multi-machine economic production quantity for...
TRANSCRIPT
1
Multi-machine economic production quantity for items with scrapped and 1
rework with shortages and allocation decisions 2
Amir Hossein Nobil a 3
a Faculty of Industrial and Mechanical Engineering, Qazvin Branch, Islamic Azad University, Qazvin, Iran 4
6
Amir Hosein Afshar Sedighb 7
b Department of Information Science, University of Otago, Dunedin, New Zealand 8
10
Leopoldo Eduardo Cárdenas-Barrónc1
11
c School of Engineering and Sciences 12
Tecnológico de Monterrey 13
Ave. E. Garza Sada 2501 Sur, C.P. 64849, Monterrey, Nuevo León, México 14
16
Abstract 17
This study considers a multi-product multi-machine economic production quantity inventory problem in an 18
imperfect production system that produces two types of defective items: items that require rework and scrapped 19
items. The shortage is allowed and fully backordered. The scrapped items are disposed with a disposal cost and the 20
rework is done at the end of the normal production period. Moreover, a potential set of available machines for 21
utilization is considered such that each has a specific production rate per item. Each machine has its own utilization 22
cost, setup time and production rate per item. The considered constraints are initial capital to utilize machines and 23
production floor space. The proposed inventory model is a mixed integer non-linear programing mathematical 24
model. The problem is solved using a bi-level approach, first, the set of machines to be utilized and the production 25
1 Corresponding author. Tel. +52 81 83284235, Fax +52 81 83284153.
E-mail address: [email protected] (L.E. Cárdenas-Barrón).
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allocation of items on each machine are obtained thru a genetic algorithm. Then, using the convexity attribute of the 1
second level problem the optimum cycle length per machine is determined. The proposed hybrid genetic algorithm 2
outperformed conventional genetic algorithm and a GAMS solver, considering solution quality and solving time. 3
Finally, a sensitivity analysis is also given. 4
Keywords: EPQ; defective item; MINLP; shortage; hybrid algorithm 5
6
1. Introduction 7
The inventory model was first considered at early twentieth century by Ford Whitman Harris [1] 8
who introduced the economic order quantity (EOQ) inventory model. Afterwards, 9
industrialization and market competition were motivations to optimize the inventory system 10
taking into consideration production rate and demand. In this direction, Taft [2] improved the 11
EOQ inventory model considering production rate, the result was the economic production 12
quantity (EPQ) inventory model. Pacheco-Velázquez and Cárdenas-Barrón [3] is an example of an 13
extension of EPQ inventory model by considering backorders and raw material inventory costs. 14
Inventory models that consider manufacturing multiple products on a machine may date back to 15
studies performed by Eilon [4] and Rogers [5]. Other pioneer studies of this problem are 16
Bomberger [6], Madigan [7], Stankard and Gupta [8], Hodgson [9], Baker [10]. Multi-product 17
single-machine problem is considered in more recent studies for example Taleizadeh et al. [11] 18
considered service level constraint when there is stochastic scrapped production rate and partial 19
backordering. At the same time, Taleizadeh et al. [12] developed the former model considering 20
random defective items and repair failure. 21
Afterwards, Taleizadeh et al. [13] studied multi-product single-machine problem with 22
backorders and rework. Simultaneously, Taleizadeh et al. [14] considered demand uncertainty 23
and a special discount situation. This study followed by Taleizadeh et al. [15] that developed an 24
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EPQ inventory model with immediate rework process. Ramezanian and Saidi-Mehrabad [16] 1
solved a mixed integer nonlinear programming (MINLP) model to optimize an unrelated parallel 2
machines scheduling problem with multi-product considering imperfect products. Later on, this 3
problem with non-linear production rate is studied by Neidigh and Harrison [17]. At the same 4
time, Taleizadeh et al. [18] considered repair failure in a system with random defective items. 5
Subsequently, Taleizadeh et al. [19] considered rework in a system with budget and service level 6
constraints. Moreover interruption in the process was considered by Taleizadeh et al. [20] when 7
the system cope with scrap and rework. On the other hand, a multi-delivery policy in a 8
production system with scrapped items is considered by Wu et al. [21]. In more recent studies, 9
Pasandideh et al. [22] studied EPQ inventory model with rework and scrap permission; they 10
considered several classes of reworks considering failure severity. Chiu et al. [23] proposed an 11
