Xnew(j) = Xa(j) where a â (1,2,â¦.,HMS) .... C12=30, Smin=600000000, W1=0.6, Imax=400000, SCmax=10000000, M=90000 and b1=0.3 that generally.
Applied mathematics in engineering, management and technology 4(2) 2016:63-75 www.amiemt-journal.com
Using Meta-heuristic Algorithms to Solve an Integrated Production Planning and Preventive Maintenance Model Neda Manavizadeh *,1, Taha Vafaeenezhad 2, Hamed Farrokhi-Asl 3 1
Department of Industrial Engineering, KHATAM University, Tehran, Iran School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran 3 School of Industrial Engineering, Iran University of Science & Technology, Tehran, Iran
2
Abstract Considering an integrated model which combines the production planning and maintenance approaches is one of the most recent and applicable issue in competitive business market as a research field. This consistency in manufacturing systems can enrich the profitability of system and decrease the price of final goods. As a result, by evaluating the capacity of production line and its accessories, a true perspective for production planning can be achieved. The study aims to find an optimal maintenance policy and to integrate maintenance policy with production planning in order to decrease backorder sales and reduce the total production and maintenance costs. A presented mathematical formulation is solved with two metaheuristic algorithms and the gained results are compared with each other. Finally, conclusions remarks are described in last section. Keywords: production planning; preventive maintenance; metaheuristic algorithms; harmony search; simulated annealing
1. Introduction To cope with the current tough competition many manufacturing companies have invested in highly automated production systems with sophisticated equipment. To be economically sustainable, this costly equipment should be exploited to the last instant of their maximum possible productive time [2]. Keeping the productivity of the company in a suitable level indicates an important success that significantly depends on the production system and reliability of it. This issue depends on a lot of factors itself, including unplanned disruption rate, capability of company and planning to reduce closes and considering this point in the production plan [9]. When an unplanned downtime, caused by a production line failure, occurs it often trims down the system‟s productivity and renders the current production plan obsolete. This failure often causes increased variability in product quality and in service level [1]. Production planning models seek typically to balance the costs of setting up the system with the costs of production and materials holding, while maintenance models attempt typically to balance the costs and benefits of sound maintenance plans in order to optimize the performance of the production system. However, the issue of combining the production and maintenance plans has received much less attention [1]. The traditional production planning models based upon the famous linear programming formulation has been well documented. However, the integration of preventive maintenance planning in the same model is a recent problem [13]. The coordination of maintenance and production is important to guarantee good system performance [3]. Also, it should be noted that occurring failure of equipment is inevitable. In addition to cause delay in completing customers‟ orders, it may increase operational costs and disrupt the production plan [12]. Maintenance should be used accurately as an effective strategy to reduce the costs and consequently to increase the profit. Considering system‟s constraints, e.g. availability could provide the possibility of better planning to match with the goals of the company [1]. Keeping equipment in good condition through maintenance activities can ensure a more reliable system [3]. Integration influences on many things, e.g. cost of finished goods that carry out by minimizing the cost of maintenance and the delay of delivering orders. By implementing this integration, scheduling under consideration of potential of production lines and availability of the equipment, it could be represented almost a real image for realization of the goals of the production system [7]. The objective is to find an integrated lot-sizing and preventive maintenance strategy of the system that satisfies the demand for all items over the entire horizon without backlogging, and which minimizes the expected sum of
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Applied mathematics in engineering, management and technology 4(2) 2016:63-75 www.amiemt-journal.com production and maintenance costs [2]. Several types of maintenance models have been created, e.g. corrective maintenance, preventive maintenance, condition-based maintenance, etc. Based on the literature review for this study, in many cases and especially in recent years, the preventive maintenance model has been used more. Nowadays, in the industry, preventive maintenance is the most useful maintenance policies that is included in all activities when the machine was active or ideal for retrieve, maintenance, failure prevention and create stability [6]. In this policy, a part is investigated by a preventive method at constant periods independent of its failure history and would be replaced in some cases. And if there are any midterm failures, it would be acted by corrective maintenance [4].
