Flexible Services and Manufacturing Journal

, Volume 23, Issue 1, pp 48–63

A production policy considering reworking of imperfect items and trade credit

Authors

    • Department of Business ManagementTatung University
  • Tsung-Hui Chen
    • Department of Marketing and Distribution ManagementNational Pingtung Institute of Commerce
  • Sheng-Min Huang
    • Department of Business ManagementTatung University
Article

DOI: 10.1007/s10696-010-9070-3

Cite this article as:
Tsao, Y., Chen, T. & Huang, S. Flex Serv Manuf J (2011) 23: 48. doi:10.1007/s10696-010-9070-3

Abstract

Making production decisions that will reduce total cost is a goal that most manufacturers pursue actively. However, the traditional production model development assumed that all products are perfect quality, which is far from reality. Since trade credit is a popular payment method in today’s business environment, this paper analyzes the production problem under trade credit and imperfect product reworking conditions. This work extends the traditional production model by considering reworking imperfect items and trade credit to cope with realistic situations. The objective of this study is to determine the optimal production lot size while minimizing the total cost. This paper develops an easy-to-use algorithm to solve the problem described, provides numerical examples to illustrate the proposed solution procedure, and discusses the impact of various system parameters.

Keywords

ProductionInventoryImperfect quality itemsReworkingTrade credit

1 Introduction

The enterprise operating process involves several production and inventory problems. Improving production efficiency and reducing total cost are objectives that most manufacturers pursue actively. The economic production quantity (EPQ) model can tackle the issues of inventory management in production-inventory systems. Osteryoung et al. (1986) indicated that the EPQ model is one of the most popular and widely used inventory control models in the industry. However, the traditional EPQ model development assumed that all products are perfect quality, which is far from reality. Producers must rework imperfect products, which introduces repair and holding costs, leading to an increase in total cost. Considering imperfect products when formulating a realistic production model is therefore essential.

In practice, systems producing perfect-quality items are largely non-existent. Several scholars have studied systems producing imperfect items. One way to handle imperfect items is to rework them into perfect items, and sell them to customers. Liu and Yang (1996) studied the problem of determining the optimal lot size in a single-stage imperfect production system with a high failure rate. They classified the defects generated as either re-workable defects or non-reworkable defects (which they discarded). Hayek and Salameh (2001) considered reworking imperfect items under non-instantaneous replenishment and permissible backlogging conditions. Their scenario reworked imperfect items immediately in the same cycle when initial production stops.

Chiu (2003) extended the research of Hayek and Salameh (2001) by considering that reworking imperfect items is not 100% successful, and that the reworking process can damage some imperfect items that eventually must be abandoned. Chan et al. (2003) mentioned that products above or below their specification limits are considered imperfect and should be reworked instantaneously and kept in stock at a cost. Chiu (2007) suggested reworking imperfect items in the same cycle, when the regular manufacturing process ends, considering the rate of failure in reworking. Chiu et al. (2004, 2006) integrated the random defective rate of products and the reworking of imperfect items into the EPQ model. Konstantaras et al. (2007) optimized the ordering lot size and obtained the optimal number of batches in two cases: (1) selling imperfect items to a secondary market, and (2) reworking imperfect items at some cost, and then using them as new items to satisfy demand. Chiu (2008) considered the production of imperfect items and the random breakdown of production equipment as inevitable. He therefore integrated the stochastic breakdown of production equipment and the reworking of imperfect items into the EPQ model to optimize production run time. The current model considers that a system producing perfect items is non-existent. Even enterprises characterized by quality control cannot avoid producing imperfect items. This study considers reworking imperfect items, and develops a model that is more realistic than the EPQ model.

Trade credit mostly exists between enterprises, and affects total cost. Because of the competitive market environment, suppliers typically delay the payment period to retailers to increase retailer demand or increase their market share. Retailers need not pay suppliers for their goods immediately; they can pay by check or negotiate the payment period with the supplier. Trade credit can decrease the buyers’ inventory cost and increase their purchase willingness. The same situation also occurs between retailers and consumers. In brief, the concept of trade credit is that a buyer receives goods instantaneously, but pays for them over time. Trade credit between enterprises is a common situation: manufacturers sell products and earn interest before they pay their suppliers for goods. They also pay the capital opportunity cost of surplus goods after the trade credit period. Therefore, systems should consider trade credit interest into total cost. The model proposed in this study integrates permissible delays in payment into the EPQ model to arrive at a more realistic trade model.

