Determination Optimum Inventory Level for Material Using Genetic Algorithm

The integration of decision-making will lead to the robust of its decisions, and then determination optimum inventory level to the required materials to produce and reduce the total cost by the cooperation of purchasing department with inventory department and also with other company , s departments. Two models are suggested to determine Optimum Inventory Level (OIL), the first model (OIL-model 1) assumed that the inventory level for materials quantities equal to the required materials, while the second model (OIL-model 2) assumed that the inventory level for materials quantities more than the required materials for the next period. This study was applied in Wasit Company for Textile Manufacturing in the Textile Factory, where it produces five products, which are printed striped, plain, poplin, dyed poplin and Naba weave. The products are made from cotton and they are passing through several stages to transfer to the final product. A genetic algorithm is used to determine the optimum quantity of the purchase a cotton and colors for each month and with minimum cost. Where the purchasing and transportation costs were either constant or variable with respect to purchased quantities while holding cost is kept constant. The results showed that the total cost of the first model is minimum than the second model because the holding cost for this model is less from the second model, while the purchasing and transportation costs from two models are equals. The percentage of purchasing cost for cotton is the biggest value, more 99% of purchasing cost for two models. Keyword: Inventory Level, Genetic Algorithm, Decision-making. ةينيجلا ةيمزراوخلا ماذختساب داوملل لثملاا نبزخلا ىوتسم ذيذحت .م لا يوايح قثاو ذب نسوس .د.م.أ دومحم سابع دومحم .د.م.أ يذيبزلا حيبص ٌداعًنأ جاخَلاا تسذُْ ىسق ٌداعًنأ جاخَلاا تسذُْ ىسق داصخقلاأ ةسادلاا تٛهك شيٕس تعياخ تٛخٕنُٕكخنا تعيادنا تٛخٕنُٕكخنا تعيادنا ةصلاخلا ٗنا ٘دؤٛس ثاساشقنا راخحأ مياكح ٍٛصشح تٛهًعن ٙفكح ٙخنأ تبٕهطًنا دإًهن مثيلاا ٍٚضخنا ٖٕخسي ذٚذسح ىث ٍئ ثاساشقنا ِزْ تكششنا واسقأ تٛقب عي كنزكٔ ٍٚضخنا ىسقٔ ثاٚشخشًنا ىسق ٍٛب قٛسُخنا للاخ ٍي تٛهكنا تفهكنا مٛهقح ٌأٔ جاخَلإا لأا ٖشخ بٕهطًنا دإًهن مثيلأا ٍٚضخنا ٖٕخسي ذٚذسخن ٍٛٛخرًَٕ ذاشخقا ىح دإًنا ٘ٔاسٚ ٍٚضخنا ٖٕخسي ٌأ ضاشخفا ىح لٔلأا جرًُٕنا ٙف ،ت تقزلانا ةشخفهن تبٕهطًنا دإًنا ٍي شثكأ ٍٚضخنا ٖٕخسي ٌأ ضاشخفا ىح َٙاثنا جرًُٕنا ٙف اًُٛب تبٕهطًنا ابنا ْٙٔ ثادخُي سًخ حخُٚ ثٛز ،حٛسُنا مًعي ٙف ثاخٕسًُنا تعاُصن طسأ تكشش ٙف جقبط تساسذنا ِزْ ٍٛهبٕبنأ ةداسنأ ةص ذٚذسخن تُٛٛدنا تناذنا جيذخخسا ذقٔ ٙئآَ حخُي ربصٛن مزاشي ةذعب شًٚٔ ٍطقنا ٍي عُصح ثادخًُنا أبُنا حٛسَٔ عٕبطًنا ٍٛهبٕبنأ ًنا تًٛكنا عي ةشٛغخي ٔأ ّخباث ايأ مقُنأ ءاششنا فهك جَاك ثٛز تفهك مقابٔ شٓش مكن ٌإنلاأ ٍطقنا ٍي ِاشخشي تًٛك مثيأ ،ةاشخش َٙاثنا جرًُٕنا ٍي مقأ جَاك لٔلأا جرًُٕهن تٛهكنا تفهكنا ٌا حئاخُنا جُٛب تخباث ظافخزلاا فهك جَاك اًُٛب ازٓن ظافخزلاا تفهك ٌٕكن ٍٛخرًُٕنا ٙف تٚٔاسخي ءاششنأ مقُنا فهك اًُٛب مقأ جرًُٕنا ثصٔادح ٍطقنا ةداًن ءاششنا تفهك تبسَ 99 ٍٛخرًُٕهنٔ ءاششنا تفهك ٍي % Journal of Engineering Volume 24 January 2018 Number 1 819 1.INTRODUCTION The companies select a single or multiple suppliers to fulfill the demands, and replenishment order quantity is split into different portions for each supplier at the same time. From the previous study, basically, there are two types of supplier selection problem. In the first type of supplier selection, a single supplier can fulfill the entire buyer's demand. In the other type of supplier selection, there exists no single supplier who can satisfy the entire buyer's needs. In this situation, the buyer has to split order quantities among suppliers for having a stable environment of competitiveness, Demirtas, and Üstün, 2008. There are several studies that deal with optimum inventory level, Park et. al. developed a mathematical model in which the retailer placed orders based on the EOQ policy and allocated them to the multiple manufacturers. In their model, production allocation ratios and the shipment frequencies at the manufacturers, as well as the purchasing cycle length at the retailer, were formulated to minimize the average total cost at the manufacturers and retailer, Park, et al., 2006. Sarker et. al. consider EOQ-like batch sizing models that account for the possibility of rework being done during cycles, as well as after a certain number of cycles. Especially the latter deals with quite some far going issues and hence provides some useful insights. Nonetheless, the paper stresses the need for flawless production, since rework will always be more expensive than the first-timeright production, Sarker, et al., 2008. Wadhwa and Ravindran introduced multiple objective multiple supplier selection models for low risk and cost products. The first objective was to minimize the total purchasing cost, which concluded total variable cost, fixed cost, inventory holding cost, and the bundling discounts. The second objective was to minimize the reject units under supplier capacity constraint. The shortage was not allowed and the multi-objective model was solved by preemptive goal programming, Wadhwa, and Ravindran, 2010. Araújo and Alencar put forward a model for selecting suppliers and evaluating the performance of those already working with a company. A simulation was conducted in a food industry. This sector has a high significance in the economy of Brazil. The model enables the phases of selecting and evaluating suppliers to be integrated. This is important so that a company can have partnerships with suppliers who are able to meet their needs. Additionally, a group method is used to enable managers who will be affected by this decision to take part in the selection stage, Araújo and Alencar, 2015. 