2. Problem description In this study, a multi-period, multi-item lot-sizing production system composed of parallel failure-prone production lines is considered. The deterioration of lines that is represented as a reduction of production lines capacities in function of time evolution is considered in a limited planning horizon that is a comprehensive vision. It is assumed that when a production line fails, a minimal repair is carried out to restore it to an „as-badas-old‟ status. Preventive maintenance is carried out, periodically to restore the production line to an „as-goodas-new‟ status [2,13]. It is also assumed that any maintenance action, performed on a production line in a given period, reduces the available production capacity on the line during that period. Hence, by failing the production line and lost time, some of production capacity be lost. All preventive maintenance is always performed at the beginning of each cycle for the complete duration of the first period. Also, we assume that if a maintenance operation is launched in a given period, it is completed by the end of the period. Selecting maintenance policy will have a considerable influence on the production cost. Because the number of maintenance activities affects the cost of maintenance, it‟ll depend on the maintenance policy [2]. This is the first research in the area of integration of production planning with maintenance planning that supposes several modes of regular and overtime work independently for each production line and also the costs and the limitation of outsourcing. In addition, supply of required labor, employ and discharge considered for constant cycles in this paper. Two methods, simulated annealing algorithm and harmony search algorithm is used for solving the mathematical model. It is supposed to produce a set of products in the production line 𝑗 ∈ 𝐿 at a planning horizon 𝑇 = 𝑁𝜏 that includes N period with constant duration 𝜏. A demand corresponds to "d" should be supplied for each product 𝑖 ∈ 𝐼 at every period 𝑡 ∈ 𝑇. Production time/cycle capacity constraint of regular time work and overtime work and the constraint of outsourcing for each production line have been defined. The maximum available labor at period t is defined by 𝑊𝑚𝑎𝑥 𝑡 and the working hours of labor at period t would be 𝑔𝑡 and the capacity of 𝑗
available machine in production line j at period t is 𝑀𝑡 . By means of these parameters, the capacity constraints would be determined and also 𝑎𝑡 would be the percentage of labor capacity that would be available for overtime 𝑗 work at period t and 𝑏1𝑡 is defined as the percentage of available production capacity of machine for overtime work in production line j at period t. For overtime work, capacity constraints of overtime work are defined and finally, the 𝑆𝐶𝑚𝑎𝑥 𝑖𝑡 parameters is determined for outsourcing that represents the maximum quantity of product i 𝑗
that could be acquired by the subsidiary contract at period t. 𝐻𝑡 is defined for determining the quantity of 𝑗 employed labor in production line j at period t and 𝐿𝑡 for the quantity of fired up labor in production line j at period t that should satisfy the constraints according to the parameters 𝑢𝑖 and 𝑢1𝑖 that represents respectively the person-hour required for producing a unit product i of regular time and person-hour required for produce a product i of overtime work. 𝑒𝑖 is defined for determining the duration of time that the machine needs for producing a unit product i of regular and overtime work. Some of these capacities would be lost because of maintenance activities that could be modeled by two parameters. 𝑘1𝑡 is defined as the percentage of machine capacity at each period that losses by preventive maintenance and 𝑘2𝑡 is defined as the percentage of machine capacity at each period that losses by emergency maintenance. 𝑊1𝑖𝑡 is defined as the percentage of the minimum forecasted demand of product i that determines the maximum shortage at period t. By the limitation of warehouse capacity, 𝐼𝑚𝑎𝑥 𝑖𝑡 would be needed to represent the capacity of warehouse for holding product i at period t. Finally, it is assumed that the failure probability density function 𝑓𝑗 𝑡 and the cumulative distribution
