Multiproduct EPQ Model with Single Machine ... - Semantic Scholar

Multiproduct EPQ Model with Single Machine, Backordering, and Immediate Rework Process

Ata Allah Taleizadeh Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran E-mail: [email protected]

Seyed Jafar Sadjadi Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran E-mail: [email protected]

Seyed Taghi Akhavan Niaki* Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran E-mail: [email protected] *Corresponding author

Abstract: Production systems with scrapped and rework items have recently become an interesting subject of research. While most attempts have been focused on finding the optimal production quantity in a simple production system, little work appears on a joint production environment. In this research, two joint production systems in a form of multiproduct singlemachine with and without rework are studied where shortage is allowed and backordered. For each system, the optimal cycle length, the backordered and production quantities of each product are determined such that the cost function is minimized. Proof of the convexity of the involved 1

objective functions of each model is provided and numerical illustrations are given to demonstrate the applicability of the proposed models. Furthermore, the results obtained by solving the models with and without rework of defective items are compared. Sensitivity analysis and some managerial insights based on the numerical illustration are provided at the end. [Received 12 January 2010; Revised 22 May 2010; Accepted 31 May 2010]

Keywords: Inventory Control; Multiproduct; Single-Machine; Rework; Limited Production Capacity, Backordering

Biographical notes: Ata Allah Taleizadeh is currently a Ph.D. student in Industrial Engineering at Iran University of Science and Technology. He received his B.Sc. and M.Sc. degrees both in Industrial Engineering. His research interest areas include inventory control, production planning, and integrated inventory–production-pricing models.

Seyed Jafar Sadjadi is an Associate Professor of Industrial Engineering at Iran University of Science and Technology. He has finished his BS and Master in Iran in the field on Industrial Engineering and did his PhD at the University of Waterloo, Canada in 1998. His research interest is Operations Research with Industrial Engineering and financial applications. Dr. Sadjadi is presently working as editor-in-chief of the International Journal of Industrial Engineering Computations, an indexed journal published by Growing Science.

Seyed Taghi Akhavan Niaki is a Professor of Industrial Engineering at Sharif University of Technology. His research interests are in the areas of Simulation Modeling and Analysis, Applied Statistics, Multivariate Quality Control and Operations Research. Before joining Sharif 2

University of Technology he worked as a systems engineer and quality control manager for Iranian Electric Meters Company. He received his Bachelor of Science in Industrial Engineering from Sharif University of Technology, his Master’s and his Ph.D. degrees both in Industrial Engineering from West Virginia University.

1

Introduction

The economic production quantity (EPQ) is a commonly used production model that has been studied extensively in the past few decades. One of the extension types of this model deals with the rework on imperfect quality products. Reworking imperfect products is known to be economically more attractive than to disposing of them. Pasandideh and Niaki (2008) extended a multiproduct EPQ model in which the orders may be delivered discretely in multiple pallet forms. Yoo et al. (2009) proposed a profit-maximizing EPQ model that incorporates both the imperfect production quality and two-way imperfect inspection. Jaber et al. (2008) introduced an EPQ model for items with imperfect quality subject to learning effects. They assumed imperfect quality items are withdrawn from inventory and sold at discount price. Jamal et al. (2004) developed an EPQ model to determine the optimum batch quantity in a single-stage system. They assumed rework is completed under two different operational policies, to minimize the total system cost, named immediate and N-Cycle rework processes. Cardenas-Barron (2009) introduced an EPQ model with a rework process at a single-stage manufacturing system with planned backorders. In this research, single-stage manufacturing system generates imperfect products, which are reworked in the same cycle. Goyal and Cardenas-Barron (2002) developed an EPQ model under a simple approach to determine economic production quantity for systems

3

that produce imperfect items. Huang (2004) extended the EPQ model in which imperfect products are allowed to be present into the produced lot sizes. Chiu et al. (2004) considered the effects of random defective rate and imperfect rework process on EPQ model. Chang (2004) investigated the effects of imperfect products on the total inventory cost associated with an EPQ model. Hayek and Salameh (2001) developed an EPQ model by considering the imperfect products and rework items, assuming all the shortages are backordered and the percentage of defective products is a random variable. Goyal and CardenasBarron (2005) extended the EPQ model by considering an imperfect production system that produces defective products, supposing all the defective items that are randomly produced are reworked. Chiu et al. (2007) investigated an EPQ model that considers scrap, rework, and stochastic machine breakdowns to determine the optimal run time and production quantity. Sarker et al. (2008) introduced an extended EPQ model dealing with the optimum production quantity in a multi-stage system in which rework is done under two different operational policies to minimize the total system cost. Liao et al. (2009) studied an integrated maintenance and production system with an EPQ model for an imperfect process involving a defective production system under increasing failure rate. Cardenas-Barron (2008) introduced a simple derivation to find the optimal production quantity of a system that includes a rework process. Choi et al. (2007) extended a production model in which demand is satisfied by recovering used products as well as new products. They assumed that a fixed proportion of the used products are collected from customers and later recovered for reuse. Taleizadeh et al. (2010a) introduced an EPQ model with scrapped items and limited production capacity as well as a multiproduct single-machine production system with stochastic scrapped production rate, partial backordering, and service level constraint (2010b). Furthermore, Taleizadeh et al. (2010c) studied a production quantity

4

model with random defective items, service level constraints, and repair failure in multiproduct single machine situations. Other related researches are: Cárdenas-Barrón (2007, 2009), Chandrasekaran et al. (2007), Liu et al. (2008), Mohan et al. (2008), and Parveen and Rao (2009).

2 Problem definition The inventory control problem of this research involves imperfect production processes, in which due to process deterioration or some other factors, defective items of n different types are generated at a rate β i ; i = 1, 2,..., n . The production rate of producing the ith item is Pi assuming perfect items' quantities, i.e., (1 − β i )Pi , satisfy the corresponding demand, d i in each cycle. The demand, production, and proportion of defective rates of the products are considered constants. A cycle consists of production uptime ( T i1 and T i 5 ), reworking time ( T i 2 ), and production downtime (T i 3 and T i 4 ). During the reworking time, no defective items are produced and the

production and rework tasks are performed utilizing shared resources. Since all of the products are manufactured on a single machine with a limited capacity, a unique cycle length for all items is considered, i.e, T1 = T 2 = " = T n = T . All of the imperfect quality items are reworked and it is assumed that there are no scrapped items at the end of a rework process. Furthermore, both the production and rework processes are accomplished using the same resource, the same cost, and at the same speed where shortage is allowed in the form of backorder. Moreover, we assume that there is a real constant production capacity limitation on the single machine on which all products are produced and that the setup cost is nonzero.