2. INVENTORY LEVEL Effective forecasting is essential to achieve service levels, to plan allocation of total inventory investment, to identify needs for additional production capacity, and to choose between alternative operating strategies, where the accurate forecast is important to increase service levels, decrease inventory levels, and operating costs, Russell and Taylor, 2009. The Just in time (JIT) methodology is far more geared toward towards the stabilization of the inventory levels throughout the supply chain than the traditionally fixed order quantity methodology, also known as the economic order (EOQ) model. Manufacturers need a strategy to decrease total costs for items and to increase customer satisfaction. The purchasing department receives the items from suppliers at the same time of the demand is one of the keys of decreasing the risk for the manufacturers. Just-In-Time (JIT) model is one of the ways for achieving this goal, but it may not be the optimal solution. The first reason is, in the JIT model the manufacturers order the items whenever they need to meet the demand thus, it covers just pull systems and short planning horizon. The second reason is, by increasing order quantities, the price and shipping cost per item will be decreased, although, in a JIT model, the price breaks for purchasing and transportation costs may not happen at all time points, Eiliat, 2013. Journal of Engineering Volume 24 January 2018 Number 1 882 3. OPERATING COSTS Operating costs consist of the following: 3.1 Ordering Costs The ordering costs is a fixed cost of tracking trucks from a supplier to inventory, labor costs of processing orders, inspection and returning of poor quality products, Onawumi, et al., 2009. Conversely to the costs fixed per unit, the inventory costs fixed per order comprises only a portion of the acquisition cost of inventory. This is the cost incurred each time a stock replenishment order is placed and includes costs such as import duties, telephone calls, stock consolidator , s fee, etc., Bredenkamp, 2005. 3.2 Holding Costs Holding cost is defined as the cost associated with having one unit in inventory for a period of time. According to them, holding cost consists of four components, Holstein and Olofsson, 2009: 1. Capital cost. 2. Inventory service cost. 3. Storage space cost. 4. Inventory risk cost. Capital cost considers as the major contributor to holding cost. The other components such as inventory service cost, storage space cost, and inventory risk cost are sometimes called out-of-pocket holding costs. 3.3 Purchasing Costs It is the primary concern of any manufacturing organization to get an item at the right price. But right price need not be the lowest price. It is very difficult to determine the right price; general guidance can be had from the cost structure of the product, Eiliat, 2013. 3.4 Transportation Costs Transportation costs will at first decline as the number of facilities increase, but will eventually increase the number of facilities increase as a result of inbound and outbound transportation costs. The total cost of transporting products must be measured and not only the cost of moving the products to the warehouse. With fewer locations saving can be obtained by making use of bulk distribution from the manufacturer or supplier. There will, however, be a certain point where there are too many warehouses and fewer inventory of the various item lines will have to be shipped to the warehouse to ensure that there are no items that are overstocked. This will lead to higher costs charged by the transporter due to smaller loads, Burger, 2003. 4. GENETIC ALGORITHM (GA) The genetic algorithm is a stochastic search method for solving both constrained and unconstrained optimization problems that are based on the natural selection process that mimics biological evaluation. It explores the solution space by using concepts taken from natural genetics and evolution theory, Baz, 2004. GA starts with an initial set of solutions which is known as a population. The individuals of the population are called chromosomes which are evaluated according to a predefined fitness function, in our case the total cost. Each chromosome includes several genes. The gene represents an order quantity of item I at time point j. For example, if there ar