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Applied mathematics in engineering, management and technology 4(2) 2016:63-75 www.amiemt-journal.com function 𝐹𝑗 𝑡 of each production line are known. The failure rate of each production line 𝑟𝑗 𝑡 is given by the equation (1): 𝑟𝑗 𝑡 =
𝑓 𝑗 (𝑡)
1−𝐹𝑗 (𝑡)
∀𝑗 ∈ 𝐿 , ∀𝑡 ∈ 𝐻𝑜𝑟𝑖𝑧𝑜𝑛 (T)
)1(
According to the goal of minimizing the repairing activities, an inhomogeneous Poisson process is used for 𝑡 modeling at duration (0, t). As a result, the expected failure could be calculated by 0 𝑟(𝑡)𝑑𝑡 . The considered maintenance policy suggests maintaining at predetermined instances 𝑇 = 𝑘𝜏, 2𝑘𝜏, 3𝑘𝜏, … in each production line for preventive maintenance would be done and to carry out a minimal repair whether an unplanned failure occurs. All maintenance actions are supposed to be perfectly performed. Indexes: i Index of products t Index of periods j Index of production lines Sets: I T L
Set of products Set of Periods Set of production lines
Parameters: Forecasted demand for family product i at period t (According to previous data) 𝑑𝑖𝑡 𝑗 Production cost of one unit of product i of regular time work at period t in production line 𝐶1𝑖𝑡 j 𝑗 Production cost of one unit of product i of overtime work at period t in production line j 𝐶2𝑖𝑡 𝑗 Cost of one unit of Man-hour labor of regular time work at period t in production line j 𝐶3𝑡 𝑗 Cost of one unit of Man-hour labor of overtime work at period t in production line j 𝐶4𝑡 Cost of one unit product i acquired by subsidiary contract at period t 𝐶5𝑖𝑡 𝑗 Preparation cost of product i at period t in production line j 𝐶6𝑖𝑡 𝑗 Preventive maintenance cost at period t in production line j 𝐶7𝑡 Holding cost of one unit product i in warehouse at period t 𝐶8𝑖𝑡 Cost of one unit shortage (back-log order) of product i at period t 𝐶9𝑖𝑡 𝑗 Discharge cost of one labor at period t that was working in production j 𝐶10𝑡 𝑗 Employ cost of one labor at period t that was working in production j 𝐶11𝑡 𝑗 Corrective maintenance cost at period t in production line j 𝐶12𝑡 Depot capacity for holding product at period t in production line j 𝐼𝑚𝑎𝑥 𝑖𝑡 Minimum forecasted demand of product i that determines maximum shortage of period t 𝑆𝑚𝑖𝑛 𝑖𝑡 Maximum quantity of product i that could be acquired by subsidiary contract at period t 𝑆𝐶𝑚𝑎𝑥 𝑖𝑡 Maximum available labor at period t 𝑊𝑚𝑎𝑥 𝑡 Working hours of each labor at period t 𝑔𝑡 Percentage of labor capacity that would be available of overtime work at period t 𝑎𝑡 Man-hour units required for producing one unit product i of regular time work 𝑢𝑖 Man-hour units required for producing one unit product i of overtime work 𝑢1𝑖 Duration required for producing one unit product i of regular and overtime work by 𝑒𝑖 machine 𝑗 Available capacity of machine of regular time work at period t in production line j 𝑀𝑡 Percentage of minimum forecasted demand of product i that determines maximum 𝑊1𝑖𝑡 shortage of period t Percentage of machine capacity that would be lost at each period because of preventive 𝑘1𝑡 maintenance Percentage of machine capacity that would be lost at each period because of emergency 𝑘2𝑡 maintenance
65
Applied mathematics in engineering, management and technology 4(2) 2016:63-75 www.amiemt-journal.com 𝑗
𝑏1𝑡 𝑗 𝐵𝑁𝑖𝑡
Percentage of machine available capacity of overtime work at period t in production line j 0-1 parameter showing that producing of product i at period t in production line j
