A comparative study of production–inventory model for determining effective production quantity and safety stock level

Applied Mathematical Modelling xxx (2015) xxx–xxx

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A comparative study of production–inventory model for determining effective production quantity and safety stock level _ Gülsßen Aydın Keskin a,⇑, Sevinç Ilhan Omurca b, Nursßen Aydın c, Ekin Ekinci b a

Department of Industrial Engineering, Kocaeli University, 41380 Kocaeli, Turkey Department of Computer Engineering, Kocaeli University, 41380 Kocaeli, Turkey c Faculty of Engineering and Natural Sciences, Sabancı University, 34956 Istanbul, Turkey b

a r t i c l e

i n f o

Article history: Received 11 December 2012 Received in revised form 9 January 2015 Accepted 19 January 2015 Available online xxxx Keywords: Safety stock level Production capacity programming Product capacity optimization Mathematical modeling Greedy algorithm Genetic algorithm

a b s t r a c t This study presents a comparative study to determine ideal stock levels of a multi-national tire manufacturing company. The conventional inventory models can not be sufficient to optimize the production, the inventory quantity and the backorder simultaneously. Therefore, it is not possible to obtain a production policy by considering these objectives for all produced parts concurrently. In this paper, a production problem with three objectives is solved with mathematical modeling, greedy algorithm and genetic algorithm considering production constraints of a company. While existing inventory models based on conventional methods were applied for safety stock level determination, our proposed model uses the mathematical programming based optimization methods based on mathematical programming. Furthermore, the production planning policy is obtained with the optimum production amount and the stock is determined by considering the constraints defined by the firm. Finally, in our numerical results, we compare each solution methodology with respect to each objective criteria.  2015 Elsevier Inc. All rights reserved.

1. Introduction Production planning considers matching available capacity with the forecasted demand for varying customer orders from 3 to 18 months. Production planning aims to set the production level of all product categories to satisfy fluctuating or uncertain demand while making decisions regarding the backorders and the inventory levels and defining suitable resources to be used [1]. Under today’s competitive conditions, customer satisfaction is one of the key points that should be addressed. Hence, the objectives of production planning can be summarized as delivering orders at the time of the customers’ request, organizing production and inventory levels parallel to sales level, optimum resource usage, and decreasing cost. Concordantly, backorder and ideal inventory level directly affects the optimum resource usage and decreasing cost/cost reduction. The inventory theory assumes that there are only two possibilities for stocking out when a customer demand arrives. Either the customer leaves the system cancelling the demand resulting in loss of profit, or the customer can wait until

⇑ Corresponding author at: Department of Industrial Engineering, Engineering Faculty, Kocaeli University, Umuttepe Campus, 41380 Kocaeli, Turkey. Tel.: +90 262 3033326; fax: +90 262 3033003. _ E-mail addresses: [email protected], [email protected] (G.A. Keskin), [email protected] (S.Ilhan Omurca), [email protected] (N. Aydın), [email protected] (E. Ekinci). http://dx.doi.org/10.1016/j.apm.2015.01.037 0307-904X/ 2015 Elsevier Inc. All rights reserved.

Please cite this article in press as: G.A. Keskin et al., A comparative study of production–inventory model for determining effective production quantity and safety stock level, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.01.037

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the stock is available and then completes the transaction. This event is referred to as ‘‘backorder’’ [2]. When unsatisfied demand occurs in real-life applications, we can generally observe that the customers refuse to wait for the backorder [3]. Therefore, the researchers have turned their attention to the models that do not allow backorders. Organizations have to keep safety stock to avoid backorder in the cases of uncertain demand, deviation in lead time and logistics, output gap and inaccurate knowledge [4],[5]. When the literature was examined regarding to these requirements, it was observed that there have been several solutions for estimation of the safety stock level based on the inventory theory from 1950s to the present [6]. Among the inventory problems, deterministic and stochastic models have been proposed. San Jose et al. [7] studied an inventory model related to backorder. Rezaei and Davoodi [8] introduced imperfect items and storage capacity in the lot sizing with supplier selection problem and formulated the problem as a mixed integer programming model to solve using genetic algorithm. Shah and Gor [9] aimed to minimize costs with the economic lot size model. Pentico and Drake [10] proposed an approach to model economic order quantity with partial backorders. Chou et al. [11] proposed a demand addicted unit cost and fuzzy economic order quantity model (EOQ) for an inventory problem with fuzzy constraints, based on the max–min operator. They reviewed their solution procedure by a method based on Kuhn–Tucker approach to point out their results. Maiti et al. [12] developed an inventory model based on stochastic lead time and price-addict demand while considering advance payment. Chang and Lo [13] proposed an approach to overcome the shortcoming of traditional methods for improving the continuous and discrete lead time with backorders. He et al. [14] developed production–inventory model for deteriorated items. Kang and Lee [15] presented an inventory model to minimize total cost that considers warehouse capacity, revenue rate, and quantity discount for a certain product. Cardenas-Barron [16] proposed a practical method for the economic production quantity and economic lot sizing models by using analytical geometry and algebra. Cardenas-Barron et al. [17] presented an alternative heuristic algorithm to solve a vendor managed inventory system with multi-product, multi-constraint based on EOQ with backorders considering linear and fixed backorders costs. Omrani and Keshavarz [18] developed an EOQ model to maximize the profit. Their model also determined the price, marketing cost and lot sizing. The studies summarized above employ deterministic inventory model as the basis. Furthermore, numerous studies based on stochastic inventory models exist in the inventory management literature. Moinzadeh and Aggarwal [19] proposed an (s,S) production policy for such systems and developed expressions for the operating characteristics of the system. Skouri and Papachristos [20] defined five costs, which are deterioration, holding, backorder, opportunity and renewal cost, to model continuous review inventory model. Ghalebsaz-Jeddi et al. [21] modeled a multi-item stochastic inventory system with backorders when the estimation of marginal backorder cost is available, and payment is due upon order arrival. Dutta et al. [22] presented a continuous review inventory system (Q,r) to find reorder point and optimum order quantity at fuzzy and random conditions. Al-Rifai and Rossetti [23] worked on the estimation of the reorder point at two echelon inventory system with one warehouse and m identical retailers. Annadurai and Uthayakumar [24] worked on probabilistic inventory model with optimum backorder related to capital investment. Taleizadeh et al. [25] defined stochastic replenishment arrivals at increased/decreased demand levels. Lee [26] studied an inventory problem for order and optimum pricing decisions subject to service level for multi-priced two products. While the suggested analytical solutions in [27–34] can be easily applied, they failed at realistic supply chain problems that consider key features such as multi resource sharing originating from multiple customers, capacity constraints and demands. Additionally, the safety stock leveling depending on the probabilistic distribution of demand and production rate is insufficient in classical inventory models [35]. In this study, we consider a real life problem of a tire production company. The main objective is to develop a production planning method which minimizes the amount of backorder and inventory by considering the production environment of the company. This problem is modeled as a mixed integer linear programming (MILP) model and solved by an optimization solver. Due to the computational time, we also propose efficient heuristic methods like genetic algorithm and greedy algorithm and assessed the performance of each solution approach. We make the following research contributions in this paper: When inventory management literature is examined in depth, we observe that the production model which optimizes production, inventory and backorder quantity together and achieves these objectives for all produced parts concurrently has not been proposed [7,34,36–39]. Rahmani et al. [40] studied a two-stage real world capacitated production system with lead time and setup decisions. In their paper, they developed a mixed-integer programming (MIP) model to formulate the robust production planning problem. Karimi-Nasab and Konstantaras [41] presented a new multi-objective production planning model for one product type. This model simultaneously optimizes three objectives by: (I) minimizing the total cost of production plan, (II) minimizing the total variations in lot sizes, and (III) minimizing the distance of lot sizes to the customer needs by mathematical model. Benjaafar et al. [42] considered the optimal control of a production inventory-system with both backorders and lost sales of a single product. Mocquillon et al. [43] optimized production quantity and safety stock level together. Sarkar [44] developed a production–inventory model in a two echelon supply chain management. Tian et al. [45] proposed an iterative approach to jointly solve the problems of tactical safety stock placement and tactical production planning. Zhou et al. [46] applied the joint replenishment strategy into an inventory system and builds a multi-product multi-echelon inventory control model by using the genetic algorithm (GA) for solving the model. In brief, most of the researchers studied only two concepts together as it is detailed above. Glasserman and Tayur [47] worked on the multi echelon production inventory systems considering base stock levels. They proposed a simulation based approach to estimate inventory costs. In our study, we propose a mathematical programming model which optimizes production, inventory and backorder quantities simultaneously for multi-product, multi period real-life problem.