algebraic method to solve multi-product problem considering multi-shipment policy with 12
rework. Shafiee-Gol et al. [24] considered pricing and production decisions with rework and 13
discrete delivery. 14
On the other hand, due to the complexity of supply chain systems some researchers benefitted 15
from heuristics or meta-heuristic algorithms. For instance, Vahdani et al. [25] employed 16
simulated annealing (SA) to obtain optimal production size and schedule considering that there 17
exist one deteriorating item and several warehouses. Pasandideh et al. [26] used non-dominated 18
sorting genetic algorithm (NSGA-II) and multi-objective particle swarm optimization (MOPSO) 19
algorithm to optimize the warehouse space and production cost in an imperfect production 20
system with rework and limited orders. Recently, Forouzanfar et al. [27] employed bee colony 21
optimization (BCO) and genetic algorithm (GA) to make decisions about capacity and 22
allocations in a closed-loop supply chain considering transportation time and production costs. 23
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Mahmoodirad and Sanei [28] used three meta-heuristics, i.e. differential evolution (DE), particle 1
swarm optimization (PSO), and gravitational search algorithm (GSA), for designing a multi-level 2
solid supply chain with several products. Also it is important to mention that multi-product 3
multi-machine problems are considered in recent studies as well, for example, Neidigh and 4
Harrison [29] who extended their former study. Moreover, Sarkar and Saren [30], Kang et al. 5
[31], and Tayyab and Sarkar [32] are instances of some recent studies that considered existence 6
of defective items in different environments. Jaggi et al. [33] and Jaggi et al. [34] are some 7
instances of recent extensions of inventory models related to non-instantaneous deteriorating items 8
in a two storage facilities. Finally, Nobil et al. [35] developed multi-product multi-machine 9
problem by considering utilization and allocation decisions. They employed a hybrid genetic 10
algorithm using convexity property of multi-product single-machine problem. 11
This study presents a multi-product multi-machine economic production quantity (EPQ) 12
inventory model for an imperfect production system considering allocation of products on 13
machines. The imperfect production system produces two types of poor-quality items which are 14
products that require rework and scraps. The rework is done after the termination of normal 15
production period and scrap products are disposed. Moreover, in this study the shortage is 16
allowed and fully backordered. Further, a potential set of production machines is at hand that 17
each one can manufacture products at a different rate. The utilization price of these machines and 18
their performance are different with distinct set up time for each product on each machine. The 19
initial capital to buy machines and the production hall are limited. So, in the studied problem 20
there exist three fundamental questions that must be answered in order to minimize the total cost, 21
including utilization, installation, production, rework, shortages, warehouse construction and 22
holding costs. These questions are as follows: 23
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1. What machines to be utilized? 2. Which items allocate to each utilized machine? 3. What are 1
the optimum amounts of production and shortage for each item? 2
A genetic algorithm is applied in order to solve this mixed integer linear programming (MINLP) 3
problem. In this paper, first the MINLP problem is transformed to a bi-level problem, where in 4
the first level there is a mixed integer linear programming problem and the second level there is a 5
continuous nonlinear problem. 6
2. Problem statement 7
This study extends two former studies Nobil et al. [35] and Pasandideh et al. [26]. On the one 8
hand, Nobil et al. [35] presented a multi-product multi-machine economic production quantity 9
problem with scrapped items. They studied a defective production system which some of its 10
products are disposed after normal production time. In their study, several potential machines for 11
allocation of items are considered to be utilized, shortage is not permitted and some constraints, 12
such as budget and initial capital to buy the machines are studied. They assumed that the items 13
that are assigned to each machine have common cycle time. The study aimed to answer the 14
following questions with regard to minimizing the total cost including the cost of utilization, 15
setup, production, maintenance and scraps: what machines to be purchased? What products to be 16
allocated to each machine? How much of each item to be produced? 17