Goyal (1985) was the first trade credit researcher to examine the effect of the credit period on the optimal inventory policy. Since the publication of Goyal’s (1985) paper almost 25 years ago, over fifty papers have appeared in the literature dealing with a variety of trade credit situations. Teng (2002) amended the model of Goyal (1985) by considering the difference between unit price and unit cost. Huang (2003) considered the situation in which the supplier offers a credit period to the retailer, and the retailer offers a credit period to the end customer. Huang (2006) adopted the models of Teng (2002) and Huang (2003) to consider limited storage space. Huang and Chung (2003) and Ouyang et al. (2005) considered the ordering policy under a cash discount and payment delay conditions. Chung and Huang (2006) considered an EOQ model to allow imperfect items under permissible delay in payments. Tsao and Sheen (2007) considered pricing and replenishment decisions for deteriorating items with lot-size and time-dependent purchasing costs under a credit period. Sheen and Tsao (2007) discussed channel coordination under trade credit with freight cost and quantity discounts. Ouyang et al. (2008) demonstrated achieving a significant profit increase for the entire supply chain by linking trade credit and freight rate policies. Tsao and Sheen (2008) determined dynamic pricing, promotion, and replenishment policies for a deteriorating item under trade credit conditions. Recently, Ouyang et al. (2009) considered deteriorating items with partially permissible delays in payment linked to order quantity. Teng (2009) considered the retailer offering either a partial or a full trade credit to customers. In summary, the issue of trade credit is very popular in this field of research.

Most trade-credit research considers the EOQ model. So far, only Chung and Huang (2003), Ouyang et al. (2006), Huang (2007), Liao (2008) and Teng and Chang (2009) have considered the EPQ model under permissible delay in payments. However, all of these studies assume a perfect production system, meaning that the production system does not produce defective products. This assumption is not practical in reality. Therefore, this paper extends the traditional production model by considering reworking imperfect items and trade credit to cope with realistic situations. This is the first study to consider reworking imperfect items and trade credit simultaneously. The proposed approach determines the optimal production lot size while minimizing total cost. This paper develops an easy-to-use algorithm to solve the problem. Computational analysis illustrates the solution procedure and discusses the impact of various system parameters. This work concludes with a computational analysis and makes a variety of managerial implications.

2 Model formulation

The following notations are used in this study.
P

Production rate in units per year

P1

Rate of rework of imperfect quality items in units per year

d

Production rate of imperfect quality items in units per year

C

Production cost per item, i.e. the cost to produce an item

λ

Demand rate in units per year

x

Percentage of imperfect quality items produced

Q

Total items produced during a production cycle (decision variable)

H

Inventory level when rework of imperfect quality items is completed

H1

Inventory level when original production is completed

CR

Repair cost per item of imperfect quality

K

Setup cost for each production run

h

Holding cost of perfect items per item per year

h1

Holding cost of the imperfect quality items being reworked per item per year

Sp

Selling price per item of good quality

M

The manufacturer’s credit period offered by the supplier

Cp

Material purchasing cost per item, i.e. the material cost that the manufacturer pays to his/her supplier

Ie

Interest rate that can be earned per dollar

Ip

Interest rate that can be charged per dollar

The mathematical model is developed under the following assumptions:

  1. 1.

    The demand rate and production rate are known constants.

     
  2. 2.

    The percentage of imperfect items produced is a known constant.

     
  3. 3.

    All imperfect items can be reworked.

     
  4. 4.

    Whenever a repair is completed, the product is added to the inventory of perfect items.

     
  5. 5.

    The unit-selling price of the products sold during the credit period is deposited in an interest bearing account with rate Ie. At the end of this period, the credit is settled and the company starts paying interest charges for the items in stock with rate IP.

     
  6. 6.

    The inspection time and rework time of defective items are very short and can be ignored.

     
  7. 7.

    Shortages are not allowed in this model.

     
Figures 1 and 2 show the on-hand inventory of perfect and imperfect items, respectively.
https://static-content.springer.com/image/art%3A10.1007%2Fs10696-010-9070-3/MediaObjects/10696_2010_9070_Fig1_HTML.gif
Fig. 1

On-hand inventory of perfect items

https://static-content.springer.com/image/art%3A10.1007%2Fs10696-010-9070-3/MediaObjects/10696_2010_9070_Fig2_HTML.gif
Fig. 2