Please cite this article in press as: G.A. Keskin et al., A comparative study of production–inventory model for determining effective production quantity and safety stock level, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.01.037

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Contrary to existing studies, considered constraints affecting the production process is mandatory without any assumption and they are adopted from practical production environment. The problem size we work in this paper is the largest among the studies we are aware of in the corresponding literature. The original contribution of this paper is based on these points that mentioned above. In this paper, besides a mathematical programming model, a greedy algorithm and a genetic algorithm are presented to solve the production inventory problem in order to obtain alternative solutions. The computational results show that the greedy algorithm performs better than the genetic algorithm and it is computationally more efficient compared to other methods. The paper is organized as follows. In Section 2, the industrial problem is presented in detail. In Section 3, a mathematical model is presented in order to clearly state the constraints and objectives of the problem. In Section 4, the heuristics with production constraints are presented. Finally, the experiments are evaluated in Section 5.

2. Industrial context Brisa is the leading industrial conglomerate in Turkey, under the license agreement signed with American BF Goodrich Company in 1974. The Company, a 100% Turkish investment, was named Lassa Tire Manufacturing and Trading Inc. at that time. The production started in 1977 at the factory and the sales were extended to 60 provinces in Turkey through 186 organized dealers. Until 1988, the Company produced tires under the Lassa brand, expanding its product range from tires for passenger cars, trucks and buses to farm and off-the-road vehicles. In response to the developments in the world tire industry, a joint venture agreement was signed between the Bridgestone Corporation of Japan and the Sabanci Group. As a result of this agreement, the Company name was changed to BRISA Bridgestone Sabanci Tire Manufacturing and Trading Inc. Today Brisa is the number one tire manufacturer in Turkey and the sixth biggest tire producer in Europe [48]. The manufacturing process of the company consists of mainly six steps, which can be described as follows: 1. Rubber Mixing and Tread Stock Extruding Process: In a banbury mixer, various rubber compounds are mixed with chemical agents to produce the type of material desired. This material is then sent to the next step in the manufacturing process in the form of tread rubber or bead rubber. The tread rubber is shaped into a tire tread by extruder, and cut into the size required by each type. 2. Rubber Coating and Cutting Process: Pre-heated and chemical treated nylon, rayon and polyester fabric cords and steel cords are passed through the calendar which coasts both sides with carcass rubber. The rubber coated cord is cut and jointed into one with a specified width and angle by a cutting machine and then sent to the next building process. 3. Bead Manufacturing Process: A predetermined number of high tensile steel bead wires are lined up to be coated with rubber while being extruded from the bead wire extruder. The rubber-covered bead wires are then formed by a bead former into looped bead wires of the specific number of strands for a given size according to the final use of the bead. After the addition of a flipper and bead filler, the finished beads are sent to the building process. 4. Building Process: In the building process, a carcass (cord) and beads are affixed on a former to make what is called a green case. Then, tread rubber is laminated onto the green case to make a green tire. The building process of radial tires is basically the same as that of bias tires. 5. Vulcanizing (Curing) Process: Each green tire is then placed in a mould in which a given pattern has been engraved and vulcanized for the required time at a specified pressure and temperature. The mould is heated by saturated steam to a temperature ranging from 120 C to 170 C. Steam (at 150 C to 180 C) is also introduced into a rubber bag called a bladder, located inside the green tire. Thus, the green tire is vulcanized from both inside and outside. 6. During the vulcanizing process, temperature, pressure, and vulcanizing time are automatically controlled by a fully automatic vulcanizer. The vulcanized tires are then inflated by post-cure inflator and allowed to cool for a measured time so that the desired tire dimension may be obtained after the rubber has set. The tires cooled by the post cure, inflator are sent to the finishing process for the removal of spews and other things and then onto the inspection process. 7. Inspection Process: In the inspection process, the tires undergo various inspections such as external appearance inspection, uniformity inspection, and balance inspection, before being shipped. The layout showing all parts of the factory is shown in Fig. 1. The material movements are shown with the help of arrows and location of all parts. The company produces 457 tire types but it is not possible to produce all of them simultaneously due to production capacity and resource constraints. The company keeps inventory for all tire types at each month to prevent unexpected backorder. However, this results in high holding cost. Therefore, a policy considering production, inventory and backorder simultaneously will prevent high inventory and backorders costs. A system providing a more efficient production and stock control model is aimed with the heuristic algorithms proposed in the paper. For greedy and genetic algorithms, the safety stock levels of tires are required to be determined monthly by calculating the production quantities. These approaches constitute acceptable and efficient solutions for the firms at different sectors because of their flexibility and adaptivity. Please cite this article in press as: G.A. Keskin et al., A comparative study of production–inventory model for determining effective production quantity and safety stock level, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.01.037