On the other hand, Pasandideh et al. [26] proposed a model for a single-machine multi-product 18
economic production quantity problem with defective items. In the studied problem, a proportion 19
of defective items require rework and the rest are scrapped items. The items requiring rework are 20
classified into different categories based on the failure severity and associated rework rate. In 21
their study, the shortage is permitted and fully backordered and the items are manufactured on a 22
single machine with limited capacity. The objective of study is to find the optimum amount of 23
6
production and shortage for each item considering total costs minimization including, setup, 1
production, rework, scrap, shortage holding and warehouse construction costs. 2
This study considers a multi-product multi machine economic production quantity problem for a 3
defective production system with respect to items allocation to each machine. The defective 4
items may be scrapped or need rework. The objective is to minimize the total cost including: 5
utilization, installation, production, rework, scraps, shortage, holding, and warehouse 6
construction costs. The constraints of this problem are: initial capital, floor space for machines, 7
and production machines capacity. The machine produces item with production rate of and 8
the defective items are percent of the produced items. The ratio of rework required by 9
produced items is and items are scrapped. In other words, . Moreover, the 10
items that require rework immediately are repaired after ordinary production cycle with a higher 11
production rate ( times faster). Other assumptions of the proposed inventory model are as 12
follows: 13
1- The shortage is permitted and fully backordered; 14
2- A proportion of defective items needs rework and the rest is scrapped; 15
3- The rework ratio is a coefficient greater than one of ordinary production ratio; 16
4- Rework costs are different from ordinary production costs; 17
5- Rework process starts immediately when ordinary production stops and no scrapped 18
items are produced during rework period; 19
6- The scrapped items should be disposed, so the system faces disposal cost for each 20
scraped item; 21
7- The disposal of scrapped items occurs when the ordinary process stops; 22
8- There are different types of production machines to manufacture items; 23
7
9- Decision maker faces maximum budget and production floor constraints for purchasing 1
production machines; 2
10- The items allocated to each machine have same production cycle, in other words, 3
. 4
11- All parameters of this problem are known; 5
12- The warehouse space for item includes storeroom plus passageways. The passageways 6
for item are a coefficient less than one of its storeroom. 7
3. Formulated problem 8
The following parameters, decision variables, and notations are employed in this paper 9
for machines and items : 10
number of machines,
number of products,
demand rate of j th
product (units/ unit time)
production rate of j th
product on machine i (units/ unit time)
setup time of ith
machine to produce j th
product (unit time)
proportion of manufactured rework items of j th
product on machine i (%)
proportion of manufactured scrapped items of j th
product on machine i (%)
proportion of manufactured imperfect quality products j on machine i (%)
binary parameter, if ( ) ; otherwise, .
ratio of the rework rate of j th
product to the j th
item production rate on machine i (
)
8
maximum on-hand inventory of j
th product, based on which the regular production
process stops (units)
maximum on-hand inventory of j
th product, based on which the rework process stops
(units)
space required per unit of j th
product for holding (ft2/ product)
ratio of the aisle space to the maximum level of on-hand inventory of j th
product
required space of i th
machine (ft2/ machine)
maximum available space for the production floor space (ft2)
maximum available budget ($)
fixed cost of utilizing for i th
machine ($/ machine)
setup cost of i th
machine to produce j th
product ($/ setup)
unit production cost of j th
product on machine i ($/ unit)
unit rework cost of j th
product on machine i ($/ unit)
disposal cost of scrapped product j per unit ($/ unit)
unit holding cost of j th
product per unit time ($/ unit/ unit time)
unit backorder cost of j th
product per unit time ($/ unit/ unit time)
unit warehouse construction cost of j th
product per unit space ($/ unit)
total cost ($)
number of cycles per unit time for i th
machine; dependent variables
production lot size of j th
product in a cycle (units); dependent variables
total shortage quantity of j th
product in a cycle (units); decision variables
cycle length of i th
machine (unit time); decision variables
9
if machine i is utilized. Otherwise, ; decision variables