Variables: 𝑗 𝑥𝑖𝑡 𝑗 𝑦𝑖𝑡 𝑗 𝑊𝑡 𝑗 𝐻𝑡 𝑗 𝐿𝑡 𝑗 𝑂𝑇𝑡 𝐼𝑖𝑡 𝐵𝑖𝑡 𝑆𝐶𝑖𝑡 𝑗 𝑃𝑀𝑡 𝑗
𝐶𝑀𝑡
Quantity of family product i of regular time work at period t in production line j Quantity of family product i of overtime work at period t in production line j Quantity of labor required at period t in production line j Quantity of labor employed at period t in production line j Quantity of labor discharged at period t in production line j Overtime work duration required at period t in production line j Inventory level of family product i at the end of period t Shortage (back-log order) level of family product i at period t Quantity of family product i that acquired by subsidiary contract at period t Maintenance is equal to 1 if preventive maintenance has been done at the beginning of the period t in production line j; Otherwise 0. Maintenance is equal to 1 if emergency maintenance has been done at the beginning of the period t in production line j; Otherwise 0.
Objective function: The goal of the problem is minimizing the total cost at the planning horizon. The first part of the objective function is the total production cost of regular and overtime work that is acquired from multiplying the quantity of production in the production cost of one unit, in addition to preparation cost of production machines, if the production line is activated. The second part is the labor cost of regular and overtime work. 𝐼
𝑇
𝐿
𝐿
𝑗 𝑗 ( 𝐶1𝑖𝑡 𝑥𝑖𝑡 𝑖=1 𝑡=1 𝑗 =1
+ 𝑇
𝑗 𝑗 𝐶2𝑖𝑡 𝑦𝑖𝑡
+
𝑇
𝑗 𝑗 ( 𝐶3𝑡 𝑊𝑡
+ 𝑡=1 𝑡=1
𝑇 𝑗 𝑗 (𝐶7𝑡 + 𝐶12𝑡 𝑛𝑗
𝐼
+
( 𝐶5𝑖𝑡 𝑆𝐶𝑖𝑡 ) 𝑡=1 𝑖=1
𝑗
𝐶8𝑖𝑡 𝐼𝑖𝑡 + 𝐶9𝑖𝑡 𝐵𝑖𝑡 + 𝑡=1 𝑖=1 𝑇 𝑇
𝑗 𝑗 + 𝐶4𝑡 𝑂𝑇𝑡 )
𝑗 =1 𝑡=1 𝐿 𝑇
𝐼
+
𝑇
𝑗 𝑗 𝐶6𝑖𝑡 𝐵𝑁𝑖𝑡 ) +
𝑗
𝑗 𝑗
(𝐶10𝑡 𝐻𝑡 + 𝐶11𝑡 𝐿𝑡 )
)2(
𝑗 =1 𝑡=1 𝑛𝑗 𝜏 0
𝑟𝑗 𝑡 𝑑𝑡)
The third part is the cost of acquiring products by subsidiary contract. The fourth part is the holding and shortage cost, and the fifth part is the change cost of labor by discharging or employing at every period. The last part of the objective function is the maintenance cost that is added to the objective function by the comments below: 𝜙 𝑗 𝑛𝑗 that is the maintenance cost in producion line j with constant periods, is a nonlinear function that consists of two parts; preventive maintenance and corrective maintenance. Supposed T=n𝜏 that 𝜏 is the duration of each period: 𝑇
𝑗
𝜙 𝑛𝑗 = 𝑡=1
𝑇 𝑗 𝑗 (𝐶7𝑡 + 𝐶12𝑡 𝑛𝑗
𝑛𝑗 𝜏 0
𝑟𝑗 𝑡 𝑑𝑡)
(3)
It could be considered 𝐹𝑗 0 = 0 that could be obtained from equation (4): 𝑎 𝑎 𝑓𝑗 (𝑡) 1 − 𝐹𝑗 (0) 𝑟𝑗 𝑡 𝑑𝑡 = 𝑑𝑡 = log( ) 1 − 𝐹𝑗 (𝑎) 0 0 1 − 𝐹𝑗 (𝑡) = log 1 − 𝐹𝑗 0
− 1 − 𝐹𝑗 𝑎
= −log 1 − 𝐹𝑗 𝑎
By replacing the above equation in 𝜙 𝑗 𝑛𝑗 the new equation is given by:
66
(4)
Applied mathematics in engineering, management and technology 4(2) 2016:63-75 www.amiemt-journal.com 𝑇
𝐿
𝑡=1 𝑗 =1
𝑇 𝑗 𝑗 (𝐶7𝑡 − 𝐶12𝑡 ∗ log (1 − 𝐹𝑗 𝑛𝑗 𝜏 ) 𝑛𝑗
Constraints:
(5)
𝐿 𝑗
𝑑𝑖𝑡 = 𝐼𝑖𝑡 −1 − 𝐵𝑖𝑡 −1 − 𝐼𝑖𝑡 + 𝐵𝑖𝑡 + 𝑆𝐶𝑖𝑡 + (
𝑗
𝑥𝑖𝑡 + 𝑦𝑖𝑡 )
(6)
𝑗 =1
𝐼
𝐼𝑖𝑡 < 𝐼𝑚𝑎𝑥 𝑖𝑡
(7)
𝑖=1
𝐵𝑖𝑡 ≤ 𝑊1𝑖𝑡 𝑆𝑚𝑖𝑛 𝑖𝑡 𝑆𝐶𝑖𝑡 ≤ 𝑆𝐶𝑚𝑎𝑥 𝑖𝑡
(8) (9)
𝐿
𝑗
𝑊𝑡 ≤ 𝑊𝑚𝑎𝑥 𝑡 𝑗 =1 𝑗 𝑊𝑡 = 𝑗 𝑗 𝐻𝑡 . 𝐿𝑡
𝑗
(10) 𝑗
𝑗
𝑊𝑡−1 + 𝐻𝑡 + 𝐿𝑡 =0 𝐼𝑖𝑡 𝐵𝑖𝑡 = 0 𝑗 𝑗 𝑂𝑇𝑡 ≤ 𝑔𝑡 𝑎𝑡 𝑊𝑡
(11) (12) (13) (14)
𝐼
𝑗
𝑢𝑖 𝑥𝑖𝑡 ≤ 𝑔𝑡 𝑊𝑡
𝑗
(15)
𝑖=1 𝐼 𝑗
𝑗
𝑢1𝑖 𝑦𝑖𝑡 ≤ 𝑂𝑇𝑡
(16)
𝑖=1 𝐼 𝑗
𝑗
𝑗
𝑒𝑖 𝑥𝑖𝑡 + 1 − 𝑃𝑀𝑡 𝐾1𝑡 𝑀𝑡 + (1 − 𝐶𝑀𝑡 )𝐾2𝑡 𝑖=1 𝐼 𝑗
𝑗
𝑗
𝑗
𝑗
𝜶𝒕 +𝝉 𝜶𝒕
𝑗
𝑟𝑗 (𝑡)𝑑𝑡 ≤ 𝑀𝑡 𝑗
𝑗
𝑒𝑖 𝑦𝑖𝑡 + 1 − 𝑃𝑀𝑡 𝐾1𝑡 𝑏1𝑡 𝑀𝑡 + (1 − 𝐶𝑀𝑡 )𝐾2𝑡 𝑏1𝑡 𝑀𝑡 𝑖=1
𝜶𝒕 +𝝉 𝜶𝒕
(17) 𝑗
𝑗
𝑟𝑗 (𝑡)𝑑𝑡 ≤ 𝑏1𝑡 𝑀𝑡
0, 𝑖𝑓 𝑎 𝑃𝑀 𝑎𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 𝑝𝑒𝑟𝑓𝑜𝑟𝑚𝑒𝑑 𝑎𝑡 𝑡𝑒 𝑏𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑜𝑓 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡 𝑓𝑜𝑟 𝑙𝑖𝑛𝑒 𝑗 1, 𝑂. 𝑊 0, 𝑖𝑓 𝑎 𝐶𝑀 𝑎𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 𝑝𝑒𝑟𝑓𝑜𝑟𝑚𝑒𝑑 𝑎𝑡 𝑡𝑒 𝑏𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔 𝑜𝑓 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡 𝑓𝑜𝑟 𝑙𝑖𝑛𝑒 𝑗 𝑗 𝐶𝑀𝑡 = 1, 𝑂. 𝑊 𝑗 𝑗 𝑗 𝑗 𝑗 𝑗 𝑂𝑇𝑡 ≥ 0, 𝑥𝑖𝑡 , 𝑦𝑖𝑡 , 𝑊𝑡 , 𝐻𝑡 , 𝐿𝑡 , 𝐼𝑖𝑡 , 𝐵𝑖𝑡 , 𝑆𝐶𝑖𝑡 ≥ 0 𝑎𝑛𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑗
𝑃𝑀𝑡 =
(18) (19) (20) (21)
Constraint (6), is the balance of production at each period that means the demand of each product is the summation of the inventory of each product at previous period and the production of each product of regular and overtime work and by the subsidiary contract minus its inventory level at the end of the current period. Constraint (7) represents that the summation of inventory of the products should be less than the maximum allowable capacity at each period. Constraint (8) shows that the shortage of each product should be less than a determined percentage of its minimum demand at each period. Constraint (9) determines that the volume of subsidiary contract for each product should be less than its maximum allowable subsidiary contract at each period. Constraint (10) represents that the quantity of labor should be less than the maximum available labor at each period. Constraint (11) declares the labor balance, hence, the labor quantity of each period equals to this quantity of the previous period and the quantity of employ and discharge of the current period. According to constraint (12) it would be either employing or discharging at each period. According to constraint (13) there would be either stock or shortage at each period. Constraint (14) determines the maximum duration capacity of overtime work at each period. Constraint (15) represents that the required duration for the production of regular work should be less than the available duration of labor of regular work. This duration is calculated by the quantity of labor of regular work in to the available duration of each labor at each period. Constraint (16) represents that the required duration for production of overtime work should be less than the available duration
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Applied mathematics in engineering, management and technology 4(2) 2016:63-75 www.amiemt-journal.com of labor of overtime work. Constraint (17) represents that the summation of the required duration for regular production plus the duration of maintenance plus the decreased duration of the system because of the failure should be less than the machine capacity at each period. Constraint (18) determines that the summation of the required duration for overtime production - supposing that if maintenance not performed, a certain percentage of available duration of the machine would be used for failure - should be less than machine capacity of overtime work. The decreased capacity is a certain percentage of regular capacity of the machine. For convenience, these two constraints could be represented as equations (22) and (23): 𝐼
𝑗
𝑗
𝑗
𝑗
𝑗
𝑒𝑖 𝑥𝑖𝑡 + 1 − 𝑃𝑀𝑡 𝐾1𝑡 𝑀𝑡 + (1 − 𝐶𝑀𝑡 )𝐾2𝑡 ∗ 𝑙𝑜𝑔 1 − 𝐹𝑗 𝛼𝑡
− 𝑙𝑜𝑔 1 − 𝐹𝑗 𝛼𝑡 + 𝜏
𝑗
≤ 𝑀𝑡
(22)
𝑖=1 𝐼 𝑗
𝑗
𝑗
𝑗
𝑗
𝑒𝑖 𝑦𝑖𝑡 + 1 − 𝑃𝑀𝑡 𝐾1𝑡 𝑏1𝑡 𝑀𝑡 + (1 − 𝐶𝑀𝑡 )𝐾2𝑡 𝑏1𝑡 𝑀𝑡 ∗ log 1 − 𝐹𝑗 𝛼𝑡 𝑖=1
− log 1 − 𝐹𝑗 𝛼𝑡 + 𝜏
𝑗
(23)
𝑗
≤ 𝑏1𝑡 𝑀𝑡
Equation (19) shows the variable of performing or not performing the maintenance at period t that if performed would be 1 and otherwise, 0. Equation (20) shows the variable of performing or not performing the emergency maintenance at period t that if performed would be 1 and otherwise, 0. The decision variables are shown by the equation (21): quantity of production of regular time work, quantity of production of overtime work, quantity of employing, quantity of discharging, duration of overtime, period inventory, period shortage, period subsidiary contract and 0-1 variable of performing two modes of maintenance at each period. All of the variables are integer except overtime duration.
3. Methodology This model has been solved by two algorithms; harmony search algorithm and simulated annealing algorithm that will presented in the next sub-sections.
3.1.