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Fig. 1. Block plan and material flow schema of the company.

3. Mathematical programming model The safety stock problem is a critical problem for a production process in different fields. The proposed stock control system in this paper can be easily adapted to different companies with few modifications. The scope of the study is shown in Fig. 2. In this section, we propose a mixed integer linear programming model (MILP) for safety stock problem in the tire manufacturing industry. First, we introduce the main points of the model and then, a detailed application is provided. Due to the production of a wide variety of tires, the company encountered backorder and/or high inventory level. While backorder will cause loss of prestige and hence, a possible profit loss, high inventory will require increasing the size of the

Mould capacity Manufacturing capacity Number of moulds

Producon constraints

Keeping in stock Backorder Proposed opmizaon model

Number of machines

Safety stock level

Product variaon

Monthly demand

Forecasngs

Fig. 2. The scope of the study.

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storage area which may cause high holding cost. The company searches out an optimum solution for this potential situation. In this problem, all constraints are determined with the production planning department of the company. The company has to plan the production of 457 type tires denoted by n for 12-month period denoted by m. Due to the machine and mould capacities and the company policy, at most 200 types of product can be produced in each month. Production of each tire requires a different type of tire mould and manufacturing machine. Let w denote the tire mould code and Ti denote the daily mould capacity. We also let Lm and Ni to denote the manufacturing machine type and associated capacity, respectively. These variables are summarized in Table 1. Moreover, the available number of mould codes is limited. We use Mw to present the number of available moulds which is presented in Table 2. Although the company has 527 different moulds, it is stated that some of them are not used even if they are used in production of many tires. For each tire, holding cost CLi and backorder cost CSi are defined and presented in Table 3. The available number of tire manufacturing machines MAs is defined as in Table 4. If we overproduce tires to prevent backorder, higher inventory may occur. On the other hand, producing tires regarding to minimize holding costs may cause backorder. Our goal is to obtain a production policy that provides the optimal trade-off between these costs. Before proceeding to model formulation, we state our assumptions as follows. Lead time is ignored. Deteriorated products are ignored. Transaction cost is not related to the quantity and product variation. Production demand is known through the planning period. Next, we present in detail the proposed mathematical formulation. Due to the company policy, we ignore the production cost in this problem. Parameters: i = 1,. . .,n: number of tires j = 1,. . ., m: month w = 1,. . ., s: number of moulds CLi, "i = 1,...,n: holding cost for one unit tire code i CSi, "i = 1,...,n: backorder cost for one unit tire code i Dj, "j = 1,...,m: working day in month j Sij, "i = 1,...,n, "j = 1,...,m: demand of tire code i at month j Ti, "i = 1,...,n: daily mould capacity of tire code i Ni, "i = 1,...,n: daily manufacturing machine capacity of tire code i Mw: number of mould code w A: number of machine type A B: number of machine type B M: big number Maxstock: maximum inventory level Maxcode: number of tire type code at the same month

aik ¼



if tire i uses machine type k; k 2 fA; Bg 0

otherwise

Variables: Pij, "i = 1,...,n, "j = 1,...,m: the number of production for tire code i at month j Eij, "i = 1,...,n, "j = 1,...,m: amount of inventory of tire code i at the beginning of month j Bij, "i = 1,...,n, "j = 1,...,m: backorder of tire code i at month j

 yij ¼

if tire i is produced at month j; 0 otherwise

Table 1 Production constraints model for ideal inventory level. Tire code

Mould code (w)

Daily mould capacity (Ti) (tire number/mould)

Tire manufacturing machine (Lm)

Tire manufacturing capacity (Ni)

X1 X10 X100 X101 .. .

K256 K291 K59 K501 .. .

94 90 101 101 .. .

B B B A .. .

693 693 728 728 .. .

Please cite this article in press as: G.A. Keskin et al., A comparative study of production–inventory model for determining effective production quantity and safety stock level, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.01.037

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G.A. Keskin et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx Table 2 Resource capacity tables. Mould code (w)

Mould number (Mw)

K1 K2 K3 K4 .. .

2 1 1 1 .. .

Table 3 Costs. Tire code

Holding cost of one unit tire

Backorder cost of one unit tire

X1 X10 X100 X101 .. .

CL1 CL2 CL3 CL4 .. .

CS1 CS2 CS3 CS4 .. .

Table 4 Tire production machine constraints. Tire manufacturing machine

Machine number (MAs)

A B

11 39

Mathematical model: P Pn minimize m j¼1 i¼1 ðCLi Eij þ CSi Bij Þ subject to:

Eij ¼ Eiðj1Þ þ Piðj1Þ  Siðj1Þ þ Biðj1Þ n X ½ðPij =Dj Þ=T i  6 M w

8i; j;

8j; w;

i¼1 n X aiA ½ðPij =Dj Þ=Ni  6 A 8j; i¼1 n X aiB ½ðPij =Dj Þ=Ni  6 B 8j; i¼1 n X Eij 6 Maxstock 8j; i¼1 n X yij 6 Maxcode 8j; i¼1

Pij 6 Myij

8i; j;

Eij ; Pij ; Bij 2 Z þ ; yij 2 f0; 1g: Available mould numbers were defined for 527 different mould codes that were essential for tire production. Constraint (1) gives the inventory balance equations of each product. Constraint (2) ensures that the daily total mould usage can 2  n  m þ m  s þ 4m not exceed the number of available mould. Tires are produced by two machines denoted as A and B codes respectively. Eleven A machines and 39 B machines are available in the company. Constraint (3) and (4) guarantee Please cite this article in press as: G.A. Keskin et al., A comparative study of production–inventory model for determining effective production quantity and safety stock level, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.01.037