On-hand inventory of imperfect items

The production rate of imperfect quality items can be written as:
$$ d = P \cdot x. $$
(1)
The production rate of good items is always greater than or equal to the sum of the demand rate and the rate at which defective items are produced, i.e.,
$$ P - d - \lambda \ge 0\;{\text \quad {or} \quad }\;0 \le x \le \left( {1 - {\frac{\lambda }{P}}} \right). $$
(2)
The cycle time is
$$ T = t_{1} + t_{2} + t_{3} \;{\text \quad {and} \quad }\;T = {\frac{Q}{\lambda }}. $$
(3)
The production time is
$$ t_{1} = {\frac{{H_{1} }}{P - d - \lambda }}. $$
(4)
The inventory level when reworking imperfect items is completed is
$$ H_{1} = (P - d - \lambda ) \cdot \frac{Q}{P}. $$
(5)
The rework time is
$$ t_{2} = {\frac{d \cdot Q}{{P \cdot P_{1} }}} = {\frac{Q \cdot x}{{P_{1} }}}. $$
(6)
The maximum inventory level is
$$ H = \left[ {1 - {\frac{{\lambda (d + P_{1} )}}{{P \cdot P_{1} }}}} \right] \cdot Q. $$
(7)
Then
$$ t_{3} = {\frac{H}{\lambda }} = Q \cdot \left[ {{\frac{1}{\lambda }} - {\frac{{P_{1} + d}}{{P \cdot P_{1} }}}} \right]. $$
(8)
Therefore,
$$ t_{a} = t_{1} = \frac{Q}{P},\quad t_{b} = t_{1} + t_{2} = Q\left( {\frac{1}{P} + {\frac{x}{{P_{1} }}}} \right)\quad {\text{and}}\quad t_{c} = t_{1} + t_{2} + t_{3} = T = Q\left( {\frac{1}{P} + {\frac{x}{{P_{1} }}} + {\frac{1}{\lambda }} - {\frac{{P_{1} + d}}{{PP_{1} }}}} \right) = {\frac{Q}{\lambda }}. $$
The following differential equations describe the perfect items inventory level of the production system:
$$ {\frac{{dI_{1} (t)}}{dt}} = P - d - \lambda ,\quad 0 \le t \le t_{a} . $$
(9)
$$ {\frac{{dI_{2} (t)}}{dt}} = P_{1} - \lambda ,\quad t_{a} \le t \le t_{b} . $$
(10)
$$ {\frac{{dI_{3} (t)}}{dt}} = - \lambda ,\quad t_{b} \le t \le t_{c} . $$
(11)
From the boundary conditions \( I_{1} (0) = 0,\;I_{1} (t_{a} ) = I_{2} (t_{a} ),\;I_{2} (t_{b} ) = I_{\text{MAX}} \;{\text{and}}\;I_{3} (t_{c} ) = 0, \) we can solve the differential equations as follows:
$$ I_{1} (t) = t(P - d - \lambda ),\quad 0 \le t \le t_{a} , $$
(12)
$$ I_{2} (t) = I_{\text{MAX}} - (t_{b} - t)(P_{1} - \lambda ),\quad t_{a} \le t \le t_{b} , $$
(13)
$$ I_{3} (t) = \lambda \left( {t_{c} - t} \right),\quad t_{b} \le t \le t_{c} . $$
(14)
Based on Eqs. 13 and 14 and the condition I2(tb) = I3(tb), we obtain the maximum inventory level \( I_{\text{MAX}}: \)
$$ I_{\text{MAX}} = \lambda \left( {t_{c} - t_{b} } \right). $$
(15)

Substituting Eq. 15 into 13 leads to \( I_{2} (t) = \lambda (t_{c} - t_{b} ) - (t_{b} - t)(P_{1} - \lambda ). \)

The following differential equations describe the inventory level of imperfect items in the production system:
$$ {\frac{{dI_{4} (t)}}{dt}} = d,\quad 0 \le t \le t_{a} , $$
(16)
$$ {\frac{{dI_{5} (t)}}{dt}} = - P_{1} ,\quad t_{a} \le t \le t_{b} . $$
(17)
Based on the boundary conditions \( I_{4} (0) = 0 \) and \( I_{5} (t_{b} ) = 0, \) we can solve the differential equations as follows:
$$ I_{4} (t) = dt,\quad 0 \le t \le t_{a} , $$
(18)
$$ I_{5} (t) = P_{1} \left( {t_{b} - t} \right),\quad t_{a} \le t \le t_{b} . $$
(19)
The total annual cost consists of the following elements
  1. (a)

    Annual production cost = C · λ.

     
  2. (b)

    Annual repair cost = CR · λ · x.

     
  3. (c)

    Annual setup cost = /Q.