5

3 Modeling Jamal et al. (2004) developed an EPQ model to determine the optimum production quantity of an item where rework is performed in two different situations. The objective function of their model was to minimize the total inventory cost of the production system under consideration. In this paper, the model of Jamal et al. (2004) is extended and applied to a more realistic inventory control problem in which several items and several constraints are available. However, before developing the model, since the problem at hand is of a multiproduct type, for i = 1, 2,..., n the following notations are first defined for the parameters and the variables of the model. qi

The production lot size of the ith product in each cycle (a decision variables)

T

The cycle length (a decision variable)

bi

The backordered quantity of the ith product in a cycle (a decision variable)

di

The demand rate of the ith product

Pi

The production (or rework) rate of the ith product

βi

The proportion of defective items of the ith product

C is

The setup cost of a production run for the ith product

Tis

The machine setup time to produce the ith product

N

Number of cycles per year

TC (q , b )

The total inventory costs per year

Ii

The maximum on-hand inventory of the ith product when a regular production process stops

I iMax

The maximum on-hand inventory of the ith product when the reworking ends

6

C iP

The production cost of the ith product per item ($/item)

C iR

The inspection and rework cost of the ith product per item ($/item)

C ih

The holding cost of the ith product per item per unit time ($/item/unit time)

C iB

The shortage cost of the ith product per item per unit time ($/item/unit time) The multiproduct single machine (MP-SM) EPQ model with rework is first developed in

section 3.1 and then a simpler model without rework is introduced in section 3.2.

3.1 MP-SM with rework

Figure (1) depicts a cycle for the inventory control problem under study. In order to model the problem, a single-product problem consisting of the ith product is first developed, then, the model will extend to include several products. be greater than or equal to the demand rate di . As a result, we have:

(1 − βi ) Pi − d i

≥0

(1)

The production cycle length (see Fig. 1) is the summation of the production uptimes, the rework time, and the production downtimes, i.e., : 5

T = ∑T i j

(2)

j =1

The basic assumption in EPQ model with rework process is that (1 − β i ) Pi must always where T i1 andT i 5 are the production uptimes (the periods in which perfect and defective items are produced), T i 2 is the reworking time, and the production downtimes are T i 3 andT i 4 .

7

Figure 1 On hand inventory of perfect quality items for MP-SM with rework Ii

I iMax = I i + β i ( Pi − d i

)

qi Pi

Pi − d i I i = ( (1 − β i ) Pi − d i

) pi q

−di

− bi

i

(1 − β i ) Pi − d i

Ti 4 Ti1

Ti

2

Ti5

t

Ti3

bi

T

In this research, a part of the modeling procedure is adopted from Jamal et al. (2004) to model the problem. As noted before, since all products are manufactured on a single machine with a limited capacity, the cycle length for all products are equal (T1 = T 2 = " = T n = T ) . Hence, based on Figure (1) we have: I i = ( (1 − β i ) Pi − d i )

qi − bi Pi

(3)

and

I iMax = I i + βi ( Pi − d i )

qi q q = ( (1 − βi ) Pi − d i ) i + βi ( Pi − d i ) i − bi Pi Pi Pi

It is obvious from Figure (1) that:

8

(4)

qi bi − Pi (1 − β i )Pi − d i

T i1 =

T i 2 = βi

(5)

qi Pi

(6)

⎛ di di ⎞ ⎜1 − − βi ⎟ qi Pi Pi ⎠ b ⎝ 3 − i Ti = di di

(7)

Ti4 =

bi di

(8)

Ti5 =

bi (1 − β i )Pi − d i

(9)

Hence, using Equation (2) the cycle length for a single product problem is: 5

T = ∑T i j = j =1

qi di

(10)

Or, q i = d iT

3.1.1

(11)

The objective function for a single product

The total cost of the system includes setup, processing, rework, shortage, and inventory carrying costs. Although the processing and rework costs are constants and do not affect the optimal solution, they are used in the objective function in order to determine the annual total cost. Defective items are produced in every batch and they are reworked within the same cycle. During the rework process, some processing and inventory holding costs are incurred for reworked quantities. Then, the total cost per year, TC (q , b ) , is obtained as:

9

Holding Cost Shortage Cost

     Rework Cost Setup Cost

  P  Max Max Ii + Ii Ii NC iB bi (T i 4 + T i 5 ) 1 2 3 P s R h Ii NC i q i + NC i + NC i β i q i + NC i [ (T i ) + (T i ) + (T i ) + ] (12) 2 2 2 2

Pr oduction Cost

TC (q , b ) =

Finally, the objective function of the joint production system (MP-SM with rework) using Equation (10) becomes: n

n

∑C

i =1

T

TC (T , b ) = ∑ C iP d i +

i =1

s i

n n ⎡I ⎤ I + I iMax I Max (T i 2 ) + i (T i 3 ) ⎥ + ∑ C iR β i d i + ∑ C ih ⎢ i (T i1 ) + i 2T 2T i =1 i =1 ⎣ 2T ⎦

(13)

C B b (T 4 + T i 5 ) +∑ i i i 2T i =1 n

3.1.2 The constraints for a single product Since the production plus rework times of the ith product is T i1 + T i 2 + T i 5 and the setup time is

T i s , the summation of the production, reworking and setup time (for all products) will n

n

i =1

i =1

be ∑ (T i1 + T i 2 + T i 5 ) + ∑T i s . It is clear that this summation must be smaller or equal to the cycle length (T ). Hence, the capacity constraint of the model becomes:

n

∑ (T i =1

n

1 i

+ T i 2 + T i 5 ) + ∑T i s ≤ T

(14)

i =1

Then, based on the Equations (3), (4), and (7) we have: n

∑ (1 + β ) i =1

3.1.3

n di T + ∑T i s ≤ T pi i =1

(15)