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that the daily total manufacturing machine usage respect to the manufacturing machine code cannot exceed the number of available machines. These constraints derive from the limited mould and machine capacity of the tire production process. Constraint (5) ensures that the total amount of inventory cannot exceed the capacity of storage area. Constraint (6) presents that the total type of tires produced at each month cannot exceed the limit defined by the company. Finally, constraint (7) guarantees that tire i can be produced if and only if the tire type is selected for the production at month j. This model involves 4  n  m variables and constraints. The objective function aims to minimize the total stock quantity and backorder. Due to the integrality constraints, this problem is difficult to solve. Therefore, in the subsequent discussion, we focus on the linear relaxation of this problem. By relaxing the integrality constraint on variables Eij ; P ij andBij , we obtain the MILP problem. We do not relax the constraint on yij to guarantee the total type of tires do not exceed the predefined company limit. The optimal solution of the resulting MILP can be non-integer. To obtain a feasible integer solution, we can simply round down the non-integer values of inventory and production (Eij andPij ) and compute the integer value of backorder with respect to the constraint (1). ^ij ; P ^ ij andB ^ ij be the optimal solution of its relaxation. Let E ; P andB be the optimal values of the mathematical model and E ij

ij

ij

ij ; P  ij andB  ij ) is feasible for the capacity constraints and balance constraint in the mathClearly the rounded down solution (E ematical model. Let the objective function values of the mathematical model, MILP and rounded down solution are denoted by z ; ^zandz, respectively. Then, we have;

z ¼

m X n X ðCLi Eij þ CSi Bij Þ; j¼1 i¼1

^z ¼

m X n X ^ij þ CSi B ^ ij Þ; ðCLi E j¼1 i¼1

z ¼

m X n X ij þ CSi B  ij Þ ðCLi E j¼1 i¼1

Since MILP is a partial relaxation of the mathematical model, we have ^z 6 z . Moreover, the rounded down solution is feasible for the mathematical model but not necessarily optimal, we have z 6 z. Consequently, we obtain ^z 6 z 6 z. As we shall see in our computational study, the gap z  ^z turns out to be quite tight for the real life case study. 4. Numerical experiments At this section, three approaches used in this paper and the obtained results are examined in depth. 4.1. Optimization model The company has to produce 457 type tires in a 12 month period. The forecasted demand and the beginning inventories of these tires are presented as in Table 5. Because of the restricted machine and mould capacities and the organization policy, only 200 of the tires can be produced in a month. Due to the defined production constraints, the company encountered backorder and/or high inventory levels. However, the capacity of current warehouse is insufficient for high level of inventory. Moreover, the company rejects backorder to prevent loss of prestige and hence, a possible profit loss. Therefore, they prefer to hold maximum possible stock and this requires usage of large scale warehouse. If our proposed policy enables the company to minimize its backorder, the production planning department of the company will then propose an automation investment in the upcoming production period. Otherwise, it will have to use the investment funds to acquire a new warehouse. In this section, safety stock problem at tire manufacturing industry is modeled as a MILP and solved by IBM ILOG CPLEX to find optimal production, backorder and inventory levels for each tire in 12 months period. Resulting monthly production quantities, possible backorders and safety stock levels are presented in Tables 6–8, respectively. One of the challenging constraints in the model is the capacity constraint for the number of tires per month that have to be produced. As a result of the application, 200 tires are determined with respect to the optimum production conditions. The mathematical model has been solved by ILOG Cplex 10.0. The experiment has been run on an Intel Core 2 Quad 2.40 GHz with 4.00 GB RAM and it takes about 1205.94 s to solve the problem optimally for 457 products at 12-month period. For the larger problem sizes (e.g. production plan over a year), CPLEX cannot efficiently be applied. Due to the long computation time required by CPLEX, in the next section, we propose a time partitioning heuristic to solve the problem efficiently with lower computation time while maintaining the quality of solution close to optimality. 4.2. Heuristic models Greedy and genetic algorithms are proposed to obtain alternative solutions. Please cite this article in press as: G.A. Keskin et al., A comparative study of production–inventory model for determining effective production quantity and safety stock level, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.01.037

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Table 5 Forecasting demands and beginning inventories of tires. Tire code

Beginning inventory

Jan

Feb

March

Apr

May

June

July

Aug

Sept

Oct

Nov

Dec

X1 X10 X100 X101 X102 X103 X104 X105 .. . X212 X213 X214 X215 X216 .. . X95 X96 X97 X98 X99

1 790 1346 149 3384 115 3220 1887 .. . 748 60 283 1599 379 .. . 724 2699 143 2524 960

1 1580 2691 298 6768 229 6440 3774 .. . 1495 120 566 3197 757 .. . 1447 5397 285 5047 1920

0 152 4630 338 1745 429 3283 643 .. . 885 87 1603 4249 449 .. . 3506 696 2691 527 1485

0 54 4169 454 4736 455 11722 2122 .. . 2757 471 1788 4153 656 .. . 2431 5048 1723 3601 20

0 0 3617 289 4263 338 7088 1968 .. . 1650 314 1451 4624 408 .. . 3479 1434 2073 3534 20

0 0 3821 449 3842 353 10663 2666 .. . 1860 466 998 3896 648 .. . 2820 1406 1949 2571 2021

0 1501 2686 496 3414 239 11564 2948 .. . 1239 511 1139 4080 719 .. . 2507 1568 1180 3185 0

0 762 2717 398 6232 215 9417 2285 .. . 1076 418 543 3036 583 .. . 1723 1120 1206 2548 1448

0 2420 1567 242 276 59 5678 3593 .. . 792 270 505 2301 350 .. . 1442 914 635 383 0

0 11653 3048 606 1515 761 14015 2766 .. . 1098 615 552 3570 896 .. . 2355 2830 1110 2269 0

0 9102 2575 491 1119 215 11125 2259 .. . 947 506 685 3033 722 .. . 1654 3 1030 3178 0