if j th
product manufactured by machine i. Otherwise, ; decision variables
Figure 1 shows the inventory on hand and shortages for item j in each cycle which is produced 1
by machine . During and
machines produce item, during rework of item , moreover 2
during and
no production or rework is done. Based on Figure 1, these periods in each cycle 3
of product j are calculated using Eqs. (1) - (5) as follows. 4
INSERT FIGURE 1 5
6
( )
( )
(1)
(2)
(3)
(4)
( )
(5)
Moreover, based on Figure 1, it is obvious that: 7
[( ) ]
(6)
And 8
( )
(7)
Therefore, the cycle length is obtained as follows: 9
10
( )
(8)
Hence, 1
( ) (9)
Total cost for the production system is the sum of total utilization cost, total setup cost, total 2
production cost, total rework cost, total disposal cost, total backorder cost, total warehouse 3
construction cost and total holding cost of all products. These costs are derived as follows: 4
Utilization cost 5
Utilization cost for machine is equal to and is the variable that indicates the utilization of 6
this machine. So the total utilization cost can be calculated as follows: 7
∑
(10)
Setup cost 8
The setup cost for machine to produce item is equal to . Therefore, the total cost of setting 9
up of machines can be obtained from the following equation: 10
∑∑
(11)
Based on a joint production policy (
⁄ ) thus: 11
∑∑
(12)
Production cost 12
The amount of items that can be produced per cycle is , the production cost for each unit of 13
item j on machine equals to . Therefore, the total production cost can be obtained as follows: 14
11
∑∑
∑∑
(13)
Inserting from Eq. (9) result in: 1
∑∑
∑∑
( )
(14)
Rework cost 2
The items requiring rework per cycle are and rework cost for item on machine is . 3
So, the rework cost can be calculated as follows: 4
∑∑
∑∑
( )
(15)
Disposal cost 5
The amount of disposed items in each cycle equals to and disposal cost per unit is . So, 6
the disposal cost can be calculated as follows: 7
∑∑
∑∑
( )
(16)
Backorder cost 8
In this study, the shortage is fully backordered and with respect to Figure 1 the backorder cost for 9
all items can be calculated as follows: 10
∑ (
)
∑
(
)
(17)
Substituting and
by Eq. (4) and Eq. (5), respectively result in: 11
∑∑( ( )
( ) ) (
)
(18)
12
Warehouse construction cost 1
The decisions about warehouse area are made based on each item’s space and aisles requirement. 2
Therefore, the warehouse construction cost for all items can be obtained from the following 3
relation: 4
∑∑ ( )
(19)
Substituting by Eq. (7) results in: 5
∑∑* ( )((( ) )
( )
( )
( )
)
( ) +
(20)
Holding cost 6
Based on Figure 1 total holding costs are as follows: 7
∑∑ (
(
)
(
)
(
))
(21)
Substituting ,
, and by Eqs. (1), (2) and Eq. (3), respectively results in: 8
∑∑ (
(
( )
)
(
)
(
))
(22)
Substituting and by Eq. (6) and Eq. (7), respectively result in: 9
13
∑∑* (( ) )
( )
( )( )
( )
(( ) )
( )
(( ) )
( )
( ( ))
( )
(( ) ) ( )
( )
( )
( ) (
)
( )
( )( )
(( ) )
( )( )+
(23)
Therefore, based on Eqs. (10), (12), (14), (15), (16), (18), (20) and (23), the total cost is 1
calculated by: 2
∑
∑∑* (
)
(
)
( )+
(24)
Where; 3
14
( )(
(( ) ) ( )
( )
)
( )
( )
( ) ( )
( )
(( ) )
( )
(25)
( )( )
( ) (26)
( )
( ) (27)
( )
* ( ) (( ) )+
( ) (28)
The problem’s constraints include allocation, setup, the maximum budget, maximum available 1
floor space, and production capacity. These constraints are expressed as follows: 2
Item allocation constraint 3
This constraint limits allocation of items such that each item type not to be produce on more than 4
one machine: 5
∑
(29)
Where is a binary coefficient that shows the availability of machine for producing jth
item. 6
, ( )
Machine utilization constraint 7
This constraint limits items to be produced on a utilized machine: 8
15
(30)
Budget constraint 1
Eq. (31) limits machines’ utilization cost such that total cost not to be greater than the maximum 2
available budget. 3
∑
(31)
Production floor space constraint 4
Eq. (32) limits decision maker to utilize machines such that the total required space not to be 5
greater than maximum available space. 6
∑
(32)
Capacity of the single machine constraint 7
Eq. (33) shows the sum of the production, rework and setup times for all items manufactured by i 8
th machine must be smaller than or equal to the common cycle length of i
th machine 9
∑(
)
(33)
Substituting and
by Eq. (1) and Eq. (2), respectively result in: 10
{
∑
∑
( )
∑ (
)
( )
}
(34)
Therefore, based on the objective function in Eq. (24) and the constraints in Eqs. (29), (30), (31), 11
(32) and (34) the proposed MINLP is formulated as: 12
16
∑
∑∑* (
)
(
)
( )+
∑
∑
∑
{ }
{ } (35)
Where is a binary variable that shows machine utilization, i.e. if machine is utilized 1
and otherwise. Also, is a binary variable that shows item allocation to machines, i.e. 2