The final model of MP-SM with rework

From Equations (3) to (11), (13) and (15), the final model of the joint production system is obtained as follows. 10

Min : TC (T , B ) = α1T 2 + α 2T +

α3 T

n

+ ∑ α 4i i =1

n n bi2 n − ∑ α 5i bi − ∑ α 6i biT + ∑ (C iP + β i C iR ) d i (16) T i =1 i =1 i =1

n

s .t . : T ≥

∑T i =1

s i

n ⎛ di ⎞ 1 − ⎜ ∑ (1 + β i ) ⎟ pi ⎠ ⎝ i =1

= T Lower

(17)

Where; n

α1 = ∑ i =1

n

α2 = ∑ i =1

2 2 Cih ⎡ ( (1 + β i )((1 − βi ) Pi − di ) ) di + βi ( Pi − di )di ⎤ ⎢ ⎥>0 2 ⎣ Pi ⎦

(18)

Cih ⎡ 3((1 − βi ) Pi − di )di 2 ( Pi − (1 + βi )di )di ⎤ + ⎢ ⎥>0 2 ⎣ Pi Pi ⎦

(19)

n

α 3 = ∑C is > 0

(20)

i =1

⎡ ⎤ (1 − β i ) Pi ⎢ ⎥>0 ⎣ ((1 − βi ) Pi − di )di ⎦

(21)

C ih ⎛ d i ((1 − βi )Pi − d i ) β i (Pi − d i ) ⎞ + + α 5i = ⎜1 + 2βi ⎟>0 2 ⎝ Pi Pi Pi ⎠

(22)

α 4i =

α 6i =

Cih 2

⎡ ((1 − βi ) Pi − 2di ) ⎤ CiB ⎢ ⎥+ ⎣ ((1 − βi ) Pi − d i )di ⎦ 2

C ih ⎡ (1 + βi )d i ⎢ 2 ⎣ Pi

⎤ ⎥>0 ⎦

n

Note that since

∑ (C i =1

P i

(23)

+ βi CiR ) di is constant, it has been removed from the objective function. In

section four, a solution method is given for the developed model.

3.2 The MP-SM without rework

In this case, according to Figure (2) the objective function of the joint production system is obtained as follows. 11

Figure 2 On hand inventory of perfect quality items for MP-SM without rework Ii

I i = ( (1 − β i ) Pi − d i

) pi q

− bi

i

−di

(1 − β i ) Pi − d i

Ti3

Ti 4 t

Ti1

Ti 2

bi

T

Shortage Cost Holding Cost     Setup Cost

  P  B 3 4 I NC b ( T + T i i i i ) NC iP q i + NC is + NC ih [ i (T i1 + T i 2 ) + ] 2 2

Pr oduction Cost

TC (q , b ) =

(24)

n

n

=

∑C i =1

P i di

+

∑C

s i

i =1

T

n

+

∑C i =1

h i

⎡ Ii 1 2 ⎤ ⎢ 2T (T i + T i ) ⎥ + ⎣ ⎦

n

∑ i =1

C iB bi

(T i + T i ) 2T 4

5

Where; T i1 =

qi bi − Pi (1 − β i )Pi − d i

(25)

⎛ di di ⎞ ⎜1 − − βi ⎟ qi Pi Pi ⎠ b ⎝ 2 Ti = − i di di Ti3 =

(26)

bi di

(27) 12

Ti4 =

bi (1 − β i )Pi − d i

(28)

The capacity constraint in this case will be: n

n

i =1

i =1

∑ (T i1 +T i 4 ) + ∑T i s ≤ T

(29)

Hence, the final model of MP-SM EPQ without rework becomes:

γ2

n bi2 n Min : TC (T , B ) = γ 1T + + ∑ γ 3i − ∑ γ 4 i bi + ∑ γ 5i T i =1 T i =1 i =1 n

(30)

n

s .t . : T ≥

∑T i =1

s i

n ⎛ di ⎞ − 1 ⎜ ∑ ⎟ ⎝ i =1 p i ⎠

= T Lower

(31)

Where; 2 C ih ⎡ ( ((1 − β i )Pi − d i ) ) d i ⎤ ⎢ ⎥>0 Pi i =1 2 ⎣ ⎦ n

γ1 = ∑

(32)

n

γ 2 = ∑C is > 0

(33)

i =1

γ 3i

⎤ ⎛ C ih + C iB ⎞ ⎡ 2(1 − βi )Pi =⎜ ⎟⎢ ⎥>0 2 ⎝ ⎠ ⎣ ((1 − βi )Pi − d i )d i ⎦

γ 4i =

C ih 2

⎛⎛ di ⎜⎜ ⎜1 − (1 + β i ) Pi ⎝⎝

γ 5i = C iP d i +

C ih 2

(34)

⎞ ((1 − βi )Pi − d i ) ⎞ ⎟⎟ > 0 ⎟+ P i ⎠ ⎠

⎛ d i ⎞ ((1 − βi )Pi − d i )d i ⎜1 − (1 + β i ) ⎟ Pi ⎠ Pi 2 ⎝

(35)

(36)

4 A solution method The methods for solving both models of sections 3.1 and 3.2 are given in sections 4.1 and 4.2, respectively. 13

4.1

A solution method for MP-SM with rework

In order to derive an optimal solution for the final model, a proof of the convexity of the objective function is first provided. Then, a classical optimization technique using partial derivatives is utilized to derive the optimal solution.

Theorem (1): The objective function TC (T , b ) in (16) is convex.

Proof: Using Equation (37), since the Hessian matrix, obtained in Appendix (1), results in positive values for all nonzero bi and T , TC (T , b ) = Z is convex.