Harmony search
Harmony search meta-heuristic algorithm was conceptualized using the improvisation process of the orchestra. Each player sounds pitches of his (or her) musical instrument in a perfect state of harmony. The goal of this process is to achieve a state that the whole of orchestra sound a pleasing harmony. In the harmony search algorithm, every solution, called a harmony and showed by a vector. This algorithm has three main phases, initial value, improvise harmony vector and update algorithm memory. The first generation of solutions is created randomly in the first phase and stored in harmony memory. Using the rules of memory considerations, pitch adjustments and making random generation, a new harmony vector or (a new solution) will be created in the second phase [8]. The harmony search algorithm is a five step procedure. 1st step is initializing the problem and harmony search parameters. These parameters are the harmony memory size, harmony memory considering rate, pitch adjusting rate and the number of improvisations. In the 2nd step, the vectors of random solution are registered in the algorithm memory. The new solution vector is created in the 3rd step. First, a random value is generated between zero and one. If this value is less than the harmony memory considering rate, a value among existing values of harmony will be chosen randomly for the variable and otherwise, a value among the set of feasible values will be chosen randomly. Figure (1) shows a pseudo code for generating new solution in harmony search algorithm. In the 4th step, if the generated solution in the previous step is better than the worst solution in the harmony memory, the new solution will be replaced and the memory will be updated. Finally, this procedure is repeated until the termination criterion is satisfied that is the number of new generated solutions [10].
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Applied mathematics in engineering, management and technology 4(2) 2016:63-75 www.amiemt-journal.com For (j=1 to n) do If (r1< HMCR) then Xnew(j) = Xa(j) where a ∈ (1,2,….,HMS) If (r2< PAR) then Xnew(j)= Xnew(j)+/- r3 * BW where r1, r2, r3 ∈ (0,1) Endif Else Xnew(j)= LBj+r*(UBj-LBj), where r ∈ (0,1) EndIf EndFor
Figure 1. Pseudo code for generation new solution in harmony search algorithm 3.2. Simulated annealing algorithm The simulated annealing algorithm is one of the meta-heuristic and local search algorithms. Its ease of implementation and convergence properties and also its capability of escaping from local optima have made it a popular technique over the past two decades [11]. This algorithm proposed by Metropolis for the first time and inspired by cooling process and annealing crystal metals. In the process of annealing metals, the crystalline solid is heated and then allowed to cool slowly. If the temperature is reduced sufficiently slowly, the crystals will configure regularly and will achieve their ideal mode (optimal solution). At each iteration of a simulated annealing algorithm a solution is generated and compared with the current best solution. Improving solutions are always accepted, while a fraction of nonimproving solutions are accepted in the hope of escaping local optima. As temperature decreases – the iterations increases – the number of non-improving solutions occur less frequently and the solution distribution associated with the inhomogeneous Markov chain that its stable situations corresponds to globally optimal solutions. Proved that the result of the simulated annealing algorithm is convergent to the globally optimal solution [4]. The steps of the simulated annealing algorithm for a minimize problem could be summarized as follows: first and before starting the algorithm, the cooling schedule should be prepared. Some terms should be determined like initial temperature, final temperature, the number of iterations of each temperature and the method of generating neighborhood solution. Then an initial solution is generated. Other methods might be used for this generation. Therefore, a solution in the neighbor of current solution is generated, if it‟s better than neighbor solution, it will be accepted and if not, the acceptance probability is used to accept or reject the new solution. If the solution is rejected, the temperature would be decreased. Finally, if the stopping criterion is met/termination condition satisfied, the algorithm would be stopped and the best result will show as the optimal solution, otherwise, a new neighbor solution is generated and the algorithm is repeated.
3.3.
Generate the initial solution 𝑗
𝑗
𝑗
𝑗
For generating the initial population for 𝑥𝑖𝑡 ، 𝑦𝑖𝑡 ، 𝑆𝐶𝑖𝑡 ، 𝑃𝑀𝑡 ، 𝐶𝑀𝑡 chromosome should be produced. E.g. for 𝑗 𝑥𝑖𝑡 for the number of production lines, matrix l*n that n is equal to the number of production periods should be determined. Also for overtime work and outsourcing the matrixes should be turned out. By the constraint of maximizing human resources and the constraint of machines‟ maintenance, the demand of each product is divided between several lines and external/outside supplier, internal production and overtime work. First of all, the production is done by the main production capacity as far as is possible via several lines. The rest of the demand would be assigned to overtime work and/or external supplier in the next phase. For the periods of maintenance, zero and one variable of the first period, once per periods is chosen for the maintenance and finally the most optimal mode of the cycle is opted. The required human resources are calculated by the size of the production of regular and overtime work. By considering planning horizon‟s periods equal to the periods of preventive maintenance T, it would be concluded that T=k and H=N. Then, 𝑛𝐼 = [𝑁 𝐾 ] would be equal to 𝑁 𝐾 if n1 has an integer value and otherwise, it would be equal to 𝑛𝐼 = 𝑁 𝐾 + 1. k represents the optimal periods of preventive maintenance in
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Applied mathematics in engineering, management and technology 4(2) 2016:63-75 www.amiemt-journal.com (T=k𝜏). According to the value of k, n1 would be calculated and therefore, the cost function of maintenance would be calculated. Then, the retained capacity of the period should be calculated and summed with the value gained from production planning. This procedure continues until the minimum cost would be gained.
