CHAPTER 1 BACKGROUND OF THE RESEARCH
1.7 Salameh and Jaber’s (2000) Model
Porteus (1986) and Lee and Rosenblatt (1987) had assumed that all defective items are reworkable. The work of Salameh and Jaber (2000) is the first model that treats imperfect quality items (not defectives) to be salvaged at a discounted price. They implied that imperfect quality items are functional items that fo not meet the quality requirements for a given product/task;
however, they do for a lower grade one. Salameh and Jaber (2000) extended the traditional EOQ model by assuming that each lot received from a vendor contains imperfect quality items. They assumed that (i) the demand is deterministic and that it occurs parallel to the inspection process and is fulfilled from goods found to be perfect by the inspection process, (ii) the orders are replenished instantaneously, (iii) there are no shortages, (iv) the lot contains a fixed fraction γ of imperfect items with known probability density function, (v) a 100% screening is performed to separate these defective items, and (vi) items of poor quality are kept in stock and sold prior to receiving the next shipment as a single batch at a discounted price. The behavior of inventory is as described in Figure 1.5.
Note that the behavior in Figure 1.5 is an average one. If the model is simulated for different values of γ, the average ending inventory by timeT is either positive or negative. The average access inventory can be assumed to be zero. A simulation study was conducted to verify this with ten thousand runs with parameters: Q=100, D=10, γ ~ U (0, 0.2) and γ ~U(0,0.5).In other words, access ending inventory in one cycle would be used to cover for a stock out in a subsequent cycle. This behavior incurs additional costs (extra holding and shortage costs). It is assumed that the value of h in the work of Salameh and Jaber (2000) and this thesis is significant enough to account for these additional costs. A detailed study of this issue may be addressed in a technical note or short communication. This will be left for a future work.
34 where Q is the lot size γ is the percentage of imperfection in Q, τ is the inspection time and T is the cycle time. The screening and cycle time are shown by τ and T respectively. To guarantee there are no shortages, Salameh and Jaber (2000) set the condition γ= 1− 𝐷/𝑥 where x is the screening rate (x > D) The number of good items in each order of size Q, is
𝑁(𝑄,𝛾) = 𝑄 −γ𝑄 = (1−γ)𝑄 (1.9)
The total revenue per cycle is given by the sum of the revenue from selling defective and nondefective items, as
𝑇𝑅(𝑄) =𝑠1𝑄(1−γ) +𝑣𝑄γ (1.10)
where s1 and 𝑣 are the unit selling price of a good item and the unit salvage price of an imperfect quality item.
Time
τ
T Inventory level
γQ
Figure 1.5 Behavior of inventory in Salameh and Jaber (2000) model Q
35 The total cost per cycle is the sum of ordering, purchasing, screening and holding costs. It is given by
𝑇𝐶(𝑄) = 𝐴𝑏 + 𝑐1𝑄 + 𝑑𝑄 + ℎ𝑏�𝑄(1−γ)𝑇 2 +γ𝑄2
𝑥 � (1.11)
where Ab is the buyer's order cost, c1 is the unit purchase cost, d is the unit screening cost, and h is the unit holding cost. Detailed derivations of Eqs. (1.9) and (1.10) are provided in Salameh and Jaber (2000). So, the total profit per cycle would be
𝑇𝑃(𝑄) =𝑇𝑅(𝑄)− 𝑇𝐶(𝑄) (1.12)
Total profit per unit time is 𝑇𝑃𝑈(𝑄) =𝑇𝑃(𝑄)𝑇 where 𝑇= (1−𝛾)𝑄𝐷 . So,
𝑇𝑃𝑈(𝑄) =𝐷 �𝑠1− 𝑣+ℎ𝑏𝑄
𝑥 �+𝐷 �𝑣 −ℎ𝑏𝑄
𝑥 − 𝑐1− 𝑑 −𝐴𝑏
𝑄 � � 1 1−γ�
−ℎ𝑏𝑄(1−γ) 2
(1.13)
Taking the fraction of defectives as a random variable, the expected annual profit would then be E[𝑇𝑃𝑈(𝑄)] =𝐷 �𝑠1− 𝑣+ℎ𝑏𝑄
𝑥 �+𝐷 �𝑣 −ℎ𝑏𝑄
𝑥 − 𝑐1− 𝑑 −𝐴𝑏
𝑄 �E� 1 1−γ�
−ℎ𝑏𝑄(1−E[γ]) 2
Refer to Salameh and Jaber (2000) for detailed derivations of TPU(Q) and E[TPU(Q)].