     
  4. (d)

    Annual holding cost = \( {\frac{h\lambda }{Q}} \cdot \left[ {{\frac{{H_{1} \cdot t_{1} }}{2}} + {\frac{{(H_{1} + H)}}{2}} \cdot t_{2} + {\frac{{H \cdot t_{3} }}{2}}} \right] + {\frac{{h_{1} \lambda }}{Q}} \cdot \frac{d}{2} \cdot t_{1}^{2} + {\frac{{h_{1} \lambda }}{Q}} \cdot {\frac{{P_{1} \cdot t_{2}^{2} }}{2}}. \)

     
  5. (e)

    There are four cases for interest charged and interest earned per year.

     

Case 1:

0 ≤ M ≤ ta

From Assumption 5, we know the items still in stock have to be financed at interest rate Ip after the credit period. When 0 ≤ M ≤ ta, the annual interest charged includes the interest charged for the on-hand inventory of perfect items in periods (M,ta), (ta,tb) and (tb,tc) and the on-hand inventory of imperfect items in periods (M,ta) and (ta,tb).
$$ \begin{aligned} {\text{Annual}}\;{\text{interest}}\;{\text{charged}} = & {\frac{{C_{p} I_{p} }}{T}}\left[ {\int\limits_{M}^{{t_{a} }} {\left( {t(P - d - \lambda )} \right)dt} + \int\limits_{{t_{a} }}^{{t_{b} }} {\left( {\lambda (t_{c} - t_{b} ) - (t_{b} - t)(P_{1} - \lambda )} \right)dt} } \right. \\ & \left. { + \int\limits_{{t_{b} }}^{{t_{c} }} {\left( {\lambda \left( {t_{c} - t} \right)} \right)dt} + \int\limits_{M}^{{t_{a} }} {\left( {td} \right)dt} + \int\limits_{{t_{a} }}^{{t_{b} }} {\left( {\left( {t_{b} - t} \right)P_{1} } \right)dt} } \right] = {\frac{{C_{p} I_{p} (P - \lambda )(Q^{2} - M^{2} P\lambda )}}{2PQ}}. \\ \end{aligned} $$
$$ {\text{Annual}}\;{\text{interest}}\;{\text{earned}} = {\frac{{S_{p} I_{e} }}{T}}\left( {\int\limits_{0}^{M} {\lambda tdt} } \right) = {\frac{{I_{e} M^{2} S_{p} \lambda^{2} }}{2Q}}. $$

Case 2:

ta < M ≤ tb

When ta < M ≤ tb, the annual interest charged includes the interest charged for the on-hand inventory of perfect items in periods (M,tb) and (tb,tc) and the on-hand inventory of imperfect items in period (M,tb).
$$ {\text{Annual}}\;{\text{interest}}\;{\text{charged}} = {\frac{{C_{p} I_{p} }}{T}}\left[ {\int\limits_{M}^{{t_{b} }} {\left( {\lambda (t_{c} - t_{b} ) - (t_{b} - t)(P_{1} - \lambda )} \right)dt + \int\limits_{{t_{b} }}^{{t_{c} }} {\left( {\lambda \left( {t_{c} - t} \right)} \right)dt + \int\limits_{M}^{{t_{b} }} {\left( {\left( {t_{b} - t} \right)P_{1} } \right)dt} } } } \right] = {\frac{{C_{p} I_{p} \left( {Q - M\lambda } \right)^{2} }}{2Q}} $$
$$ {\text{Annual}}\;{\text{interest}}\;{\text{earned}} = {\frac{{S_{p} I_{e} }}{T}}\left( {\int\limits_{0}^{M} {\lambda tdt} } \right) = {\frac{{I_{e} M^{2} S_{p} \lambda^{2} }}{2Q}} $$

Case 3:

tb < M ≤ tc

When tb < M ≤ tc, the annual interest charged only includes the interest charged for the on-hand inventory of perfect items in period (M,tc).
$$ {\text{Annual}}\;{\text{interest}}\;{\text{charged}} = {\frac{{C_{p} I_{p} }}{T}}\left[ {\int\limits_{M}^{{t_{c} }} {\left( {\lambda \left( {t_{c} - t} \right)} \right)dt} } \right] = {\frac{{C_{p} I_{p} \left( {Q - M\lambda } \right)^{2} }}{2Q}}. $$
$$ {\text{Annual}}\;{\text{interest}}\;{\text{earned}} = {\frac{{S_{p} I_{e} }}{T}}\left( {\int\limits_{0}^{M} {\lambda tdt} } \right) = {\frac{{I_{e} M^{2} S_{p} \lambda^{2} }}{2Q}}. $$

Case 4:

