Mathematical Model

CHAPTER 3 ECONOMIC ORDER QUANTITY (EOQ) FOR ITEMS WITH IMPERFECT

3.3 Mathematical Model

Figure 3.1 shows how the inventory behaves with a buyer. It should be noted here that Salameh and Jaber (2000) suggested this behavior. The screening and consumption of the inventory continues until time τ₁, after which all the defectives (B1) are withdrawn from inventory as a single batch and are sold to the secondary market. The consumption process continues at the demand rate until the end of cycle time T. Due to inspection error, some of the items used to fulfill the demand would be defective. These defective items are later returned to the inventory and are shown in Figure 3.1 as B2. The position of B2 is chosen to ensure the disposal of two defective batches (B1 and B2) at the same time. The cycle inventory is kept at a minimum level this way. It may be beneficial to sell the defective lot in two batches (_𝐵2^′ and 𝐵₂^′′) in a cycle, as shown in Figure 3.2, where the holding cost of the returned inventory is less;

however, implicitly, this may incur additional cost such as transportation (fixed and variable cost), which makes the savings in inventory less than the additional costs incurred. So, in this

Q

B2

Inventory level

Z2

Z1

B1

Time

τ₁ t2

T

T

79 chapter, it is assumed here that the two types of defective items (screened and misclassified) are sold at the same time, as shown in Figure 3.1. The model in Figure 3.2 may be addressed in a technical note sometime in the near future.

Figure 3.2 Selling the defective items twice in a cycle

To avoid shortages, it is assumed that the number of nondefective items is at least equal to the adjusted demand, that is the sum of the actual demand (D) and items that are replaced for the ones returned (γm2 D) from the market over T. Thus

𝑄 − 𝑄(1− 𝛾)𝑚₁− 𝑄𝛾(1− 𝑚₂)≥ 𝐷𝑇+𝛾𝑚₂𝑄 𝑄(1− 𝛾)− 𝑄𝑚₁(1− 𝛾)≥ 𝐷𝑇

𝑄(1− 𝛾)(1− 𝑚1)≥ 𝐷𝑇

So, for the limiting case, the cycle length (which includes the screening time) can be written as 𝑇= 𝑄(1− 𝛾)(1− 𝑚₁)

𝐷 (3.1)

Inventory level

Z2

Z1

B1

τ₁ t2

T

𝐵₂^′ 𝐵₂^′′

Time

80 It should be noted that the above expression is unaffected by the Type II error and reduces to the cycle length in Salameh and Jaber (2000) if the Type I error becomes zero.

Consider now the different cases of misclassifications that an inspection process can have.

There are four possibilities in such an inspection process. Those are: Case (1) A nondefective item is classified as nondefective; Case (2) A nondefective item is classified as defective; Case (3) A defective item is classified as nondefective and Case (4) A defective item is classified as defective. This scenario is depicted in Figure 3.3 below. The number of items going into different categories following these cases is given by:

Case (1): 𝑄(1− 𝛾)(1− 𝑚1) Case (2): 𝑄(1− 𝛾)𝑚₁ Case (3): 𝑄𝛾𝑚₂

Case (4): 𝑄𝛾(1− 𝑚2)

Now B1 and B2 are given by 𝐵₁ =𝑄(1− 𝛾)𝑚₁+𝑄𝛾(1− 𝑚₂) 𝐵₂ = 𝑄𝛾𝑚₂

Figure 3.3 Four possibilities in the inspection process

Inspection/Decision Picking an Item for Inspection

Nondefective (1 – γ)

Defective (γ)

Classify an item as defective (1 – m2)

Classify a bad item as Nondefective Classify an item as Nondefective (1 – m1)

Classify a good item as defective

81 The items in batch B2 are returned from the market at the rate 𝑄𝛾𝑚₂/𝑇 and are taken from the inventory with batch B1. Therefore the revenue from salvaging B = B1 + B2 items within a cycle are given by:

𝑅₁ =𝑣(𝐵₁+𝐵₂) =𝜈𝑄(1− 𝛾)𝑚₁+𝜈𝑄𝛾(1− 𝑚₂) +𝜈𝑄𝛾𝑚₂ or

𝑅₁ =𝜈𝑄(1− 𝛾)𝑚₁+𝜈𝑄𝛾

The revenue from selling the good items is computed as 𝑅2 = 𝑠1𝑄(1− 𝛾)(1− 𝑚1) +𝑠1𝑄𝛾𝑚2

So, the total revenue is given as 𝑅 =𝑅₁+𝑅₂

𝑅 =𝑠₁𝑄(1− 𝛾)(1− 𝑚₁) +𝑠₁𝑄𝛾𝑚₂+𝜈𝑄(1− 𝛾)𝑚₁+𝜈𝑄𝛾 (3.2)

Consider now the different costs of the inventory system. The procurement cost per cycle is

𝑃𝐶 =𝐴_𝑏+𝑐₁𝑄 (3.3)

where 𝑐₁ is the variable cost. The screening cost per cycle is the sum of the costs of inspection and misclassifications which is given by

𝐼𝐶(𝑄) =𝑑𝑄+𝑐_𝑟(1− 𝛾)𝑄𝑚₁+𝑐_𝑎𝑄𝛾𝑚₂ (3.4)

The holding cost per cycle is the cost of carrying the (i) nondefective lot, (ii) defective lot and (iii) returned lot. So, from Figure 3.1 the holding cost for a cycle can be written as

𝐻𝐶(𝑄) =ℎ �(𝑄 − 𝑍₁)𝜏₁

2 +𝑍₁𝜏₁+𝑍₂𝑡₂

2 �+ℎ �𝐵₂𝑇 2 �

where Z represents the stock level before and after the screening/inspection process. Replacing B2 with Q𝛾m2, Z2 with Z1 – B1, Z1 with Q – Dτ1 and τ₁ with D/x, it can be written as

82 𝐻𝐶(𝑄) =ℎ_𝑏𝑄²

2 �2 𝑥 −

𝐷 𝑥²+𝑉²

𝐷 �+ℎ �𝑄𝛾𝑚₂𝑇

2 � (3.5)

where 𝑉 = 1−^𝐷_𝑥−(𝑚₁− 𝛾) +𝛾(𝑚₁+𝑚₂)

Therefore the total cost per cycle is given by summing up Eq. (3.3), Eq. (3.4) and Eq. (3.5), i.e.

𝐶_𝑏= 𝑃𝐶+𝐼𝐶+𝐻𝐶 or

𝐶_𝑏(𝑄) =𝐴_𝑏+𝑐₁𝑄+𝑑𝑄+𝑐_𝑟(1− 𝛾)𝑄𝑚₁+𝑐_𝑎𝑄𝛾𝑚₂+ℎ𝑏

2 ��2 𝑥 −

𝐷 𝑥² +𝑉²

𝐷 � 𝑄²+𝑄𝛾𝑚₂𝑇�

The term 𝑉² is simplified in Appendix 2. Figure 3.1 depicts the behavior of different types of inventory in the order cycle. The triangle at the bottom represents the defective lot that is returned by the market and is accumulated into the salvaged lot.

The total profit per cycle can now be written as the difference between the total revenue and total cost per cycle, that is

𝑇𝑃(𝑄) =𝑠1𝑄(1− 𝛾)(1− 𝑚1) +𝑠𝑄𝛾𝑚2+𝜈𝑄(1− 𝛾)𝑚₁+𝜈𝑄𝛾 − �𝐴_𝑏+𝑐₁𝑄+𝑑𝑄+𝑐_𝑟(1− 𝛾)𝑄𝑚₁+𝑐_𝑎𝑄𝛾𝑚₂ +ℎ𝑏

2 ��2 𝑥 −

𝐷 𝑥²+𝑉²

𝐷 � 𝑄²+𝑄𝛾𝑚₂𝑇��(3.6) Since 𝛾, 𝑚1 and 𝑚2 are assumed in this thesis to be independent and identically distributed random variables with probability density functions 𝑓₁(𝛾), 𝑓₂(𝑚₁) and 𝑓₃(𝑚₂), respectively. The expected total profit can be written as