(1.14)
To find the optimal value of the above profit, they differentiated the above equation w.r.t Q and equated that to zero. The second derivative of the above equation remains negative for all
36 values Q which implies that there exists a unique value Q* that maximizes the above profit. That value would be
𝑄∗ =� 2𝐴𝑏𝐷E[1/(1−γ)]
ℎ𝑏[1−E[γ]−2𝐷(1−E[1/(1−γ)])/𝑥] (1.15)
Note that when γ=0, the denominator reduces to h and the numerator reduces to 2𝐴𝑏𝐷.
It should be noted that the above equation reduces to the traditional EOQ model when percentage defective γ is zero. They concluded that economic lot size quantity tends to increase as the average percentage of imperfect quality items increase. The number “2” in the denominator was missing in the above equation in Salameh and Jaber (2000) which was pointed out by Cárdenas-Barrón (2000). For a detailed derivation of the above model, refer to Salameh and Jaber (2000), page 61. This model can be referred to as a base EOQ model for imperfect items. This model was extended in a number of directions. Few of these extended models are outlined here: Goyal et al. (2003) used the base model to develop a two level supply chain model for imperfect items, Chan et al. (2003) introduced a number of quality classifications in the base model, Papachristos and Konstantaras (2006) developed the sufficient conditions for the base model and that of Chan et al. (2003), Wee et al. (2007) incorporated backordering, and Maddah and Jaber (2008) suggested using renewal reward theorem to estimate the expected annual profit.
Goyal and Cárdenas-Barrón (2002) simplified the above base model by ignoring the screening and purchasing costs. They showed that this simplification results in almost zero penalty. Maddah and Jaber (2008) used this simplified and corrected a flaw in the base model.
They noticed that the annual profit in the base model is not exact as the cycle profit is a renewal process. They suggested using renewal reward theory to compute it. That is, the annual profit should be calculated as a ratio of profit per cycle and the cycle time, in the presence of imperfect items in the lot being screened. So, the expected time unit profit function is written as
E[𝑇𝑃𝑈(𝑄)] =E[𝑇𝑃(𝑄)]
E[𝑇]
= 1
1−E[𝛾]�[𝑠1(1−E[𝛾]) +𝑣𝐸[𝛾]− 𝑐1− 𝑑]−𝐴𝑏𝐷 𝑄 −
ℎ𝑏𝑄
2 E[(1− 𝛾)2]−ℎ𝑏𝐷 𝑥 E[𝛾]�
(1.16)
37 Thus, they obtained simpler expressions for the annual profit and the optimal order size while the penalty for using their annual profit function instead of that in Salameh and Jaber (2000) was negligible. They also showed that the optimal order size has a direct relation with the screening rate and the fraction of defectives.
A possible issue with the base model is if it addresses the uncertainty in a supplier’s quality well with a uniform distributed fraction of defectives. Numerous examples were tested to study the difference in the annual profit in Maddah and Jaber (2008) for (i) uniformly and (ii) normally distributed fraction of defectives. The mean and variance of the uniform distribution were used as input parameters for normal distribution. A snapshot of this experiment is shown in Appendix 1. It was observed that the difference in expected cost and the lot size quantity were insignificant, which reasonably justifies the use of the uniform distribution in the base model. Besides, a uniform distribution was adopted in all the studies that extended or modified the model of Salameh and Jaber (2000). The same distribution will be used for the numerical analysis throughout this thesis.