T < M

When T < M, this means the account is settled after all items have sold out. Therefore, no annual interest is charged in this case.
$$ {\text{Annual}}\;{\text{interest}}\;{\text{charged}} = 0. $$
$$ {\text{Annual}}\;{\text{interest}}\;{\text{earned}} = {\frac{{S_{p} I_{e} }}{T}}\left( {\int\limits_{0}^{T} {\lambda tdt} } \right) + S_{p} I_{e} \lambda T\left( {M - T} \right)/T = {\frac{{S_{p} I_{e} \left( {2M\lambda - Q} \right)}}{2}}. $$

Therefore, the total annual cost TVC(Q) has four different values, as follows:

Case 1:

0 ≤ M ≤ ta
$$ \begin{aligned} TVC_{1} (Q) = & C\lambda + C_{R} \lambda x + {\frac{K\lambda }{Q}} + {\frac{Q\lambda h(P - d - \lambda )}{{2P^{2} }}} + {\frac{{Q\lambda xh\left[ {P_{1} \left( {2P - d - 2\lambda } \right) - d\lambda } \right]}}{{2PP_{1}^{2} }}} + {\frac{{dQh_{1} \lambda }}{{2P^{2} }}} \\ & + {\frac{{Qh\left[ {d\lambda + P_{1} \left( {\lambda - P} \right)} \right]^{2} }}{{2P^{2} P_{1}^{2} }}} + {\frac{{Q\lambda x^{2} h_{1} }}{{2P_{1} }}} + {\frac{{C_{p} I_{p} (P - \lambda )(Q^{2} - M^{2} P\lambda )}}{2PQ}} - {\frac{{I_{e} M^{2} S_{p} \lambda^{2} }}{2Q}}; \\ \end{aligned} $$
(20)

Case 2:

ta < M ≤ tb
$$ \begin{aligned} TVC_{2} (Q) = & C\lambda + C_{R} \lambda x + {\frac{K\lambda }{Q}} + {\frac{Q\lambda h(P - d - \lambda )}{{2P^{2} }}} + {\frac{{Q\lambda xh\left[ {P_{1} \left( {2P - d - 2\lambda } \right) - d\lambda } \right]}}{{2PP_{1}^{2} }}} \\ & \quad + {\frac{{Qh\left[ {d\lambda + P_{1} \left( {\lambda - P} \right)} \right]^{2} }}{{2P^{2} P_{1}^{2} }}} + {\frac{{dQh_{1} \lambda }}{{2P^{2} }}} + {\frac{{Q\lambda x^{2} h_{1} }}{{2P_{1} }}} + {\frac{{C_{p} I_{p} \left( {Q - M\lambda } \right)^{2} }}{2Q}} - {\frac{{I_{e} M^{2} S_{p} \lambda^{2} }}{2Q}}; \\ \end{aligned} $$
(21)

Case 3:

tb < M ≤ tc
$$ \begin{aligned} TVC_{3} (Q) = & C\lambda + C_{R} \lambda x + {\frac{K\lambda }{Q}} + {\frac{Q\lambda h(P - d - \lambda )}{{2P^{2} }}} + {\frac{{Q\lambda xh\left[ {P_{1} \left( {2P - d - 2\lambda } \right) - d\lambda } \right]}}{{2PP_{1}^{2} }}} \\ & \quad + {\frac{{Qh\left[ {d\lambda + P_{1} \left( {\lambda - P} \right)} \right]^{2} }}{{2P^{2} P_{1}^{2} }}} + {\frac{{dQh_{1} \lambda }}{{2P^{2} }}} + {\frac{{Q\lambda x^{2} h_{1} }}{{2P_{1} }}} + {\frac{{C_{p} I_{p} \left( {Q - M\lambda } \right)^{2} }}{2Q}} - {\frac{{I_{e} M^{2} S_{p} \lambda^{2} }}{2Q}}; \\ \end{aligned} $$
(22)

Case 4:

T < M
$$ \begin{aligned} TVC_{4} (Q) =\, & C\lambda + C_{R} \lambda x + {\frac{K\lambda }{Q}} + {\frac{Q\lambda h(P - d - \lambda )}{{2P^{2} }}} + {\frac{{Q\lambda xh\left[ {P_{1} \left( {2P - d - 2\lambda } \right) - d\lambda } \right]}}{{2PP_{1}^{2} }}} \\ & + {\frac{{Qh\left[ {d\lambda + P_{1} \left( {\lambda - P} \right)} \right]^{2} }}{{2P^{2} P_{1}^{2} }}} + {\frac{{dQh_{1} \lambda }}{{2P^{2} }}} + {\frac{{Q\lambda x^{2} h_{1} }}{{2P_{1} }}} - {\frac{{S_{p} I_{e} \left( {2M\lambda - Q} \right)}}{2}}. \\ \end{aligned} $$
(23)

Figure 1 assumes that the rework rate of imperfect items is larger than the demand rate, i.e. P1 > λ. When P1 < λ or P1 = λ, following the same modeling steps mentioned above, the results are the same as when P1 > λ. Therefore, the models (Eq. 2023) are the same, no matter P1 > λ, λ > P1 or P1 = λ.