E[𝑇𝑃(𝑄)] =𝑠₁𝑄(1−E[𝛾])(1−E[𝑚₁]) +𝑠𝑄E[𝑚₂]E[𝛾] +𝜈𝑄(1−E[𝛾])E[𝑚1] +𝜈𝑄E[𝛾]− 𝐴𝑏− 𝑐1𝑄 − 𝑑𝑄 − 𝑐𝑟(1−E[𝛾])𝑄E[𝑚₁]− 𝑐𝑎𝑄E[𝑚2]E[𝛾]

−ℎ𝑏

2 ��2 𝑥 −

𝐷

𝑥²+E[𝑉²]

𝐷 � 𝑄²+𝑄E[𝑚₂]E[𝑇]E[𝛾]�

(3.7)

Now from Eq. (3.1), the expected cycle length would be:

E[𝑇] =𝑄(1−E[𝛾])(1−E[𝑚1])

𝐷 (3.8)

83 Maddah and Jaber (2008b) corrected the approach in the Salameh and Jaber (2000) model to determine the annual profit. They suggested using renewal reward theorem. Using this new approach, the expected annual profit for our model, is written as

E[𝑇𝑃𝑈(𝑄)] = E[𝑇𝑃(𝑄)]

E[𝑇]

or

E[𝑇𝑃𝑈(𝑄)] =𝑠₁𝐷+ 𝑠𝐷E[𝛾]E[𝑚2]

(1−E[𝛾])(1−E[𝑚₁]) +

𝑣𝐷E[𝑚1] (1−E[𝑚₁]) +

𝑣𝐷E[𝛾]

(1−E[𝛾])(1−E[𝑚₁])−

𝐷 �𝐴𝑄^𝑏 +𝑐₁+𝑑+𝑐_𝑟(1−E[𝛾])E[𝑚1] +𝑐𝑎E[𝛾]E[𝑚2] +ℎ𝑏

2 ��2 𝑥 − 𝐷

𝑥²+ E[𝑉²] 𝐷 � 𝑄��

(1−E[𝛾])(1−E[𝑚₁]) −ℎ_𝑏𝑄E[𝛾]E[𝑚₂]

2

(3.9)

It should be noted that Eq. (3.9) converges to the expected annual profit equation in Maddah and Jaber (2008b) once the value of errors goes to zero. It can be demonstrated that this expected annual profit follows a concave function. The first derivative of Eq. (3.9) is given by

d

d𝑄E[𝑇𝑃𝑈(𝑄)] =−𝐷 �− 𝐴𝑄^𝑏2 +ℎ𝑏

2 ��2 𝑥 − 𝐷

𝑥²+ E[𝑉²] 𝐷 ���

(1−E[𝛾])(1−E[𝑚₁]) −ℎE[𝛾]E[𝑚₂] 2 The second derivative of Eq. (3.9) is

d²

d𝑄²E[𝑇𝑃𝑈(𝑄)] = − 2𝐴𝑏𝐷

𝑄³(1−E[𝛾])(1−E[𝑚₁])

Since 𝐴_𝑏> 0, D> 0, 0 < E[𝛾] < 1, and 0< E[m1] < 1, then ^d2

d𝑄²E[𝑇𝑃𝑈(𝑄)]< 0 for every 𝑄 > 0, suggesting that the annual profit in Eq. (3.9) is concave. This fact is also demonstrated graphically in the next section. Thus, the optimal order size that represents the maximum annual profit, is determined by setting the first derivative equal to zero and solving for Q to get

84

𝑄^∗ =� 2𝐴𝑏𝐷

ℎ_𝑏E[𝛾]E[𝑚2](1−E[𝛾])(1−E[𝑚₁]) +ℎ_𝑏𝐷 �2 𝑥 − 𝐷

𝑥²+ E[𝑉²] 𝐷 �

(3.10)

It should be noticed that the denominator in Eq. (3.10) remains always positive, since x > D, 0 , E[𝛾]< 1, 0< E[𝑚1]< 1, ℎ𝑏 > 0, and E[𝑉²] > 0 from Appendix 2. Note also that when γ = m1 = m2

= 0, Eq. (3.10) reduces to the traditional EOQ formula.