⎡T ⎤ ⎢b ⎥ n ⎢ 1⎥ 2α [T , b1 ,b2 ,",bn ] × H × ⎢b2 ⎥ = 2α1T 2 + 3 − 2∑α6i biT > 0 T ⎢ ⎥ i =1 ⎢#⎥ ⎢⎣bn ⎥⎦

(37)

To find the optimal production period length T and the optimal backorder quantities bi s, partial differentiations of Z with respect to T and bi are obtained in Equations (38) and (39). n

α ∑ 4i ∂Z = 2α1T + α 2 − 32 − i =1 2 T T ∂T

α bi2

n

− ∑ α 6i bi = 0

(38)

i =1

n

n

i =1

i =1

→ 2α1T 3 + (α 2 − ∑ α 6i bi )T 2 − (α 3 + ∑ α 4i bi2 ) = 0 ∂Z 2α 4i bi = − α 5 i − α 6 iT = 0 ∂bi T

→ 2α 4i bi − α 5iT − α 6iT 2 = 0 →

bi =

α 5 iT + α 6 iT 2α 4i

2

(39)

Then the following solution procedure is used to solve and ensure the feasibility of the problem:

14

The solution procedure: Step1: Check for feasibility:

If (1 − βi ) Pi − di ≥ 0 and

n

di

∑ (1 + β ) P < 1 , go to step 2. Otherwise, the problem is infeasible. i =1

i

i

Step2: Find a solution point:

Find the optimal solution using equations (38) and (39) and by an iterative approach. To do this, start with bi = 0 and insert bi = 0 into Equation (25). Then, the new real positive values of T are obtained. This iterative search will continue until the difference between two consecutive values of T is smaller than a given Δ . Step3: Check the constraint:

Check the constraint based on the obtained value of T . If T > TLower then T * = T . Otherwise,

T * = TLower and go to step 4. Step4: Obtain the optimal solution:

Based on the obtained value of T * and using q i * = d iT * , bi* will be derived by Equation (39). Then calculate the objective function value and go to step 5. Step5: Terminate the procedure.

4.2

A solution method for MP-SM without rework

Similar to the previous case, the objective function of the MP-SM without rework model is convex too. To find the optimal production period length T and the optimal backorder quantities

bi s, partial differentiations of Z with respect to T and bi are obtained as are given in Equations (40) and (41).

15

n

γ ∂Z = γ 1T − 22 − ∂T T

∑γ i =1

n

2 3i b i

T

2

∂Z 2γ 3i bi = − γ 4i = 0 ∂bi T

=0 → T 2 =

→

bi =

γ 2 + ∑ γ 3i bi2 i =1

(40)

γ1

γ 4i T 2γ 3i

(41)

By substituting equation (41) in equation (40), we have:

γ2

T =

n

γ1 − ∑ i =1

bi =

( γ 4i )

(42)

2

4γ 3i

γ 4i T 2γ 3i

(43)

In this case, the following solution procedure is used to solve and ensure the feasibility of the problem:

The solution procedure: Step1: Check for feasibility: n (γ ) d If (1 − β i ) Pi − d i ≥ 0 and ∑ i < 1 and γ 1 > ∑ 4i , go to step 2. Otherwise, the problem is i =1 Pi i =1 4γ 3i 2

n

infeasible.

Step2: Find a solution point: Using equations (42) and (43) find the optimal solution.

Step3: Check the constraint: Check the constraint based on the obtained value of T . If T > T Lower thenT * = T . Otherwise,

T * = T Lower and go to step 4.

Step4: Obtain the optimal solution:

16

Based on the obtained value ofT * and using q i * = d iT * , bi* will be obtained by Equation (43). Then calculate the objective function value and go to step 5.

Step5: Terminate the procedure.

5 Numerical Examples Consider two multiproduct EPQ problems consisting of ten products with breakdown and capacity constraint. In these examples the demand, production, and proportion of defective rates of each product are considered constant. Furthermore, the production rate of non-defective items is considered constant and is greater than the demand rate for each product. For each example, both the immediate rework and no rework situations are considered. Moreover, there are no scrapped or defective items during the rework process. The production and rework are accomplished using the same resource at the same speed and shortages are allowed as backorders. The general and the specific data of these examples are given in Tables (1) and (2), respectively. Tables (3) and (4) show the best results for the two examples considering immediate rework using the first solution procedure. It should be noted that a value of

Δ = 10−12 is assumed in the solution procedure. Using the second solution procedure, Tables (5) and (6) show the best results for the two examples when rework is not performed.

17

Table 1 General Data for Example #1 Product

di

Pi

T is

C is

1 2 3 4 5 6 7 8 9 10

100 150 200 250 300 350 400 450 500 550

5000 5500 6000 6500 7000 7500 8000 8500 9000 9500

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

700 650 600 550 500 450 400 350 300 250

Table 2

C iR

Cih

C iB

βi

24 22 20 18 16 14 12 10 8 6

24 22 20 18 16 14 12 10 8 6

10 9 8 7 6 5 4 3 2 1

20 18 16 14 12 10 8 6 4 2

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

CiP

C iR

Cih

C iB

βi

24 22 20 18 16 14 12 10 8 6

24 22 20 18 16 14 12 10 8 6

10 9 8 7 6 5 4 3 2 1

20 18 16 14 12 10 8 6 4 2

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

General Data for Example #2

Product

di

Pi

T is

C is

1 2 3 4 5 6 7 8 9 10

100 200 300 400 500 600 700 00 900 1000

5000 5500 6000 6500 7000 7500 8000 8500 9000 9500

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

700 650 600 550 500 450 400 350 300 250

T

T*

qi

bi

Z

0.7794

TLower

0.7794

Product

The Best Results for Example #1 with Immediate Rework

0.1246

Table 3

1 2 3 4 5 6 7 8 9 10

CiP

77.9442 116.9163 155.8884 194.8605 233.8362 272.8047 311.7768 350.7489 389.7210 428.6931

12.9534 19.4380 25.9523 32.5130 39.1414 45.7818 52.7732 60.0117 68.1049 79.7102

63,610

18

Table 4

T

0.6014

1 2 3 4 5 6 7 8 9 10

TLower

0.8931

Product

T*

qi

bi

Z

0.8931

The Best Results for Example #2 with Immediate Rework

89.3094 178.6178 267.9281 357.2374 446.5468 535.8562 625.1655 714.4749 803.7843 893.936