if item is allocated to machine is utilized and otherwise. Moreover, is a 3
continuous decision variable that represents the backorder quantity of item . Finally, is a 4
continuous decision variable that represents the cycle time for machine . 5
6
7
4. Hybrid solution procedure 8
17
In this study a heuristic method is employed to solve proposed mixed integer non-linear 1
programming model. It uses a genetic algorithm and the convexity attribute of single machine 2
problem to find the near optimal solution. In this method, for the first step the problem (35) is 3
converted to a bi-level problem as follows: 4
∑
∑
∑
∑
{ }
{ } (36)
And 5
∑∑* (
)
(
)
( )+
∑ ∑
( )
∑ (
)
( )
(37)
18
The first level of this problem is a mixed integer non-linear model that it can be solved using a 1
genetic algorithm. In this level, the decision variables are and so the number of variables 2
equals to and the number of constraints is . After obtaining and 3
values the second level problem is solved considering as an input parameter. The problem to 4
be solved in second level (37) is a non-linear continuous problem that can be optimized using 5
derivatives. The decision variables in the second level are and , so the number of variables 6
and constraints are and , respectively. 7
Using the bi-level procedure, utilization and allocation are obtained randomly (using GA rules) 8
in (36) then, knowing that (37) is a convex NLP (see Appendix A) the optimum cycle length per 9
machine and shortage value per item are calculated employing derivatives method. Using this 10
method, we can eliminate the necessity of searching for optimal solution through continuous 11
variables. A general scheme for obtaining new solutions’ structure (chromosomes) is represented 12
in Figure 2. Also, the following steps for derivatives method is employed: 13
1- If ∑ ∑
( )
and ∑ (
)
( )
are simultaneously 14
either positive or negative, then go to Step 2. Otherwise, the solution is infeasible, and go to Step 15
7. 16
2- If (∑
∑
( )
) is positive, then go to Step 3. Otherwise, the solution is 17
infeasible, and go to Step 7. 18
3- The following calculates and based on the value: (a detail calculation is 19
represented in Appendix B) 20
19
√
∑
(∑
∑(
)
)
(38)
∑
(39)
4- The lower bound of is obtained as follows: 1
∑ ∑
( )
∑ (
)
( )
(40)
5- If , then go to
. Otherwise,
and go to Step 6. 2
6- Based on the value of , obtain
using Eq. (39) and go to Step 7. 3
7- Terminate the procedure 4
INSERT FIGURE 2 5
Regarding Figure 2, and are obtained randomly using GA, then optimum values for and 6
are calculated based on the employing derivatives method. Then the inventory system’s 7
total cost is determined based on these values. To solve this MINLP problem, we employ a 8
hybrid genetic algorithm (HGA) that combines GA and the derivatives method. The solution 9
procedure of the proposed inventory model using hybrid genetic algorithm is as follows: 10
1- In the genetic algorithm the chromosomes are produced randomly in the first step of the 11
algorithm, this step is called as the initial generation production. 12
2- After producing initial generation, the crossover and mutation operators are used to 13
reproduce new chromosomes. Further, it is saved a percentage of better chromosomes of last 14
generation. 15
20
3- Elite selection transfers best chromosomes to next generation per iteration. 1
4- This procedure repeats until solution convergence is met. 2
A flowchart and a general procedure of the proposed hybrid genetic algorithm are proposed in 3
Figure 3 and Figure 4, respectively. 4
INSERT FIGURE 3 5
INSERT FIGURE 4 6
7
4.1. Initial definitions 8
Chromosome: solutions of model are called chromosomes. 9
Generation Population: number of chromosomes in an iteration (npop). 10
Crossover Probability (Pc): crossover chance per chromosome. Therefore, is the 11
number of crossovers per generation. 12
Mutation Probability (Pm): mutation chance per chromosome. Therefore, is the 13
number of mutations per generation. 14
Maintenance Probability (Pr): the probability of maintenance per generation. Therefore, 15
is the number of maintained chromosomes per generation. 16
4.2. Chromosome scheme 17
Each chromosome composed of a number of genes. These problem solutions, i.e. chromosomes, 18
consist of two types of genes, ijx and iy . A schematic overview of an arbitrary chromosome is 19
represented in Figure 5. 20
INSERT FIGURE 5 21
4.3. Initial population 22
21
chromosomes are generated randomly and stored in to form the initial generation. To 1
do so, random numbers from {0,1} are generated considering (31) and (32) constraints to create 2
genes related to . Afterwards, random numbers from {0,1} are generated considering (29) and 3
(30) to create genes related to . Moreover, some rules to improve feasibility and optimality are 4
employed as follows: 5