1.INTRODUCTION
The companies select a single or multiple suppliers to fulfill the demands, and replenishment order quantity is split into different portions for each supplier at the same time.From the previous study, basically, there are two types of supplier selection problem.In the first type of supplier selection, a single supplier can fulfill the entire buyer's demand.In the other type of supplier selection, there exists no single supplier who can satisfy the entire buyer's needs.In this situation, the buyer has to split order quantities among suppliers for having a stable environment of competitiveness, Demirtas, and Üstün, 2008.There are several studies that deal with optimum inventory level, Park et.al. developed a mathematical model in which the retailer placed orders based on the EOQ policy and allocated them to the multiple manufacturers.In their model, production allocation ratios and the shipment frequencies at the manufacturers, as well as the purchasing cycle length at the retailer, were formulated to minimize the average total cost at the manufacturers and retailer, Park, et al., 2006.
Sarker et.al. consider EOQ-like batch sizing models that account for the possibility of rework being done during cycles, as well as after a certain number of cycles.Especially the latter deals with quite some far going issues and hence provides some useful insights.Nonetheless, the paper stresses the need for flawless production, since rework will always be more expensive than the first-timeright production, Sarker, et al., 2008.
Wadhwa and Ravindran introduced multiple objective multiple supplier selection models for low risk and cost products.The first objective was to minimize the total purchasing cost, which concluded total variable cost, fixed cost, inventory holding cost, and the bundling discounts.The second objective was to minimize the reject units under supplier capacity constraint.The shortage was not allowed and the multi-objective model was solved by preemptive goal programming, Wadhwa, and Ravindran, 2010.
Araújo and Alencar put forward a model for selecting suppliers and evaluating the performance of those already working with a company.A simulation was conducted in a food industry.This sector has a high significance in the economy of Brazil.The model enables the phases of selecting and evaluating suppliers to be integrated.This is important so that a company can have partnerships with suppliers who are able to meet their needs.Additionally, a group method is used to enable managers who will be affected by this decision to take part in the selection stage, Araújo and Alencar, 2015.