0 6768 2496 446 3144 331 10339 2064 .. . 851 462 550 2764 652 .. . 1916 2201 784 2456 0

0 4230 2088 493 744 162 11407 2933 .. . 840 510 905 2149 732 .. . 1379 1821 766 2160 0

Table 6 Production quantities. Tire code

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

M11

M12

X1 X10 X100 X101 X102 X103 X104 X105 .. . X212 X213 X214 X215 X216 .. . X95 X96 X97 X98 X99

0 790 1345 0 3384 0 3220 1887 .. . 747 0 283 1598 378 .. . 723 2698 0 2523 960

0 0 4630 0 1745 0 3283 643 .. . 885 0 1603 4249 0 .. . 3506 696 2691 0 1485

0 0 4169 0 4736 0 11722 2122 .. . 2757 0 1788 4153 656 .. . 2431 5048 1723 3601 0

0 0 3617 0 4263 0 7088 1968 .. . 1650 0 1451 4624 0 .. . 3479 1434 2073 3534 0

0 0 3821 0 3842 0 10663 2666 .. . 1860 0 998 3896 648 .. . 2820 1406 1949 2571 2021

0 1501 2686 0 3414 0 11564 2948 .. . 1239 0 1139 4080 719 .. . 2507 1568 1180 3185 0

0 762 2717 398 6232 215 9417 2285 .. . 1076 418 0 3036 583 .. . 1723 1120 1206 2548 1448

0 2420 1567 0 0 0 5678 3593 .. . 792 0 0 2301 0 .. . 1442 914 635 0 0

0 11653 3048 0 1515 0 14015 2766 .. . 1098 0 0 3570 896 .. . 2355 2830 1110 2269 0

0 9102 2575 0 1119 0 11125 2259 .. . 947 0 0 3033 0 .. . 1654 0 1030 3178 0

0 6768 2496 0 3144 0 10339 2064 .. . 851 0 0 2764 0 .. . 1916 2201 784 2456 0

0 4230 2088 0 744 0 11407 2933 .. . 840 0 905 2149 732 .. . 1379 1821 766 2160 0

4.2.1. Greedy algorithm Greedy algorithm is generally used at scheduling problems for production planning. In this paper, we adapt the algorithm to an inventory and a backorder problem in order to obtain the desired production level considering the defined inventory and backorder constraints. As mentioned in Section 3, certain types of the tires have to be produced in a year period and due to the machine and mould capacity limits, all of them cannot be produced in a month. This case may result in an excessive amount of inventory or backlogged demand. This is a typical trade-off problem. Due to the mould and machine capacity restrictions, maximum 200 tires have to be produced in a month. On the other hand, a warehouse capacity is limited to 800,000 tires. Different than the proposed mathematical model, greedy algorithm decides the production plan for each month. Since it does not consider the following months, this approach presents a myopic solution. Greedy algorithm starts with the first month. While designing an algorithm, first 200 tires with the highest backorder cost are considered for production due to the nature of greedy algorithm. By controlling the remaining mould and machine capacities, we choose the 200 tires for production. We determine their production amount with respect to the demand of each tire for that month. Then, we compute the amount of inventory and backorder for all tires. Next month’s production is planned by considering the first 200 tires

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G.A. Keskin et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx Table 7 Backorders. Tire code

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

M11

M12

X1 X10 X100 X101 X102 X103 X104 X105 .. . X212 X213 X214 X215 X216 .. . X95 X96 X97 X98 X99 Total

0 0 0 149 0 114 0 0 .. . 0 60 0 0 0 .. . 0 0 142 0 0 13395

0 152 0 338 0 429 0 0 .. . 0 87 0 0 449 .. . 0 0 0 527 0 32612

0 54 0 454 0 455 0 0 .. . 0 471 0 0 0 .. . 0 0 0 0 20 37267

0 0 0 289 0 338 0 0 .. . 0 314 0 0 408 .. . 0 0 0 0 20 57703

0 0 0 449 0 353 0 0 .. . 0 466 0 0 0 .. . 0 0 0 0 0 32160

0 0 0 496 0 239 0 0 .. . 0 511 0 0 0 .. . 0 0 0 0 0 49409

0 0 0 0 0 0 0 0 .. . 0 0 543 0 0 .. . 0 0 0 0 0 125682

0 0 0 242 276 59 0 0 .. . 0 270 505 0 350 .. . 0 0 0 383 0 50088

0 0 0 606 0 761 0 0 .. . 0 615 552 0 0 .. . 0 0 0 0 0 139806

0 0 0 491 0 215 0 0 .. . 0 506 685 0 722 .. . 0 3 0 0 0 72210

0 0 0 446 0 331 0 0 .. . 0 462 550 0 652 .. . 0 0 0 0 0 101950

0 0 0 493 0 162 0 0 .. . 0 510 0 0 0 .. . 0 0 0 0 0 51740

Table 8 Safety stock levels. Tire code

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

M11

M12

X1 X10 X100 X101 X102 X103 X104 X105 .. . X212 X213 X214 X215 X216 .. . X95 X96 X97 X98 X99 Total

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 3641

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 962

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 2170

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 1877

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 2659

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 52574

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 0

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 22836

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 4332

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 2279

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 2798

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 0

with the highest net forecasted demand after updating the initial inventory. This process continues until we reach the final month. As a result, we obtain inventory, production and backorder matrices by considering the minimum stock and backorder. The pseudo code of the algorithm is given in Appendix A. The outputs of greedy algorithm such as production quantities, possible backorders and safety stock levels are presented in Tables 9–11, respectively. 4.2.2. Genetic algorithm In production planning, genetic algorithm (GA) is generally used in scheduling problems like greedy algorithm. Different than the literature, we implemented the algorithm to an inventory and a backorder problem to obtain simultaneously the desired production, backorder and inventory quantities for each product by considering the capacity constraints. The algorithm has been executed with the following parameters under the same constraints: Iteration: 200 Population size: 70 Mutation probability: 0.05 Please cite this article in press as: G.A. Keskin et al., A comparative study of production–inventory model for determining effective production quantity and safety stock level, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.01.037

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Table 9 Production quantities. Tire code