1. If , then at least an item should be produced on machine . 6
2. An item should be setup only on one machine. 7
3. Machines investment ∑ should not exceed the total budget. 8
4. Whole floor space that machines occupy, i.e. ∑ should be less than or equal to 9
available floor space. 10
4.4. Crossover operator 11
One point crossover is applied to generate two offspring (new chromosomes) from two parents 12
(randomly chosen chromosomes). To do so, a random number from { } is generated and 13
parents are cut from there for both and . This method is able to generate 14
new chromosomes such that all members are feasible. 15
16
17
18
4.5. Mutation operator 19
A randomly chosen offspring is mutated by reversing a random gene of based on Eq. (41). 20
Then, considering (29) and (30) constraints genes are generated randomly 21
from { } to form , such that all members are feasible. 22
(41) 23
22
4.6. Maintenance operator 1
chromosomes from former generation are maintained. 2
4.7. Fitness function 3
and are computed for a chromosome based on and employing derivatives method (the 4
second level). Moreover, chromosomes’ fitness function can be calculated as follows: 5
∑
∑∑* (
)
(
)
( )+
(42)
It is possible to have infeasible solutions during calculation of and using derivatives 6
method, i.e. the first step. As a result, for infeasible solutions (chromosomes) the following 7
penalty function to the fitness function is added. 8
(43) 9
Where is a sufficiently large number. 10
4.8. Selection operator 11
The P2, P3 and P4 are merged per iteration and better solutions, i.e. have smaller fitness 12
function, form next generation. Finally, if there is no obvious enhancement in solutions for 13
iterations, then the algorithm is terminated. 14
5. Numerical examples 15
In this section the numerical results for proposed mixed integer linear programming problem and 16
sensitivity analysis are discussed. The model is solved using three different approaches, i.e. 17
hybrid genetic algorithm, conventional genetic algorithm and a software, i.e. GAMS solver 18
Couenne, for 10 sample problems of different sizes proposed in Table 1. In these 10 instances, 19
23
the input parameters are randomly selected from Table 2. As it can be seen from Table 1, for 1
small sized instances three methods obtain almost equal solutions. However, increasing problem 2
dimension affects the accuracy and running time and solutions’ quality decrease dramatically. 3
Contrary to GAMS, the conventional GA and proposed hybrid GA can obtain an acceptable 4
solution for large dimensions of this MINLP problem. Comparing results of two GAs, it is 5
obvious that proposed hybrid GA outperforms conventional GA in terms of solution quality and 6
solving time. So, based on Table 1 it can be concluded that proposed hybrid genetic algorithm 7
has an appropriate performance for this MINLP. 8
INSERT TABLE 1 9
INSERT TABLE 2 10
There is a vast literature for sensitivity analysis of multi-product non-linear problems on effect of 11
production rate, demand rate and cost parameters on inventory system total costs (see Nobil et al. 12
[35], Pasandideh et al. [22], Taleizadeh et al. [18, 19].) Therefore, this problem focuses on the 13
parameters that are important for production system and have an impact on total costs. These 14
parameters are as follows: rework ratio, disposal ratio and rework speed. To do so, a 15
problem is considered and these three parameters changed based on the Table 3 to study their 16
impact on the total cost and final solution. Based on Table 3, it is obvious that incrementing 17
rework speed decreases total cost. On contrast, increasing rework and scrap ratio leads to more 18
system costs. 19
INSERT TABLE 3 20
21
6. Conclusion and future research directions 22
This paper proposes a mixed integer non-linear programing (MINLP) mathematical model and 23
its optimization procedure for a multi-product multi-machine economic production quantity 24
24
(EPQ) inventory problem in an imperfect production environment. The considered system 1
produced two types of defective items, items that need rework and scrapped items. The shortage 2
is allowed in a way that demands for unavailable items are totally backordered. The scrapped 3
items are disposed with a disposal cost and rework is done afterward finishing the normal 4
production period. Moreover, it is considered the system when there is a potential set of available 5
machines with a specific production rate with its utilization cost and setup time per item. This 6
system is studied under some constraints such as: initial capital for machines’ utilization and 7
production floor space. A bi-level method is used to solve this problem. First, a set of machines 8
to be utilized and a production allocation of items on each machine are obtained by genetic 9