3 Solution approach

The proposed model assumes that the manufacturer wants to determine the optimal production quantity \( Q^{*} \) to minimize the total cost TVC(Q). The problem is to minimize
$$ \begin{gathered} TVC\left( Q \right) = \left\{ \begin{gathered} TVC_{1} (Q)\quad {\text{if}}\quad 0 \le M \le t_{a} \hfill \\ TVC_{2} (Q)\quad {\text{if}}\quad t_{a} < M \le t_{b} \hfill \\ TVC_{3} (Q)\quad {\text{if}}\quad t_{b} < M \le t_{c} \hfill \\ TVC_{4} (Q)\quad {\text{if}}\quad T < M \hfill \\ \end{gathered} \right. \hfill \\ \hfill \\ \end{gathered} $$
which is a four-branch function with one variable. Since
$$ TVC_{1} (MP) = TVC_{2} (MP),\quad TVC_{2} \left({\frac{M}{{1/P + x/P_{1} }}}\right) = TVC_{3} \left({\frac{M}{{1/P + x/P_{1} }}}\right) \quad {\text{and}} \quad TVC_{3} (M\lambda ) = TVC_{4} (M\lambda ),\quad TVC(Q), $$
is continuous at Q = MP, \( Q = {\frac{M}{{1/P + x/P_{1} }}} \) and Q = .
Since \( \frac{dTVC}{dQ} = 0 \) is the essential condition to find the optimal total items produced during a production cycle, the first derivatives of TVCi(Q) with respect to Q are
$$ \begin{aligned} {\frac{{dTVC_{1} }}{dQ}} = \, & {\frac{h\lambda (P - d - \lambda )}{{2P^{2} }}} + {\frac{{d\lambda h_{1} }}{{2P^{2} }}} + {\frac{{x^{2} \lambda h_{1} }}{{2P_{1} }}} + {\frac{{hx\lambda \left[ {P_{1} \left( {2P - 2\lambda - d} \right) - d\lambda } \right]}}{{2PP_{1}^{2} }}} \\ & + {\frac{{h\left[ {d\lambda + P_{1} \left( {\lambda - P} \right)} \right]^{2} }}{{2P^{2} P_{1}^{2} }}} - {\frac{K\lambda }{{Q^{2} }}} + {\frac{{C_{p} I_{p} (P - \lambda )}}{P}}\left\{ {1 - {\frac{{\left( {Q^{2} - M^{2} P\lambda } \right)}}{{2Q^{2} }}}} \right\} + {\frac{{M^{2} \lambda^{2} I_{e} S_{p} }}{{2Q^{2} }}} = 0; \\ \end{aligned} $$
(24)
$$ \begin{aligned} {\frac{{dTVC_{2} }}{dQ}} = \, & {\frac{h\lambda (P - d - \lambda )}{{2P^{2} }}} + {\frac{{d\lambda h_{1} }}{{2P^{2} }}} + {\frac{{x^{2} \lambda h_{1} }}{{2P_{1} }}} + {\frac{{hx\lambda \left[ {P_{1} \left( {2P - 2\lambda - d} \right) - d\lambda } \right]}}{{2PP_{1}^{2} }}} \\ & + {\frac{{h\left[ {d\lambda + P_{1} \left( {\lambda - P} \right)} \right]^{2} }}{{2P^{2} P_{1}^{2} }}} - {\frac{K\lambda }{{Q^{2} }}} + {\frac{{C_{p} I_{p} \left( {Q - M\lambda } \right)}}{Q}}\left\{ {1 - {\frac{{\left( {Q - M\lambda } \right)}}{2Q}}} \right\} + {\frac{{M^{2} \lambda^{2} I_{e} S_{p} }}{{2Q^{2} }}} = 0; \\ \end{aligned} $$
(25)
$$ \begin{aligned} {\frac{{dTVC_{3} }}{dQ}} = \, & {\frac{h\lambda (P - d - \lambda )}{{2P^{2} }}} + {\frac{{d\lambda h_{1} }}{{2P^{2} }}} + {\frac{{x^{2} \lambda h_{1} }}{{2P_{1} }}} + {\frac{{hx\lambda \left[ {P_{1} \left( {2P - 2\lambda - d} \right) - d\lambda } \right]}}{{2PP_{1}^{2} }}} + {\frac{{h\left[ {d\lambda + P_{1} \left( {\lambda - P} \right)} \right]^{2} }}{{2P^{2} P_{1}^{2} }}} - {\frac{K\lambda }{{Q^{2} }}} \\ & + {\frac{{C_{p} I_{p} \left( {Q - M\lambda } \right)}}{Q}}\left\{ {1 - {\frac{{\left( {Q - M\lambda } \right)}}{2Q}}} \right\} + {\frac{{M^{2} \lambda^{2} I_{e} S_{p} }}{{2Q^{2} }}} = 0; \\ \end{aligned} $$
(26)
$$ \begin{aligned} {\frac{{dTVC_{4} }}{dQ}} = & {\frac{h\lambda (P - d - \lambda )}{{2P^{2} }}} + {\frac{{d\lambda h_{1} }}{{2P^{2} }}} + {\frac{{x^{2} \lambda h_{1} }}{{2P_{1} }}} + {\frac{{hx\lambda \left[ {P_{1} \left( {2P - 2\lambda - d} \right) - d\lambda } \right]}}{{2PP_{1}^{2} }}} \\ & + {\frac{{h\left[ {d\lambda + P_{1} \left( {\lambda - P} \right)} \right]^{2} }}{{2P^{2} P_{1}^{2} }}} - {\frac{K\lambda }{{Q^{2} }}} + {\frac{{S_{p} I_{e} }}{2}} = 0. \\ \end{aligned} $$
(27)