9.9875 19.9476 29.9332 39.9866 50.1512 60.4890 71.1199 82.3392 95.09655 114.3833

101,890

Table 5 The Best Results for Example #1 without Rework Product 1 2 3 4 5 6 7 8 9 10

TLower

0.1294

T

7.2282

T*

qi

bi

Z

0.1294

12.9417 19.4125 25.8834 32.3542 38.8251 45.2959 51.7668 58.2376 64.7085 71.1793

0.7356 1.0236 1.2616 1.4511 1.5934 1.6893 1.7397 1.7452 1.7501 1.7589

77,215

19

Table 6 The Best Results for Example #2 without Rework Product 1 2 3 4 5 6 7 8 9 10

TLower

0.0779

T

3.6794

T*

qi

bi

Z

0.0779

7.7880 15.5761 23.3641 31.1521 38.9402 46.7282 54.5162 62.3043 70.0923 77.8804

0.4427 0.8034 1.0892 1.3051 1.4549 1.5417 1.6483 1.7173 1.7839 1.8369

127,020

A comparison study based on the results given in Tables (3)-(6) shows that lower optimum total costs are obtained for both examples in which reworking is allowed. Furthermore, the optimum cycle length obtained for the systems where rework is permitted is greater than those of the non-reworking systems.

6 A Sensitivity Analysis The sensitivity analysis of this section is performed only for two previously mentioned examples of the immediate rework case. To study the effects of the parameter changes on the optimal result derived by the proposed method, the sensitivity analysis involves increasing or decreasing the parameters one at a time, by 20% and 40%. Tables (7) and (8) show the results of the sensitivity analysis for the first and second numerical example, respectively. For the first numerical example, based on the results in Table (7), T * = T = 0.7794 ,

TLower does not have any effect on the optimal value, and the capacity constraint is satisfied 20

( T > TLower ). Hence, when Pi increases by 20 and 40 percents, the changes in the values of

T *and T are identical. This result is valid for a 20 percents decrease in the values of Pi as well. However, when Pi is decreased by 40 percents, T decreases from 0.7794 to 0.7540 and

TLower increases from 0.1246 0.7998. Therefore, T < TLower and the optimal value of the cycle length becomesT * = T Lower = 0.7998 . In this case, changes in the value of T * are considered with the situations of the three previous states (See Table 7). Since T *and T are the same, changing the values of C is has the same effect on T *and T . Moreover, because C is has not been used to calculate TLower , the changes in the values of C is dose not have any effect on TLower . By changing the values of βi , the values of Z , TLower , and T are changed. Since the optimal value of the cycle length is equal to T , when βi changes, the percentage of changes in the values of T *and T are identical. To study the effects of changes inT i s , because T i s is only used to calculate the values of TLower , whenT i s is changed, the optimal value does not change. Since an increase of even 40 percents in the value of T i s does not cause TLower to become greater than T , T * and Z remain unchanged. In the second example, based on the results in Table (8), for a 40% increase on Pi ,

TLower = 0.1668 and T = 0.6141 which results in T * = 0.6141 . Therefore, T increases while T * decreases. Despite the fact that in the second example T < TLower which made TLower and T * interchangeable (see Table 8), a 40% increase in Pi results in T > TLower and then T and T * will be interchangeable. Such a result is obtained in a 20% increase in Pi as well. In cases where Pi n

decreases 20% or 40%, since

d ∑ (1 + β ) p becomes greater than one, the problem becomes i

i =1

i

i

21

infeasible. Because changes in the values of C is do not have any effect on TLower and since in this problem T * = TLower , no changes occur in the values of T * . For a 40% increase in βi , however n

when

d ∑ (1 + β ) p becomes greater than one, the problem becomes infeasible. Nevertheless, with i

i =1

i

i

a 20% increase in βi , T * = TLower , both T * and TLower will increase 306.69 %. By a 40% decrease on βi , TLower decreases significantly (to 0.3561), while T lowers to 0.5933. In this case, due to

T > TLower , T * is equal to T and there are no significant changes in the values of T * . In cases where T i s increases by 20% or 40%, because T i s directly affects TLower and T * = TLower , changes in the values of T * and TLower are identical. However, since T i s is not used to calculate T , changes in the values of T are not proportional to changes in T i s .

Table 7 Effects of Parameter Changes in Example #1 % Changes in Parameters

Pi

C is

βi

T is

+40 +20 -20 -40 +40 +20 -20 -40 +40 +20 -20 -40 +40 +20 -20 -40

% Changes in

TLower

T

T*

Z

-26.54 -17.4 +46.37 +541.91 0 0 0 0 +13.84 +6.49 -5.68 -10.78 +40 +20 -20 -40

+1.67 +0.95 -1.31 -3.25 +17.68 +9.23 -10.26 -21.99 +0.84 +0.41 -0.37 -0.72 0 0 0 0

+1.67 +0.95 -1.31 +2.62 +17.68 +9.23 -10.26 -21.99 +0.84 +0.41 -0.37 -0.72 0 0 0 0

-0.25 -0.14 +0.2 +0.51 +3.39 +1.76 -1.95 -4.15 +7.09 +3.54 -3.55 -7.09 0 0 0 0

22

A 20% decrease in T i s has exactly the same effect as a 20% increase. Furthermore, when

T i s decreases 40%, TLower becomes 0.5359 and T remains untouched ( T = 0.6014 ). Hence, T becomes greater than TLower , and T * decreases from 0.8931 to 0.6014.

7 Managerial Insight In the first example, according to the results in Table (7), while Z and T are slightly-sensitive to the changes in the values of Pi , TLower is sensitive to them. Since T * = T , T * is slightly-sensitive to the changes in the values of Pi . Generally, Z and T are slightly-sensitive and sensitive to the changes in the values of C is , respectively, TLower is insensitive to the changes in values of C is ,

T *and T are slightly-sensitive, and T * is sensitive to the changes in values of βi .