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

M11

M12

X1 X10 X100 X101 X102 X103 X104 X105 .. . X212 X213 X214 X215 X216 .. . X95 X96 X97 X98 X99

0 790 1345 0 3384 0 3220 1887 .. . 747 0 283 1598 378 .. . 723 2698 0 2523 960

0 0 4630 0 1745 0 3283 643 .. . 885 0 1603 4249 0 .. . 3506 696 2833 0 1485

0 0 4169 941 4736 998 11722 2122 .. . 2757 0 1788 4153 1105 .. . 2431 5048 1723 4128 0

0 0 3617 0 4263 0 7088 1968 .. . 1650 0 1451 4624 0 .. . 3479 1434 2073 3534 0

0 0 3821 0 3842 0 10663 2666 .. . 1860 1398 998 3896 1056 .. . 2820 1406 1949 2571 2061

0 1707 2686 1234 3414 0 11564 2948 .. . 1239 0 1139 4080 0 .. . 2507 1568 1180 3185 0

0 0 2717 0 6231 292 9417 152 .. . 0 0 0 3036 1302 .. . 1723 292 292 2548 1448

0 3182 1567 0 0 0 5678 5725 .. . 1868 1199 0 2301 0 .. . 1442 1741 1548 0 0

0 11653 3048 0 1790 1672 14015 2766 .. . 0 0 1600 3570 0 .. . 1416 2830 0 2652 0

0 9102 2575 1737 0 0 11125 2259 .. . 2045 0 0 3033 1969 .. . 2592 0 2140 3178 0

0 6768 2496 0 4262 0 10339 1016 .. . 0 1583 0 2764 0 .. . 1916 2203 0 2456 0

0 4230 2088 0 0 0 11407 3981 .. . 1691 0 2140 2149 1384 .. . 0 1820 1550 2160 0

Table 10 Backorders. Tire code

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

M11

M12

X1 X10 X100 X101 X102 X103 X104 X105 .. . X212 X213 X214 X215 X216 .. . X95 X96 X97 X98 X99 Total

0 0 0 149 0 114 0 0 .. . 0 60 0 0 0 .. . 0 0 142 0 0 13395

0 152 0 487 0 543 0 0 .. . 0 147 0 0 449 .. . 0 0 0 527 0 44093

0 206 0 0 0 0 0 0 .. . 0 618 0 0 0 .. . 0 0 0 0 20 66426

0 206 0 289 0 338 0 0 .. . 0 932 0 0 408 .. . 0 0 0 0 40 97579

0 206 0 738 0 691 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 99749

0 0 0 0 0 930 0 0 .. . 0 511 0 0 719 .. . 0 0 0 0 0 122831

0 762 0 398 0 852 0 2132 .. . 1076 929 543 0 0 .. . 0 827 913 0 0 210525

0 0 0 640 275 911 0 0 .. . 0 0 1048 0 350 .. . 0 0 0 383 0 152170

0 0 0 1246 0 0 0 0 .. . 1098 615 0 0 1246 .. . 938 0 1110 0 0 238280

0 0 0 0 1119 215 0 0 .. . 0 1121 685 0 0 .. . 0 2 0 0 0 237055

0 0 0 446 0 546 0 1047 .. . 851 0 1235 0 652 .. . 0 0 784 0 0 249364

0 0 0 939 744 708 0 0 .. . 0 510 0 0 0 .. . 1379 0 0 0 0 201878

Cross over operator: one point crossover Mutation operator: inversion The solution steps of the algorithm are established as follows: Step 1. Generating an initial population: Initial population is generated randomly. It includes 70 feasible chromosomes with 457 bits. Each bit denotes a tire type. As we mentioned before, at each month at most 200 tires can be produced. Therefore, while 200 bits represent the produced tires, the remaining 257 bits present the ones that are not selected for the production. During the selection process of 200 bits in a chromosome, the production amount of each tire is determined by considering the remaining mould and machine capacities. Step 2. Evaluating the fitness: Fitness value of each chromosome is computed. In this study, the fitness value is formulated P P457 as FV ¼ 457 i¼1 Inv entoryi Cost þ i¼1 Backorder i Cost. Then, total fitness value of the population is computed. Step 3. Selection: Chromosomes are selected from the current population to be the parents of generated offspring for the new generation according to their fitness values. In this paper, roulette wheel selection method is used as the selection technique. Since the objective is to minimize the total amount of inventory and backorder costs, the quantile of each chromosome is computed by taking the inverse of the ratio of its fitness value and the total fitness. Please cite this article in press as: G.A. Keskin et al., A comparative study of production–inventory model for determining effective production quantity and safety stock level, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.01.037

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G.A. Keskin et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx Table 11 Safety stock levels. Tire code

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

M11

M12

X1 X10 X100 X101 X102 X103 X104 X105 .. . X212 X213 X214 X215 X216 .. . X95 X96 X97 X98 X99 Total

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 0

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 0

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 0

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 0

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 0

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 0

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 35

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 0

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 0

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 0

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 7

0 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 0

Table 12 Production quantities. Tire code

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

M11

M12

X1 X10 X100 X101 X102 X103 X104 X105 .. . X212 X213 X214 X215 X216 .. . X95 X96 X97 X98 X99

1 0 2691 298 6768 229 6440 3774 .. . 0 0 0 3197 0 .. . 1447 0 0 0 1920

0 942 3284 0 0 314 63 0 .. . 0 147 1886 2650 0 .. . 2782 0 0 3050 0

0 54 0 643 0 455 11722 0 .. . 4389 471 0 4153 1483 .. . 2431 8442 0 3601 0

0 0 7786 0 7360 338 7088 2846 .. . 1650 314 3239 4624 0 .. . 3479 0 6629 3534 565

0 0 3821 738 3842 0 10663 2666 .. . 0 0 998 3896 1056 .. . 2820 2840 1949 2571 2021