INVENTORY LEVEL
Effective forecasting is essential to achieve service levels, to plan allocation of total inventory investment, to identify needs for additional production capacity, and to choose between alternative operating strategies, where the accurate forecast is important to increase service levels, decrease inventory levels, and operating costs, Russell and Taylor, 2009.The Just in time (JIT) methodology is far more geared toward towards the stabilization of the inventory levels throughout the supply chain than the traditionally fixed order quantity methodology, also known as the economic order (EOQ) model.Manufacturers need a strategy to decrease total costs for items and to increase customer satisfaction.The purchasing department receives the items from suppliers at the same time of the demand is one of the keys of decreasing the risk for the manufacturers.Just-In-Time (JIT) model is one of the ways for achieving this goal, but it may not be the optimal solution.The first reason is, in the JIT model the manufacturers order the items whenever they need to meet the demand thus, it covers just pull systems and short planning horizon.The second reason is, by increasing order quantities, the price and shipping cost per item will be decreased, although, in a JIT model, the price breaks for purchasing and transportation costs may not happen at all time points, Eiliat, 2013.

OPERATING COSTS
Operating costs consist of the following:

Ordering Costs
The ordering costs is a fixed cost of tracking trucks from a supplier to inventory, labor costs of processing orders, inspection and returning of poor quality products, Onawumi, et al., 2009.Conversely to the costs fixed per unit, the inventory costs fixed per order comprises only a portion of the acquisition cost of inventory.This is the cost incurred each time a stock replenishment order is placed and includes costs such as import duties, telephone calls, stock consolidator , s fee, etc., Bredenkamp, 2005.

Holding Costs
Holding cost is defined as the cost associated with having one unit in inventory for a period of time.According to them, holding cost consists of four components, Holstein and Olofsson, 2009: 1.Capital cost.2. Inventory service cost.3. Storage space cost.4. Inventory risk cost.Capital cost considers as the major contributor to holding cost.The other components such as inventory service cost, storage space cost, and inventory risk cost are sometimes called out-of-pocket holding costs.

Purchasing Costs
It is the primary concern of any manufacturing organization to get an item at the right price.But right price need not be the lowest price.It is very difficult to determine the right price; general guidance can be had from the cost structure of the product, Eiliat, 2013.

Transportation Costs
Transportation costs will at first decline as the number of facilities increase, but will eventually increase the number of facilities increase as a result of inbound and outbound transportation costs.The total cost of transporting products must be measured and not only the cost of moving the products to the warehouse.With fewer locations saving can be obtained by making use of bulk distribution from the manufacturer or supplier.There will, however, be a certain point where there are too many warehouses and fewer inventory of the various item lines will have to be shipped to the warehouse to ensure that there are no items that are overstocked.This will lead to higher costs charged by the transporter due to smaller loads, Burger, 2003.

GENETIC ALGORITHM (GA)
The genetic algorithm is a stochastic search method for solving both constrained and unconstrained optimization problems that are based on the natural selection process that mimics biological evaluation.It explores the solution space by using concepts taken from natural genetics and evolution theory, Baz, 2004.GA starts with an initial set of solutions which is known as a population.The individuals of the population are called chromosomes which are evaluated according to a predefined fitness function, in our case the total cost.Each chromosome includes several genes.The gene represents an order quantity of item I at time point j.For example, if there are 12 items and 12-time points, we will have 144 genes (order quantity) in one chromosome as in Fig. 1.The chromosomes evolve through successive iterations called generations, Li, et al, 2010.
A new generation is created by changing chromosomes in the existing population through crossover and mutation, Baz, 2004, as shown in Fig. 2.

APPLICATION OF PROPOSED METHODOLOGY
This paper was applied in Wasit company for Textile Industries as a case study to determine optimum inventory level for material.Textile Factory produces five products (N=5) which are (printed striped, poplin, Nuba, and dyed poplin weaves), their representation with symbols is ( A, B, C, D, and E ) respectively, that will make on the same production line inside factory and need setup time to change arrangements this production line when altering the production to another product.The materials required for manufacturing of products (meter) are cotton (ton) and colors (gram) by used bill of materials as shown in Fig. 3. Determination of the materials required depend on quantities forecasted in the marketing department, where requested quantities from cotton to the year 2016 is shown in Table .1,making an approximation to near integer number more than requested quantities and also for color as shown in Table 2.The purchasing department will make the plan to purchase the materials required for the entire year with minimum total cost (holding, purchasing, and transportation costs) depend on forecasting.