Table 8

Effects of Parameter Changes in Example #2

% Changes in Parameters

Pi

C is

βi

Tis

+40 +20 -20 -40 +40 +20 -20 -40 +40 +20 -20 -40 +40 +20 -20 -40

% Changes in

TLower

T

T*

Z

-81.32 -71.75 Infeasible Infeasible 0 0 0 0 Infeasible +306.69 -42.99 -60.13 +40 +20 -20 -40

+2.11 +1.19 +17.57 +9.18 -19.34 -21.89 +0.74 -0.69 -1.34 0 0 0 0

-31.24 -31.87 0 0 0 0 +306.69 -33.12 -33.56 +40 +20 -20 -32.66

-1.38 -1.27 +2.03 +1 -1.56 -1.98 +41.91 -4.88 -8.65 +3.29 +1.48 -0.97 -1.12

23

Furthermore, the results in Table (7) show that TLower is very sensitive to the changes in the values ofT i s . In the second example, the results in Table (8) show that while Z and T are slightlysensitive to the changes in the values of Pi , T * and TLower are sensitive and very sensitive to the changes in the values of Pi , respectively. Because T * = TLower , while both T * and TLower are insensitive to C is , Z and T are slightly-sensitive and sensitive to the changes in values of C is , respectively. In general, while Z and TLower are very sensitive, T is slightly-sensitive to the changes in values of βi ; furthermore, since T * = TLower in the second example, T * is also very sensitive to the changes in values of βi .

8 Conclusion In this research, two EPQ models with production capacity limitation and backordered quantities were developed under the assumptions of immediate rework and no-rework situations. The primary aim of this research has been to determine the optimal period lengths, backorder, and production quantities. The objective functions of the proposed mathematical models were proved convex and two solution procedures were proposed. Then, two numerical examples with ten products were used to illustrate the implementation of the proposed methods. The results obtained by solving the two models were next compared and finally, sensitivity analyses have been performed to demonstrate the applicability of the proposed methodology and to provide some managerial insights for practitioners. Future researches may focus on the partial backordering strategies and multiproduct multi-constraint problems in an uncertain environment. 24

Furthermore, an unequal cycle length consideration for all products may reduce the total cost. This may be investigated in a future research.

9 Acknowledgement The authors are thankful for the constructive comments of the anonymous reviewers that significantly improved the presentation of the paper.

References Jamal, A.M.M., Sarker, B.R. and Mondal, S. (2004) ‘Optimal manufacturing batch size with rework process at a single-stage production system’, Computers and Industrial Engineering, Vol. 47, pp.77–89. Cardenas-Barron, L.E. (2009) ‘Economic production quantity with rework process at a singlestage manufacturing system with planned back-orders’, Computers & Industrial Engineering, Vol. 57, pp.1105–1113. Goyal, S.K. and Cardenas-Barron, L.E. (2002) ‘Note on economic production quantity model for items with imperfect quality–A practical approach’, International Journal of Production Economics, Vol. 77, pp.85–87. Chang, H-C. (2004) ‘An application of fuzzy sets theory to the EOQ model with imperfect quality items’, Computers and Operations Research, Vol.31, pp. 2079–2092. Huang, C-K. (2004) ‘An optimal policy for a single-vendor single-buyer integrated productioninventory problem with process unreliability consideration’, International Journal of Production Economics, Vol. 91, pp. 91–98.

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Hayek, P.A. and Salameh, M.K. (2001) ‘Production lot sizing with the reworking of imperfect quality items produced’, Production Planning and Control, Vol.12, pp.584–590. Goyal, S.K. and Cárdenas-Barrón, L.E. (2005) ‘Economic production quantity with imperfect production system’, Industrial Engineering Journal, Vol. 34, pp. 33–36. Sarker, B.R., Jamal, A.A.M. and Mondal, S. (2008) ‘Optimal batch sizing in a multi-stage production system with rework consideration’, European Journal of Operational Research, Vol.184, pp. 915–929. Liao, G.L., Chen, Y.H. and Sheu, S.H. (2009) ‘Optimal economic production quantity policy for imperfect process with imperfect repair and maintenance’, European Journal of Operational Research, Vol.195, pp. 348–357. Cárdenas-Barrón, L.E. (2009) ‘On optimal batch sizing in a multi-stage production system with rework consideration’, European Journal of Operational Research, Vol.196, pp.1238-1244. Cárdenas-Barrón, L.E. (2007) ‘On optimal manufacturing batch size with rework process at single-stage production system’, Computers & Industrial Engineering, Vol. 53, pp.196-198. Chandrasekaran, C., Rajendran, C., Krishnaiah-Chetty O.V. and Hanumanna, D. (2007) ‘Meta heuristics for solving economic lot scheduling problems (ELSP) using time-varying lot-sizes approach’, European Journal of Industrial Engineering, Vol.1, pp. 152-181. Cardenas-Barron, L.E. (2008) ‘Optimal manufacturing batch size with rework in a single-stage production system–A simple derivation’, Computers and Industrial Engineering, Vol. 55, pp.758–765. Liu, J., Wu, L. and Zhou, Z. ‘A time-varying lot size method for the economic lot scheduling problem with shelf life considerations’, European Journal of Industrial Engineering, Vol. 2, pp.337-355.

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Mohan, S., Mohan, G. and Chandrasekhar, A. (2008) ‘Multi-item, economic order quantity model with permissible delay in payments and a budget constraint’, European Journal of Industrial Engineering, Vol.2, pp.446-460. Parveen, M. and Rao, T.V.V.L.N. ‘Optimal batch sizing, quality improvement and rework for an imperfect production system with inspection and restoration’, European Journal of Industrial Engineering, Vol. 3, pp.305-335. Choi, D.W., Hwang, H. and Koh, S.G. (2007) ‘A generalized ordering and recovery policy for reusable items’, European Journal of Operational Research, Vol. 182, pp.764–774. Taleizadeh, A.A., Najafi, A.A. and Niaki, S.T.A. (2010a) ‘Economic production quantity model with scraped items and limited production capacity’, Scientia Iranica (2010a), In Press. Taleizadeh, A.A., Niaki, S.T.A. and Najafi, A.A. (2010b) ‘Multi-product single-machine production system with stochastic scrapped production rate, partial backordering and service level constraint’, Journal of Computational and Applied Mathematics, Vol. 233, pp.18341849. Taleizadeh, A.A., Wee, H.M. and Sadjadi, S.J. (2010c) ‘Multi-product production quantity model with repair failure and partial backordering’, Computers & Industrial Engineering, Vol. 59, pp.45-54. Chiu, S.W., Wang, S.L. and Chiu, Y.S.P. (2007) ‘Determining the optimal run time for EPQ model with scrap, rework, and stochastic breakdowns’, European Journal of Operational Research, Vol. 180, pp.664-676. Chiu, S.W., Gong, D.C. and Wee, H.M. (2004) ‘Effects of random defective rate and imperfect rework process on economic production quantity model’, Japan Journal of Industrial and Applied Mathematics, Vol. 21, pp.375-389.