We use the following proposition to discuss the condition of unique solution.

Proposition 1

  1. (1)

    If\( K\lambda - C_{p} I_{p} (P - \lambda )M^{2} \lambda - M^{2} \lambda^{2} I_{e} S_{p} > 0, \)thereexists a unique solution (Q > 0) of\( {\frac{{dTVC_{1} }}{dQ}} = 0; \)

     
  2. (2)

    If\( 2K\lambda + C_{p} I_{p} M^{2} \lambda^{2} - M^{2} \lambda^{2} I_{e} S_{p} > 0, \)thereexists a unique solution (Q > 0) of\( {\frac{{dTVC_{2} }}{dQ}} = 0 \) and of \( {\frac{{dTVC_{3} }}{dQ}} = 0; \)

     
  3. (3)

    There exists a unique solution (Q > 0) of\( {\frac{{dTVC_{4} }}{dQ}} = 0. \)

     

Proof: Please see proofs in Appendix 1.

Since it is difficult to judge whether or not \( {\frac{{d^{2} TVC}}{{dQ^{2} }}} \) is greater than zero, we develop a search-based algorithm to find \( Q^{*} \). First, we find the extreme values and the boundary values of the four cases. If the extreme value does not fit its restricted condition, we exclude the extreme value. Finally, we compare the boundary values and the extreme values that fit the restricted condition to determine the optimal solution.

4 Numerical study

The goals of the numerical study in this research are as follows:

  1. 1.

    To illustrate the proposed solution approach;

     
  2. 2.

    To discuss the impact of related parameters on decisions and cost

     

4.1 Numerical example

Most values of the parameter setting follow the example of Hayek and Salameh (2001). Consider a manufactured product with a constant demand rate of 1,200 units per year (λ = 1,200). The machine used to manufacture this item has a production rate of 1,600 units per year P = (1,600). The production cost per item is $50 (C = 50). The machine setup cost is $1,500 (K = 1,500). The holding cost per perfect item is $20 per year (h = 20), and the holding cost per imperfect item is $22 per year (h1 = 22). The repair cost per imperfect item is $8 (CR = 8). The imperfect items are reworked at a rate of 1,300 units per year (P1 = 1,300). The percentage of imperfect items produced is 5% (x = 0.05). The production rate of imperfect items is 80 (d = 80). The manufacturer’s trade credit period offered by the supplier is 0.1 per year (M = 0.1). The annual interest rate earned per dollar is 0.1 (Ie = 0.1). The annual interest rate charged per dollar is 0.15 (Ip = 0.15). The material purchasing cost per item is $80 (Cp = 80). The selling price per good quality item is $200 (Sp = 200).