algorithm, then using the convexity attribute of the second level problem the final solution is 10
calculated. Furthermore, a comparison among the proposed hybrid genetic algorithm, a 11
conventional genetic algorithm and a GAMS solver is presented. The outcomes suggest that the 12
proposed method outperforms other methods considering both solution quality and solving time. 13
The proposed inventory model is applicable in real-world instances where corporate’s managers 14
deal with procurement of facilities as well as allocating them to corporate operations. In 15
aforementioned environment, a manager should make a decision based on technological 16
advantages of available facilities, i.e. production rate, failure rate, rework rate and so forth. 17
Moreover, another concern that a decision maker should take into the account is demand rate of 18
each item, shortage costs, budget constraint, desired quality, and facilities production and 19
procurement costs. In this study, a mathematical model is proposed to make a near optimal 20
decision about facilities procurement and product allocation considering system costs. To do so, 21
a multi-item production system with defective products and shortage with a set a potential 22
machine is optimized by benefiting from single machine specifications and GA altogether. 23
25
It is significant to suggest considering other objectives such as maximizing the profit or 1
minimizing the warehouse space as some extensions of the proposed problem. Moreover, 2
considering some environmental considerations in form of manager’s preference or internalizing 3
the externalities in choosing production facilities is another possible future extension of this 4
model. The solution procedure, may be extended by composing combinatorial optimizations and 5
convexity attribute of the problem, as well. However, it requires branching on both y and x 6
variables, i.e. binary variables , then a convex problem same as mentioned problem will emerge. 7
It is worth mentioning that the proposed method can provide an estimation of each branch 8
solution and its quality. On the other hand, studying this problem under different conditions, for 9
instance perishable items, several classes of rework, aggregate rework, i.e. all reworks of items 10
produced by machine be done in a separate cycle, interruption in machines manufacturing 11
process, maintenance policy consideration, stochastic production failure, discount for imperfect 12
products and fuzzy or stochastic demands or capacity or a combination of these are some 13
interesting future research directions. 14
15
Appendix A. Proof of the convexity of the objective function in Eq. (37) 16
Based on the objective function in Eq. (37): 17
∑∑* (
)
(
)
( )+
So, 18
∑
∑(
)
∑
(
)
26
∑
∑
∑
[
]
[ ∑
( )
∑ (
)
∑ (
)
∑
∑
∑
]
[ ]
[
]
∑∑(
)
27
Since and are greater than or equal to zero so the is greater than zero. As a result, 1
the hessian matrix of objective function Eq. (37) is greater than or equal to zero and is a convex 2
function. 3
4
Appendix B. Finding the optimal value of the decision variables 5
Calculating the derivative of objective function Eq. (37) with respect to , thus: 6
∑
∑
√∑(
)
(B-1)
Moreover, calculating derivative with respect to , we have: 7
∑
∑
∑(
)
(B-2)
Substituting (B-1) by (B-2) leads to: 8
√
∑
(∑
∑
( )
)
And
∑(
)
(B-3)
(B-4)
9
References 10
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29
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30 List of Figures: 31 Figure 1. The inventory position of j th item in a cycle that is produced by machine i 32
Figure 2. The structure of each solution (chromosome) 33
Figure 3. The flowchart of the proposed hybrid genetic algorithm (HGA) 34
Figure 4. The proposed hybrid genetic algorithm’s general procedure 35
Figure 5. The example of chromosome 36
37
List of Tables: 38
Table 1. Comparison of algorithms 39
Table 2. Input parameters of the proposed problem 40
Table 3. Sensitivity analysis of three parameters: rework ratio, disposal ratio and rework speed 41
42
30
tj1 tj
2 tj3
tj4 tj
5
Tij=Ti
Bj
Ij
Hj
time
inventory
1
Figure 1. The inventory position of j th item in a cycle that is produced by machine i 2
3
yi and xij Randomly
generation- GA rules
Ti and Bj
TC
Derivative method
Total cost
4
Figure 2. The structure of each solution (chromosome) 5
6
31
Start
Crossover:
One-point
Mutation:
Randomly
Maintenance:
Elite previous
generations
Elitism selection
Next generation
Initial populationRandomly
generation
Stop condition Show final solutionNo
1
Figure 3. The flowchart of the proposed hybrid genetic algorithm (HGA) 2
Preliminary
Initialize POP(K) ; POP(K): population of chromosome of the Kth generation.
Evaluate all chromosomes in POP(K)
Chromosomes of the POP(1) are generated randomly (initial population).