27

Pasandideh, S.H.R. and Niaki, S.T.A. (2008) ‘A genetic algorithm approach to optimize a multiproducts EPQ model with discrete delivery orders and constrained space’, Applied Mathematics and Computation, Vol.195, pp.506–514. Yoo, S.H., Kim, D.S. and Park, M.S. (2009) ‘Economic production quantity model with imperfect-quality items, two-way imperfect inspection and sales return’, International Journal of Production Economics, Vol. 121, pp.255–265. Jaber, M.Y., Goyal, S.K. and Imran M. (2008) ‘Economic production quantity model for items with imperfect quality subject to learning effects’, International Journal of Production Economics, Vol. 115, pp.143– 150.

Appendix 1: The Proof of the Convexity Consider the objective function as n C3 n bi2 n Min : Z = C1T + C2T + + ∑ C4i − ∑ C5i bi − ∑ C6i bT i T i =1 T i =1 i =1 2

Differentiating the objective function with respects to T and bi and setting the differentials equal to zero results in n

C ∂Z = 2C 1T + C 2 − 32 − T ∂T

∑C i =1

T

4i

bi2

2

n

− ∑ C 6 i bi = 0 i =1

n

n

i =1

i =1

→ 2C 1T 3 + (C 2 − ∑C 6i bi )T 2 − (C 3 + ∑C 4i bi2 ) = 0 ∂Z 2C 4i bi = − C 5i − C 6 iT = 0 T ∂B i → 2C 4i bi − C 5iT − C 6iT 2 = 0 →

bi =

C 5iT + C 6 iT 2C 4i

Then, the Hessian matrix H becomes

28

2

⎡ ∂2Z ⎢ 2 ⎢ ∂T ⎢ ∂2Z ⎢ ⎢ ∂b1∂T H = ⎢ ∂2Z ⎢ ⎢ ∂b 2 ∂T ⎢ ⎢ # ⎢ ∂2Z ⎢ ⎣⎢ ∂b n ∂T

∂2Z ∂T ∂b1

∂2 Z ∂T ∂b 2

∂2Z ∂ 2b1

∂2 Z ∂b1∂b 2

∂2Z ∂b 2 ∂b1

∂2 Z ∂ 2b 2

#

#

∂ Z ∂b n ∂b1

∂ Z ∂b n ∂b 2

2

2

n ⎡ 2∑C 4i bi2 ⎢ 2C ⎢ 2C 1 + 33 + i =1 3 T T ⎢ ⎢ 2C 41b1 − − C 61 ⎢ T2 ⎢ = ⎢ 2C b − 422 2 − C 62 ⎢ T ⎢ # ⎢ ⎢ 2C b ⎢ − 4 n2 n − C 6 n ⎣⎢ T

∂2 Z ⎤ ⎥ ∂T ∂b n ⎥ ∂2 Z ⎥ " ⎥ ∂b1∂b n ⎥ ⎥ ∂2 Z ⎥ " ∂b 2 ∂b n ⎥ ⎥ # # ⎥ ∂2 Z ⎥ ⎥ " ∂ 2b n ⎦⎥ "

2C b − 412 1 − C 61 T 2C 41 T 0 # 0

⎤ ⎥ 2C 42b 2 2C 4 n b n ⎥ " − − − − C C 62 6 n T2 T2 ⎥ ⎥ " 0 0 ⎥ ⎥ ⎥ 2C 42 " 0 ⎥ T ⎥ # " # ⎥ ⎥ 2C 4 n ⎥ " 0 T ⎦⎥

Hence, ⎡T ⎤ ⎡T ⎤ ⎢b ⎥ ⎢b ⎥ ⎢ 1⎥ ⎢ 1⎥ 2C3 n [T , b1 , b2 ," , bn ] × H × ⎢b2 ⎥ = [2C1T + 2 − ∑ C6i bi , − C61T , − C62T , " , − C6 nT ] × ⎢b2 ⎥ T ⎢ ⎥ ⎢ ⎥ i =1 ⎢#⎥ ⎢#⎥ ⎢⎣bn ⎥⎦ ⎢⎣bn ⎥⎦ n 2C = 2C1T 2 + 3 − 2∑ C6i biT T i =1

Substituting bi =

C 5 iT + C 6 iT 2 we will have 2C 4i

⎡T ⎤ ⎢b ⎥ ⎢ 1⎥ n ⎛ C C T 2 + C6 i T 3 ⎞ [T , b1 , b2 ," , bn ] × H × ⎢b2 ⎥ = 2 ⎜ C1T 2 + 3 − ∑ C6i 5i ⎟ T i =1 2C4i ⎢ ⎥ ⎝ ⎠ ⎢#⎥ ⎢⎣bn ⎥⎦

We need to prove that C 1T 2 +

n C3 C T 2 + C 6iT 3 > ∑ C 6 i 5i . Depending on the value of T 2C 4i i =1

may happen.