4. Numerical results The two above algorithms has been run by the same inputs. Supposed three production lines that produce four products in four periods. The predicted demand for each product is shown in the Table (1) for every period respectively. The maximum available human resource is 500 person for whole the periods. The final results were as follows: Table 1. The demand of product i in period t dit t=1 t=2 t=3 t=4 i=1 652000 654000 666000 668000 i=2 652000 667000 667000 668000 i=3 653000 668000 673000 688000 i=4
635000
640000
640000
645000
The costs are C1=30, C2=40, C3=300, C4=500, C5=45, C6=26, C7=35, C8=25, C9=20, C10=24, C11=20, C12=30, Smin=600000000, W1=0.6, Imax=400000, SCmax=10000000, M=90000 and b1=0.3 that generally assigned for all of the production lines and periods and products regarding the index of the parameters. The other parameters have shown in the tables 2 and 3. Table 2. The value of u, u1, and e parameters ui 0.00113 0.151 0.002 0.0219
i=1 i=2 i=3 i=4
u1i 0.00123 0.16459 0.00218 0.02387
ei 0.001 0.002 0.001 0.001
Table 3. The value of a, g, k1, and k2 parameters at
gt
k1t
k2t
t=1
0.4
65
0.05
0.05
t=2
0.3
65
0.05
0.05
t=3
0.3
65
0.05
0.05
t=4
0.4
65
0.05
0.05
Both of harmony search and simulated annealing algorithms have been run by MATLAB. The results represented in the Tables 4 to 11. Table 4. Amount of production in regular time work SA
HS
j=1
t=1
t=2
t=3
t=4
t=1
t=2
t=3
t=4
i=1
0
0
22914
0
0
0
0
0
i=2
0
0
16351
0
37706
0
0
0
70
Applied mathematics in engineering, management and technology 4(2) 2016:63-75 www.amiemt-journal.com
j=2 j=3
i=3
0
179464
577961
0
66552
56502 8
0
0
i=4
0
0
0
0
537145
0
29092 1
0
i=1
510337
55524
480420
0
216861
i=2
70464
45383
185668
161246
8862
19437 8 0
i=3
353511
268914
0
0
0
0
i=4
309816
152417
0
195084
0
0
i=1
33571
0
0
0
0
0
i=2 i=3
0 0
19173 0
0 0
0 0
5915 434696
i=4
296478
439863
0
143313
0
0 0 35358 7
0 0 11578 6 25878 9 50346 1 67866 0 0
56880 8 0 16644 4 0 0 0 70336 63624 5
Table 5. Amount of production in overtime work SA
j=1 j=2 j=3
i=1 i=2 i=3 i=4 i=1 i=2 i=3 i=4 i=1 i=2 i=3 i=4
HS
t=1 t=2 t=3 0 0 162983 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11764 0 222408 0 0 0 0 0 0 0 0 0 0 0 0 0 20474 0 0
t=4 0 0 0 0 0 0 0 114493 0 0 0 26078
t=1 0 0 0 0 0 0 0 0 0 0 0 0
t=2 0 0 0 0 0 0 0 0 0 0 0 0
t=3 0 0 0 0 0 0 0 0 0 0 0 0
t=4 0 0 0 0 0 0 0 0 0 0 0 0
Table 6. Amount of outsourcing SA i=1 i=2 i=3 i=4
t=1 108092 581536 299489 20544
t=2 598476 602444 28785 47720
HS t=3 2560 451426 95039 646495
t=4 131139 490892 598249 132926
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t=1 435139 599517 151752 97855
t=2 459622 781777 102972 286413
t=3 162539 484357 557214 90290
t=4 99192 711288 451220 8755
Applied mathematics in engineering, management and technology 4(2) 2016:63-75 www.amiemt-journal.com
Table 7. Amount of inventory level at the end of each period SA i= 1 i= 2 i= 3 i= 4
HS
t=1
t=2
t=3
0
0
2877
0
0
0 12312
t=4
t=1
t=2
t=3
t=4
0
0
0
0
0
0
0
0
114777
0
43288
31571
31571
0
0
0
0
0
12312
18807
0
0
0
0
0
Table 8. Backorder sales at the end of each period SA i=1 i=2 i=3 i=4
t=1 0 0 0 0
t=2 0 0 0 0
HS
t=3 0 1791 0 0
t=4 533984 17653 58180 14299
t=1 0 0 0 0
t=2 0 0 0 0
t=3 0 0 0 0
t=4 0 0 0 0
Table 9. Hiring, firing and number of staffs at the end of each period SA
HS
hiring firing Staffs of the end of period
j=1 j=2
t=1 0 0
t=2 6 0
t=3 51 273
t=4 0 1
t=1 0 0
t=2 0 0
t=3 81 87
t=4 0 0
j=3
0
92
0
49
0
92
47
50
j=1 j=2 j=3
300 12 199
0 121 0
0 0 193
57 0 0
29 275 272
253 21 0
0 0 0
99 75 0
j=1
0
6
57
0
271
18
99
0
j=2
288
167
440
441
25
4
91
16
j=3
101
193
0
49
28
120
167
217
Table 10. Preventive maintenance schedule SA t=1
t=2
HS t=3
t=4
72
t=1
t=2
t=3
t=4
Applied mathematics in engineering, management and technology 4(2) 2016:63-75 www.amiemt-journal.com j=1 j=2 j=3
1 1 1
0 0 0
0 0 1
0 1 0
73
1 1 1
0 0 0
0 0 1
0 1 0
Applied mathematics in engineering, management and technology 4(2) 2016:63-75 www.amiemt-journal.com Table 11. Production and maintenance cost production Cost Maintenance Cost Total Cost
SA
HS
397605378 3236380.01214544 400841758.012146
408988853 1594.05538543746 408990447.055386