Figure 3 shows the results computed using the proposed searched-based algorithm. The optimal solution in Case 1 is \( Q_{1}^{*} = 6 3 4. 6 5 9 \), and its total cost is 65,607.8; the optimal solution in Case 2 is \( Q_{2}^{*} = 160 \), and its total cost is 71,296.4; the optimal solution in Case 3 is \( Q_{3}^{*} = 150.725 \), and its total cost is 71,887.3; the optimal solution in Case 4 is \( Q_{4}^{*} = 120 \), and its total cost is 74,584.8. Therefore, the optimal solution is \( Q^{*} = Q_{1}^{*} = 634.659 \) and \( TVC(Q^{*} ) = TVC_{1} (Q_{1}^{*} ) = 65607.8 \). Figure 4 provides a graphic illustration of TVC(Q).
https://static-content.springer.com/image/art%3A10.1007%2Fs10696-010-9070-3/MediaObjects/10696_2010_9070_Fig3_HTML.gif
Fig. 3

Graphic illustration of \( TVC_{1} (Q_{1} ),TVC_{2} (Q_{2} ),TVC_{3} (Q_{3} )\, {\text{and}}\, TVC_{4} (Q_{4} ) \)

https://static-content.springer.com/image/art%3A10.1007%2Fs10696-010-9070-3/MediaObjects/10696_2010_9070_Fig4_HTML.gif
Fig. 4

Graphic illustration of TVC(Q)

4.2 Numerical analysis

Investigating the influence of interest charged Ip, interest earned Ie and the percentage of imperfect items x on production decision and total cost is interesting. Tables 1 and 2 show the effects of Ip and Ie on Q and TVC. The tables show that Q decreases as Ip increases, while TVC increases. Both Q and TVC decrease as Ie increases. When the interest charged Ip increases, the manufacturer’s production quantity Q will decrease. This implies the manufacturer will produce fewer items in every cycle to decrease loss. The manufacturer’s production quantity Q will decrease as the interest earned Ie increases. The manufacturer will shorten the cycle time to take advantage of the higher interest earned more times.
Table 1

Effects of Ip and Ie on Q

Q

Ip

0.125

0.150

0.175

0.200

Ie

0.090

659.1

637.5

617.7

599.6

0.095

657.7

636.1

616.3

598.2

0.100

656.2

634.7

615.0

596.9

0.105

654.8

633.2

613.6

595.6

Table 2

Effects of Ip and Ie on TVC

TVC(Q)

Ip

0.125

0.150

0.175

0.200

Ie

0.090

65,475.8

65,630.4

65,779.6

65,923.9

0.095

65,464.9

65,619.1

65,768.0

65,911.9

0.100

65,453.9

65,607.8

65,756.3

65,899.8

0.105

65,442.9

65,596.4

65,744.6

65,887.7

Table 3 presents the effects of x on Q and TVC. When the percentage of imperfect quality items x increases, the manufacturer’s production quantity Q will decrease, but the total annual cost of TVC will increase.
Table 3

Effects of x on Q and TVC

 

x

0.03

0.04

0.05

0.06

0.07

Q

641.9

638.4

634.7

630.9

627.0

TVC(Q)

65,357.8

65,482.3

65,607.8

65,734.5

65,862.2

Figures 5 and 6 show the result of sensitivity analysis. Experiments were conducted by increasing these parameters up to ±67%. Figure 5 shows that the manufacturer’s production quantity Q is more sensitive to the interest charged Ip. Figure 6 indicates that the total annual cost is more sensitive to the interest charged Ip and the percentage of imperfect quality items x. Therefore, the manufacturer should especially notice the change of Ip and x.
https://static-content.springer.com/image/art%3A10.1007%2Fs10696-010-9070-3/MediaObjects/10696_2010_9070_Fig5_HTML.gif
Fig. 5

Impact of parameters on optimal production quantity

https://static-content.springer.com/image/art%3A10.1007%2Fs10696-010-9070-3/MediaObjects/10696_2010_9070_Fig6_HTML.gif
Fig. 6

Impact of parameters on total annual cost

5 Concluding remarks

This paper addresses the production problem under the conditions of trade credit and reworking imperfect items and extends the traditional production model by considering reworking imperfect items and trade credit to cope with a more realistic situation. Unlike previous models, the proposed model calculates interest earned based on the retail price. This research determines the optimal production lot size while minimizing total cost. Numerical analysis reveals the influences of interest charged, interest earned, and the percentage of imperfect items on production and total cost.

Future research can consider several production and inventory situations, such as deteriorating items, shortage allowed, and non-re-workable defects which must be thrown away.

Acknowledgment

We thank the Editor and two referees for their thoughtful comments which have aided us in the improvement of this manuscript. This paper was supported by the National Science Council under grant NSC 98-2410-H-036-004.

Copyright information

© Springer Science+Business Media, LLC 2010