29

T two cases

First case: 0 < T < 1 n

In this case since

∑ C6i i =1

prove C1T 2 +

C5i T 2 + C6i T 2 2C4i

n C3 C T 2 + C6i T 2 ≥ ∑ C6i 5i 2C4i T i =1

n

is greater than

∑ C6i i =1

C5i T 2 + C6i T 3 2C4i

, we need to n

⎛ C5i + C6i ⎞ ⎟ . ⎝ 2C4i ⎠

. To do this, we will show that C1 ≥ ∑ C6i ⎜ i =1

Substituting for C1 , C4i , C5i , C6i in the right hand sight of the current inequality results in: n

∑ ⎡⎣((1 − β i =1

2 i

3 2 2 3 ⎤ ) Pi 2 di2 − Pd i i ) + β i Pi d i − β i Pd i i ⎦ >

3 3 4 ⎡⎣(1 − βi 2 ) Pi 2 di2 + (1 − β i 2 ) β i Pd ⎤ i i − (1 + β i ) Pd i i − (1 + β i ) β i d i ⎦ ∑ h B Ci ((1 − β i ) Pi − di ) + Ci (1 − β i ) Pi i =1 n

n

2 ⎤ ⇒ ∑ ⎡⎣((1 − β i 2 ) Pi 2 − Pd i i ) + β i Pi − β i Pd i i⎦ > i =1

2 ⎡⎣(1 − βi 2 ) Pi 2 + (1 − β i 2 ) β i Pd ⎤ i i − (1 + β i ) Pd i i − (1 + β i ) β i d i ⎦ ∑ h B Ci ((1 − β i ) Pi − di ) + Ci (1 − βi ) Pi i =1 n

n

2 ⎤ ⇒ ∑ ⎡⎣(( Pi 2 − β i 2 Pi 2 − Pd i i ) + β i Pi − β i Pd i i⎦ > i =1

3 2 2 2 ⎡⎣ Pi 2 − β i 2 Pi 2 + β i Pd ⎤ i i − β i Pd i i − Pd i i − β i Pd i i − β i di − β i di ⎦ ∑ Cih ((1 − β i ) Pi − di ) + CiB (1 − β i ) Pi i =1 n

2 2 2 ⎡⎣ Pi 2 − β i 2 Pi 2 − β i 3 Pd ⎤ i i − Pd i i − βi di − β i di ⎦ ⎤ ⇒ ∑ ⎡⎣ Pi − β i Pi − Pd i i + β i Pi − β i Pd i i⎦ > ∑ Cih ((1 − β i ) Pi − di ) + CiB (1 − β i ) Pi i =1 i =1 n

n

2

2

2

2

n

n

(

)

⇒ ∑ Pi − β i Pi − Pd i i + β i Pi − β i Pd i i > i =1

2

2

2

2

∑ (P i

i =1

2

2 2 2 − β i 2 Pi 2 − β i 3 Pd i i − β i d i − Pd i i − βi di )

Cih ((1 − β i ) Pi − di ) + CiB (1 − β i ) Pi

It is clear that

∑(P n

i =1

i

2

)

n

2 2 2 2 3 2 2 2 − β i 2 Pi 2 − Pd i i + β i Pi − β i Pd i i > ∑ ( Pi − β i Pi − β i Pd i i − β i d i − Pd i i − βi di ) i =1 n

>

∑ (P i =1

i

2

2 2 2 − β i 2 Pi 2 − βi 3 Pd i i − β i d i − Pd i i − β i di )

Cih ((1 − β i ) Pi − di ) + CiB (1 − β i ) Pi

In the denominator of the last term of the current inequality, (1 − βi ) Pi being the production rate during Ti1 is greater than di i.e., ((1 − βi ) Pi − di ) > 0 and that CiB (1 − βi ) Pi is a relatively big 30

positive. Hence,

∑ ( βi Pi 2 − βi Pdi i ) > ∑ (− βi3 Pdi i − βi di 2 − βi 2 di 2 ) n

n

i =1

i =1

Therefore, we need to show that

∑( n

i =1

)

n

2 2 2 Pi 2 − Pd i i > ∑ ( − β i Pd i i − di − βi di ) i =1

The right hand site of the above inequality is negative and the left hand site is positive (because

((1 − β i ) Pi − di ) > 0 → Pi > di → Pi 2 > Pdi ). Hence, the theorem is proved.

Second Case: T ≥ 1 n

In this case since

∑C

6i

i =1

C1T 2 +

C5iT 3 + C6i T 3 is greater than 2C4i

n

∑C i =1

6i

C5i T 2 + C6i T 3 , if we show that 2C4i

n n C3 C T 3 + C6iT 3 (C + C ) or C1T 2 ≥ ∑ C6i 5i 6i T 3 the theorem holds. In other words we ≥ ∑ C6i 5i 2C4i 2C4i T i =1 i =1

n

( C5i + C6i )

i =1

2C4i

need to show that C1 ≥ ∑ C6i

T . Similar to the first case, substituting C1 , C4i , C5i , C6i and

doing simplification, we have n

i

2

2 2 2 − β i 2 Pi 2 − βi 3 Pd i i − β i d i − Pd i i − βi di )

2

Since

T ≤ 1 , (The elements of the denominator ((1 − βi ) Pi − di ) > 0 , C ((1 − β i ) Pi − d i ) + CiB (1 − β i ) Pi

i =1

i

)

∑ (P

∑(P n

2 − β i 2 Pi 2 − Pd i i + β i Pi − β i Pd i i >

i =1

Cih ((1 − β i ) Pi − di ) + CiB (1 − β i ) Pi

T

h i

CiB (1 − β i ) Pi is a relatively big positive, and hence T cannot be greater than the denominator) therefore,

31

∑(P n

i =1

i

2 2 2 2 3 2 2 2 − β i 2 Pi 2 − Pd i i + β i Pi − β i Pd i i ) > ∑ ( Pi − β i Pi − β i Pd i i − β i d i − Pd i i − β i di ) n

2

i =1 n

>

∑ (P i =1

i

2

2 2 2 − β i 2 Pi 2 − β i 3 Pd i i − β i d i − Pd i i − βi di )

Cih ((1 − β i ) Pi − di ) + CiB (1 − βi ) Pi

Thus, similar to the first case, we need to show that

∑( n

i =1

)

n

2 2 2 2 3 2 2 2 Pi 2 − β i 2 Pi 2 − Pd i i + β i Pi − β i Pd i i > ∑ ( Pi − β i Pi − β i Pd i i − β i d i − Pd i i − β i di ) i =1

Alternatively, based on what we showed in the first case, we need to show that

∑( n

i =1

)

n

2 2 2 Pi 2 − Pd i i > ∑ ( − β i Pd i i − di − βi di ) i =1

Which has been shown to hold.

32

T