The simulated annealing algorithm has been converged to the minimum cost 400841758 at 305.94 seconds and it would be 408990447 at 588.65 seconds by the harmony search algorithm. Because of the less minimum cost of the simulated annealing algorithm result and its almost two times faster response time in the same conditions, it would be more suitable for the model. The preventive maintenance periods as shown in Table (14) are determined independently for each production line. E.g. for the third production line, both of the algorithms ordered to perform preventive maintenance every two periods. Figures (2) and (3) are shown the improvement process of both algorithms for the model with distinctive iterations. The condition of same iteration of algorithms is that the multiplication of 𝑛𝑀𝑜𝑣𝑒 ∗ 𝑀𝑎𝑥𝐼𝑡 in the simulated annealing algorithm should be equal to the multiplication of 𝑛𝑁𝑒𝑤 ∗ 𝑀𝑎𝑥𝐼𝑡 in harmony search algorithm. However, the advantages of harmony search algorithms are determined in Tables (11) and (12). The shortage of all periods is zero and the inventory is held in less periods.
Figure 2. The performance of Simulated Annealing algorithm
Figure 3. The performance of Harmony Search algorithm
5. Conclusion
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Applied mathematics in engineering, management and technology 4(2) 2016:63-75 www.amiemt-journal.com In this research, the integration of production planning with maintenance planning has been considered and a mathematical model is proposed. The model is solved by two meta-heuristic algorithms: simulated annealing and harmony search, and a numerical example is proposed also. The goal of the model is minimizing the costs while the system reliability that is potentially able to be failed, is kept at desirable situation by the optimal maintenance. Generally failures occur for the machines in each production line that would be returned to the previous situation by minimum repair and not to change the failure rate function. Also, any types of maintenance actions decrease the production capacity. An iterative solution framework is proposed for solving the model. A numerical example that is used by the two algorithms has close results, especially in the part of the preventive maintenance periods and the total cost that shows the high authenticity and accuracy of the model and the solution. The example data have been selected hypothetically. Multi product multi period multi parallel production line that performs independently has been considered in this research that is a comprehensive vision. References: [1] Aghezzaf, E.H., Jamali, M.A., Ait-Kadi, D. (2007). An integrated production and preventive maintenance planning model, European Journal of Operational Research, 181 (2007) 679–685. [2] Aghezzaf, E.H., Najid, M.N. (2008). Integrated production planning and preventive maintenance in deteriorating production systems, Information Sciences, 178 (2008) 3382–3392. [3] Aramon Bajestani, M. (2014). Integrating Maintenance Planning and Production Scheduling: Making Operational Decisions with a Strategic Perspective, University of Toronto, PhD thesis. [4] Garg, A., Deshmukh, S.G. (2006), Maintenance management: literature review and directions, Journal of Quality in Maintenance Engineering, Vol. 12 Iss: 3 pp. 205 – 238. [5] Gendreau, M., Potvin, J.Y., (2010), Handbook of metaheuristics, Springer, Berlin. [6] Hongzhou Wang. (2002). A survey of maintenance policies of deteriorating systems, European Journal of Operational Research, 139, 469–489. [7] Lee, C.Y., Chen, Z.L. (2000). Scheduling jobs and maintenance activities on parallel machines, Naval Research Logistics 47,145–165. [8] Lee, K. S. and Geem, Z. W., (2005), “A New Meta-Heuristic Algorithm for Continuous Engineering Optimization: Harmony Search Theory and Practice,” Computer Methods in Applied Mechanics and Engineering, 194, pp. 3902-3933. [9] Nourelfath, M., Nahas, N., & Ben-Daya, M. (2016). Integrated preventive maintenance and production decisions for imperfect processes. Reliability Engineering & System Safety, 148, 21-31. [10] Omran, M.G.H., and Mahdavi. M. (2008). Global-best harmony search, Applied mathematics and computation. 198, 643–656. [11] Papadimitriou, C. H., & Steiglitz, K. (1982). Combinatorial optimization: algorithms and complexity. Courier Corporation. [12] Purohit, B. S., & Lad, B. K. (2016). Production and maintenance planning: an integrated approach under uncertainties. The International Journal of Advanced Manufacturing Technology, 1-13. [13] Yalaoui, A., Chaabi, K., Yalaoui, F. (2014). Integrated production planning and preventive maintenance in deteriorating production systems, Information Sciences, 278 (2014) 841–861.
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