0 1501 0 0 3414 0 11564 0 .. . 0 0 1139 0 719 .. . 2507 0 1180 3185 0

0 762 5403 0 0 0 0 0 .. . 0 1395 543 0 0 .. . 0 0 1206 2548 1448

0 2420 0 0 6507 866 15095 0 .. . 4967 270 0 9417 0 .. . 3165 3601 635 0 0

0 11653 0 0 0 761 14015 0 .. . 1098 615 0 0 0 .. . 0 0 0 2652 0

0 9102 7190 0 2633 215 11125 0 .. . 0 506 0 6603 0 .. . 0 2832 2140 3178 0

0 0 2496 2679 3143 0 10339 15915.59 .. . 1798 462 0 2764 0 .. . 5925 0 0 0 0

0 0 0 0 0 493 0 0 .. . 0 510 3197 0 0 .. . 1379 4021 0 0 0

Step 4. Cross over: Genes are swapped between two chromosomes to produce a pair of offsprings. We use one-point crossover method. Generated offsprings are checked whether they are feasible or not according to the problem constraints. If they are not feasible, the production amount of randomly selected offspring is set to 0. Step 5. Mutation: Mutation is usually defined as flipping of randomly selected genes of an offspring. In our paper, mutation rate is 0.05. Step 6. Accepting: In this step, we first check whether each generated offspring is feasible with respect to capacity constraints or not. If a tire type in an offspring could not be produced due to the insufficient capacity, it is marked and its production amount is set to zero. This process continues until we check all 200 bits in an offspring. We randomly select a tire from the set of unproduced tires to complete the total produced tire type amount to 200. During the selection of new tires, we also check their feasibility. At the end of this process, the fitness value of each offspring is computed as in step 2. Then, their fitness values are compared with the maximum fitness value in the population. If it is lower than the maximum value, that offspring is replaced with the corresponding chromosome. Step 7. Stopping criteria: Assign the chromosome with the lowest fitness value in the population as the production program if the number of iterations is completed. Otherwise, turn back to step 2. Please cite this article in press as: G.A. Keskin et al., A comparative study of production–inventory model for determining effective production quantity and safety stock level, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.01.037

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Table 13 Backorders. Tire code

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

M11

M12

X1 X10 X100 X101 X102 X103 X104 X105 .. . X212 X213 X214 X215 X216 .. . X95 X96 X97 X98 X99 Total

0 790 0 0 0 0 0 0 .. . 747 60 283 0 378 .. . 0 2698 142 2523 0 77004

0 0 0 189 0 0 0 0 .. . 1632 0 0 0 827 .. . 0 3394 2833 0 525 99133

0 0 4169 0 3097 0 0 878 .. . 0 0 1788 0 0 .. . 0 0 4556 0 545 186163

0 0 0 289 0 0 0 0 .. . 0 0 0 0 408 .. . 0 1434 0 0 0 197204

0 0 0 0 0 353 0 0 .. . 1860 466 0 0 0 .. . 0 0 0 0 0 184898

0 0 2686 496 0 592 0 2948 .. . 3099 977 0 4080 0 .. . 0 1568 0 0 0 247442

0 0 0 894 6231 807 9417 5233 .. . 4175 0 0 7116 583 .. . 1723 2687 0 0 0 326983

0 0 1567 1136 0 0 0 8826 .. . 0 0 505 0 933 .. . 0 0 0 383 0 281614

0 0 4615 1742 1514 0 0 11600 .. . 0 0 1057 3570 1830 .. . 2355 2830 1110 0 0 456688

0 0 0 2233 0 0 0 13851 .. . 947 0 1742 0 2552 .. . 4009 0 0 0 0 459032

0 6768 0 0 0 331 0 0 .. . 0 0 2292 0 3205 .. . 0 2200 784 2456 0 424742

0 10998 2088 493 744 0 11407 2933 .. . 840 0 0 2149 3937 .. . 0 0 1550 4616 0 384148

Table 14 Safety stock levels. Tire code

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

M11

M12

X1 X10 X100 X101 X102 X103 X104 X105 .. . X212 X213 X214 X215 X216 .. . X95 X96 X97 X98 X99 Total

1 0 1346 149 3384 115 3220 1887 .. . 0 0 0 1599 0 .. . 724 0 0 0 960 238485

1 0 0 0 1639 0 0 1244 .. . 0 0 0 0 0 .. . 0 0 0 0 0 27220

1 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 10806

1 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 10434

1 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 9529

1 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 3951

1 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 2265

1 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 1399

1 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 21267

1 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 16715

1 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 15686

1 0 0 0 0 0 0 0 .. . 0 0 0 0 0 .. . 0 0 0 0 0 14871

The obtained outputs such as production quantities, possible backorders and safety stock levels are presented in Tables 12–14, respectively. The optimization model and heuristic algorithms are compared with respect to backorder levels, safety stock levels and computation times. In our numerical experiments, we denote the mathematical model with MathMod. As it is seen in Table 15, monthly inventories do not exceed the maxstock constraint of the model each of the three methods. The pivot table of the results of each model are compared in Table 15. As it is seen in Tables 15 and 16, in terms of the backorder and inventory criteria, greedy algorithm is considerably favorable than GA. The performances of the mathematical model and the greedy algorithm are 77% and 47% higher than GA in backorder scale, respectively. When we check the results in terms of inventory cost, it is observed that greedy outperforms all methods. It performs 99% higher than GA. Next, we examine the error incurred by solving the relaxed mathematical model instead of solving the integer model. Table 17 presents the error gap for each cost parameter. Comparing the total cost of mathematical model in Table 15 against the cost parameters given in Table 17, we notice that the resulting error is significantly small.

Please cite this article in press as: G.A. Keskin et al., A comparative study of production–inventory model for determining effective production quantity and safety stock level, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.01.037

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G.A. Keskin et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx Table 15 The pivot table of the results of mathematical model, greedy and genetic algorithm.