While (termination condition is not true) do:
//Select probabilistically of yi and xij
Crossover: crossover the pairs of chromosomes in POP(K) to yield P1(K) by employing a crossover probability, then go to
second level
Mutation: mutation of the genes of POP(K) to yield P2(K) by employing a mutation probability, then go to second level
Maintenance: the elite chromosomes of POP(K) to yield P3(K) by employing a maintenance probability then, go to second level
Each chromosome (yi, xij) is fed to the second level to compute φ (xij, Ti, Bj)
Start (second level)
For each chromosome
Running the derivatives method
Compute φ (xij, Ti, Bj)
End
Objective function is calculated for each chromosome TC (yi, xij, Ti, Bj)
Evaluate pop1(K) ; pop2(K) ; pop3(K) and pop4(K)
End while
Output the resulting of the best chromosome
Termination
3
Figure 4. The proposed hybrid genetic algorithm’s general procedure 4
5
6
Figure 5. The example of chromosome 7
32
Table 1. Comparison of algorithms 1
Size
n m
Proposed HGA Conventional GA GAMS solver Couenne
Total cost ($)
CPU time
(Seconds)
Total cost ($)
CPU time
(Second)
Total cost ($)
CPU time
(Seconds)
1 2 2 127565.7781971 45.6279 127565.7781971 87.6438 127565.7781971 0.0202
2 2 3 136978.5888335 49. 8007 136982.7001745 92. 2613 136980.9735568 0.0989
3 2 5 165920.4156557 56.4961 165997. 3489096 102.2991 - -
4 3 6 284632.6823419 59.5632 288915.0912641 118.2832 - -
5 3 10 289473.8272197 61.7463 292038.0537104 147.2121 - -
6 4 10 494736.1758372 68.5492 529472.6951114 167.0090 - -
7 5 12 542800.5102909 85.5382 581082.0475678 249.3129 - -
8 6 15 905370.2278785 96.1931 951113.2076686 320.4512 - -
9 6 20 1022337.5288482 107.7371 1126859.6852671 352.6391 - -
10 7 25 1291611.2514883 156.0021 1380281.4415222 423.8735 - -
2
3
Table 2. Input parameters of the proposed problem 4
* U: Uniform distribution 5
6
Table 3. Sensitivity analysis of three parameters: rework ratio, disposal ratio and rework speed 7
Parameter % changes The proposed HGA
Total cost ($) % changes in
Initial problem 0 165920.41565577 0
33
ij
+100 166180.509967279 1.00156758473923
-50 165788.089543842 0.999202472393738
-100 165654.260980893 0.998395889536408
ij
+100 166016.570453697 1.00057952360803
-50 165872.699892375 0.999712417768444
-100 165825.222680232 0.999426273281912
ij
+100 165858.860829722 0.999629009933439
+50 165879.354287614 0.999752523714495
-50 166044.198464805 1.00074603724048
1 2
Amir Hossein Nobil is currently a PhD candidate in Industrial Engineering at Qazvin Islamic Azad 3
University, Qazvin, Iran. He received his B.Sc. degree in Industrial Engineering and then his M.Sc. 4
degree in Industrial Engineering both from Qazvin Islamic Azad University, Qazvin, Iran. His research 5
interests include Supply Chain Management, Inventory Control and Production Planning. 6
7
Amir Hosein Afshar Sedigh is currently a PhD student in Information Science at University of Otago, 8
Dunedin, New Zealand. He received a B.Sc. degree from Qazvin Islamic Azad University, Qazvin, Iran, 9
in Industrial Engineering. Thereafter, he received his M.Sc. degree in Industrial Engineering Sharif 10
University of Technology, Tehran, Iran. His research interests include Supply Chain Management, 11
Inventory Control and Queuing Theory. 12
13
Leopoldo Eduardo Cárdenas-Barrón is currently a Professor at School of Engineering and Sciences at 14
Tecnológico de Monterrey, Campus Monterrey, México. He is also a faculty member in the Department 15
of Industrial and Systems Engineering at Tecnológico de Monterrey. He was the associate director of the 16
Industrial and Systems Engineering programme from 1999 to 2005. Moreover, he was also the associate 17
director of the Department of Industrial and Systems Engineering from 2005 to 2009. His research areas 18
34
include primarily related to inventory planning and control, logistics, and supply chain. He has published 1
papers and technical notes in International Journal of Production Economics, Computers and Industrial 2
Engineering, Journal of the Operational Research Society, Production Planning and Control, 3
Transportation Research Part E: Logistics and Transportation Review, European Journal of Operational 4
Research, Mathematical Problems in Engineering, Computers and Mathematics with Applications, 5
Applied Mathematical Modelling, Applied Mathematics and Computation, Mathematical and Computer 6
Modelling, Expert Systems with Applications, International Journal of Systems Science, European 7
Journal of Industrial Engineering, Journal of Manufacturing Systems, Journal of Industrial and 8
Management Optimization, Annals of Operations Research, TOP, Scientia Iranica, Applied Soft 9
Computing, among others. He has co-authored one book in the field of Simulation in Spanish. He is also 10
editorial board member in several international journals. 11