduction

14,793.23

As findings from previous research, integrating parties in supply chain system in managing inventory always results in reducing total cost. Therefore, we analyse the performance of both independent and integrated models to show the savings of moving from making inventory decisions individually to making the decisions jointly. If each party in supply

chain

system agrees to adopt integrated model, they would share informations and

determine their shipment and production jointly. The comparison of the integrated model and the independent model is reported in table 6. Clearly, the integrated model performs better in reducing joint total cost compared to the independent model. The savings that can be realised are about $14,793.23 or 2.71%. The retailer and supplier are better off by 12.39% and 20.87%, respectively, while the manufacturer is worse off by 0.012%. The manufacturer, however, always in disadvantage position since his cost increases. We observe that the decreases in retailer cost and supplier cost are always greater than the increase in manufacturer cost, thus there is significant improvement on joint total cost. The mechanism such as profit sharing can be applied to induce the manufacturer to join the integrated policy.

7. Conclusions and future research directions In this paper, we develop an integrated three-stage inventory model for single supplier, single manufacturer and single retailer system under deterministic demand, imperfect product quality and inspection errors. The production process is imperfect and produces some defective items. To guarantee that the items sold to the end customers is always a good quality item, the retailer conducts inspection process. However, the inspection process is imperfect, thus the inspector may incorrectly classify the items. We consider two types of inspection errors, that is type I for the condition that if the inspector incorrectly classify non-defective item as defective and type II for the condition that if the inspector incorrectly classify a defective item as non-defective. We seek to minimise the joint total cost by simultaneously determining shipment size, production batch, cycle time and number of raw material’s shipment. We suggest an iterative procedure to find the solution of the proposed model. Numerical examples are performed to see the impact of the defect rate, the probability of type I inspection error, the probability of type II inspection error, manufacturer holding cost and ordering cost. The results obtained from numerical example show that the defect rate and inspection errors affect the cost component. Clearly, the type I inspection error gives much more impact on total cost than that of type II inspection error. The changes in manufacturer holding cost and ordering cost affect the behavior of the proposed model. Further, the results also show that a saving can

be realised if there is a movement from making inventory decisions individually to making the decisions jointly. However, when adopting integrated policy, the manufacturer is always at disadvatage position since his annual cost increases. Above all, there should be a win-win solution for all parties involved in supply chain system. This research can be extended by incorporating some aspects. An interesting future study would be to develop different assumptions to the nature of supplier, manufacturer and retailer relationship. For example, if the demand is assumed to be stochastic, how would this affect the solutions? Studying an inventory model which consider a situation where manufacturer and retailer conduct inspection process together may provide valuable managerial insigths. By conducting inspection process together, the defective items in each shipment lot can be minimised thus minimising the total cost. Future research may also look into studying the impact of returned product or learning process in both manufacturer and supplier. Further, modeling inventory system in more complex environment such as considering multi suppliers, multi manufacturers or multi retailers, may also result interesting insights.

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