Backorders MathMod Greedy GA Inventory MathMod Greedy GA

M1

M2

M3

M4

M5

M6

M7

M8

M9

M10

M11

M12

Total

13395 13395 77004 3641 0 238485

32612 44093 99133 962 0 27220

37267 66426 186163 2170 0 10806

57703 97579 197204 1877 0 10434

32160 99749 184898 2659 0 9529

49409 122831 247442 52574 0 3951

125682 210525 326983 0 35 2265

50088 152170 281614 22836 0 1399

139806 238280 456688 4332 0 21267

72210 237055 459032 2279 0 16715

101950 249364 424742 2798 7 15686

51740 201878 384148 0 0 14871

764022 1733348 3325055 96128 42 372628

Table 16 Percentage gap with respect to genetic algorithm. Percentage Gap (%) Backorders Inventory

MathMod Greedy MathMod Greedy

77.02 47.87 74.20 99.99

Table 17 Error gap in solving the relaxation of the mathematical model. Error gap in costs z  ^z

Backorder cost 26.93

Inventory cost 6.65

Total cost 20.28

We conclude this section by presenting the results related to the computation time of each method. Greedy has the fastest computation time between the solution methods. Mathematical model takes about 1000.5 s to solve the problem for 457 industrial instances provided by the company at a 12-month period. While greedy heuristic takes about 1.05 s, GA solves the same problem at 781.47 s. 5. Conclusion While excess in production stocks may create additional burden like extra cost and new warehouse for the company, scarcity in inventory causes out of stock because of fluctuating demands and machinery breakdown and hence, customer dissatisfaction may occur. Many organizations in Turkey have not enhanced any effective stock policy to avoid this dilemma and for these reasons, this study is beneficial. In this paper, we have developed an optimum production policy for a leader tire manufacturing company’s production and stock control model. Then, the effectiveness of this solution approach is presented by comparing its results with greedy algorithm and GA solution separately. While the greedy algorithm provides better results regarding to the computation time and the inventory level, the solution of the proposed mathematical model provides the lowest backorder level compared to other methods. Since the greedy algorithm determines the production level of each product according to the total demand and backorder level, it nearly keeps zero inventory at the end of each month. However, this production policy results in high amount of backorder. Although, backorder level of greedy algorithm is high, it is always quite lower than the GA. The performance of GA is poor compared to the other methods. The efficiency of the GA depends on the number of iterations. However, as the number of iterations increases, its computation time increases exponentially. The contribution of this paper is based on three points which are summarized below. Computing optimum production quantity based on the limitations and capacity of the company and product range: The best solution cannot be found intuitively due to the large number of constraints and the constant capacity and hence, the development of a successful mathematical model has become mandatory. To the best of our knowledge, multi-product ranges and all production constraints have never been included in any model proposed in the literature. Developing an effective safety stock control model: Contrary to the existing studies, constraints of the production process are real and mandatory without any assumption. Accordingly, in multi-product companies, optimum production quantity per product is determined by considering the production capacity. Correlatively, safety stock levels per product that minimizes backorder are calculated separately by regarding the demand. In addition, it should be considered that the problem size, we work in this paper, is the largest we are aware of in the corresponding literature. Consequently, it is obvious that the companies using the stock control models efficiently, can increase their profit rates and minimize their payments through restricting their expenditures and controlling their costs. The greedy algorithm takes into account multiple resource share, capacity constraints and total demand as the key features to solve real supply chain management problems efficiently in different fields. Please cite this article in press as: G.A. Keskin et al., A comparative study of production–inventory model for determining effective production quantity and safety stock level, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.01.037

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Acknowledgement This study was funded by the University of Kocaeli Research Fund. Appendix A The pseudo code of Greedy Algorithm: Input MaxStock = 800000 MaxCode = 457 tire = 457 month = 12 Demandtiremonth: demand of tire code i at month j Dmonth: working day in month j Ttire: daily mould capacity of tire code i Ntire: daily manufacturing machine capacity of tire code i Output Ptiremonth Inventorytiremonth Backordertiremonth Begin Input InvBeg: the beginning inventory totalP = 0 j=1 for each tire i 2 [1,tire] minP(i,j) = Demand(i,j)-InvBeg(i) end SminP = Sort(minP) for each tire i 2 [1,tire] if totalP<200 if SminP(i,j) > 0 if Mw(i)>0 Q1 = min(SminP(i,j), D(j) T(i) Mw(i)) else Mw(i) = 0 Q1 = 0 end if MAs(i)>0 Q2 = min(SminP(i,j), D(j) N(i) MAs(i)) else MAs(i) = 0 Q2 = 0 end if min(Q1, Q2) > 0 P(i,j) = min(Q1, Q2) Mw(i) = Mw(i)-(P(i,j) /D(j)/T(i)) MAs(i) = MAs(i)-(P(i,j) /D(j)/N(i)) totalP = totalP + 1 else P(i,j) = 0 end if P(i,j)-Demand(i,j) + InvBeg(i,j))> = 0 backorder(i,j) = 0 else backorder(i,j) = P(i,j)-Demand(i,j) + InvBeg(i,j) end inventory(i,j) = P(i,j)-Demand(i,j) + InvBeg(i,j)

Please cite this article in press as: G.A. Keskin et al., A comparative study of production–inventory model for determining effective production quantity and safety stock level, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.01.037

G.A. Keskin et al. / Applied Mathematical Modelling xxx (2015) xxx–xxx

15

end end end for each month j 2 [2,month] totalP = 0 Input Mw: number of mould code w Input MAs: number of machine code for each tire i 2 [1,tire] minP(i,j) = Demand(i,j) + backorder(i,j-1) - inventory(i,j-1) end SminP = Sort(minP) for each tire i 2 [1,tire] if totalP<200 if (P(i,j-1)-Demand(i,j-1) + inventory(i,j-1)-Demand(i,j)>0) P(i,j) = 0 else if SminP(i,j) > 0 if Mw(i)>0 Q1 = min(SminP(i,j), D(j) T(i) Mw(i)) else Mw(i) = 0 Q1 = 0 end if MAs(i)>0 Q2 = min(SminP(i,j), D(j) N(i) MAs(i)) else MAs(i) = 0 Q2 = 0 end if min(Q1, Q2) > 0 P(i,j) = min(Q1, Q2) Mw(i) = Mw(i)-(P(i,j)/D(j)/T(i)) MAs(i) = MAs(i)-(P(i,j)/D(j)/N(i)) totalP = totalP + 1 else P(i,j) = 0 end end if P(i,j)-Demand(i,j)-backorder(i,j-1) + inventory(i,j-1) > = 0 backorder(i,j) = 0 else backorder(i,j) = P(i,j)-Demand(i,j) + inventory(i,j-1)-backorder(i,j-1) end inventory(i,j) = P(i,j)-Demand(i,j) + inventory(i,j-1)-backorder(i,j-1) end end if totalP> = 200 P(i,j) = 0 end End

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Please cite this article in press as: G.A. Keskin et al., A comparative study of production–inventory model for determining effective production quantity and safety stock level, Appl. Math. Modell. (2015), http://dx.doi.org/10.1016/j.apm.2015.01.037