PROPOSED ALGORITHM TO DETERMINE OPTIMUM INVENTORY LEVEL
Purchasing department study purchasing of materials and determine the best order quantity depending on purchasing, transportation, and holding costs for materials, that can be illustrated in Tables 3, 4 and 5 respectively according to plans of this company for the year 2016.The assumptions that are used in this algorithm are: 1. Items are always available for shipment.2. Each item has constant holding and ordering costs.3. The purchase and transportation costs vary with order quantity or constant.4. The demands are known and non-constant.5.The period between time points of planning horizon could be measured in hours, days, months, etc.The selection of the best order quantity in the textile factory can be classified into two models depending on inventory amounts.The first model will attempt reducing the inventory level, therefore will lead to reducing holding cost.This model will be explained in the section (6.1), that assumed the inventory amounts will equal the demand for next month and will be symbolled as (OIL-Model 1).The second model will assume the inventory amounts will exceed the demand for next month to reduce purchasing and transportation costs.This model will be explained in the section (6.2), and the symbol to this model is (OIL-Model 2).

OIL-Model 1
In this paper used through a hybrid algorithm that compared advantages JIT and EOQ to reduce all costs together to determine the optimum inventory level and it solve by GA where calculate from equation ( 1) and (2).
Q i j = Quantity order for material i at time j.
= Demand for material i at time j.
The company warehouse has a limited stock capacity for each material  ∈  depend on lower and an upper number of units for all materials.The inventory level of material i should be greater than or equal to the demand of production department at each time point j when there is no shortage of materials, thus:    ≥    , ∀ ∈  , ∀ ∈  (3) The price of each material decreases when the number of material increases.The purchasing cost of order quantity is: Where: = Purchasing cost for material i in time j.
= The set of price breaks of material i, where k={1,2,3,….} The transportation cost for shipping the materials decrease when the number of materials increases, therefore transportation cost of order quantity is: Where: = Transportation cost for material i in time j.
Material i has a unit holding cost h i per time period.The total holding cost for storing order quantities of material i between time points j and j+1 is: Let (   ) be the total cost, that is the summation of purchasing, holding and transportation costs.
Form equations ( 4), ( 5) and ( 6) we have: Minimize Z= (   ) = ∑ ∑ ((   ) + (   ) + (   )) ∈ ∈ (7) The solutions are given after 500 runs in MATLAB program.Each run gives various total cost with a various set of order quantities, then compares them to give best order quantities with minimal total cost, that equal to1563661500 dinars at run number 178 as shown in Fig. 4, the order quantities for this factory can be shown in Table .6and Tables 7 shows inventory levels (    ).Tables 8, 9, and 10 show the holding, purchasing, and transportation costs for all items in a year 2016.Figure ( 5) gives a summarized the percentage of the holding, purchasing and transportation costs.

OIL-Model2
This model uses the same equations as the first model except that the equation number (1) has changed to become: The order quantities for this factory can be shown in Table .11.The solutions are given after 500 runs in MATLAB program.Each run gives various total cost with a various set of order quantities, then compares them to give best order quantities with minimal total cost, that equal to 1592049000 dinars at run number 151 as shown in Fig. 6.Tables 12 shows inventory levels.Tables 13, 14, and 15 show the holding, purchasing, and transportation costs respectively for all items in the year 2016.Fig. 7 gives a summarized the percentage of the holding, purchasing and transportation costs.The percentage of purchasing cost is the biggest value, 87% and the percentage of purchasing cost for cotton is the biggest value, more 99% of purchasing cost.Decision maker in production department will make the plan to execute accepted demands with minimum setup time for the entire year and determine the best sequence of products and to all demands by coordination with another department to integrate decision making inside the factory.

CONCLUSIONS
The best order quantity in the textile factory depends on reducing each holding, purchasing, and transportation costs together by using GA, can be classified to two models depend on inventory amounts.The main conclusions of this paper are: 1.The total cost of (OIL-Model 1) less than (OIL -Model 2), where total cost for (OIL -Model 1) equals to1563661500 and for (OIL-Model 2) equal to 1592049000 dinars.2. The holding cost of (OIL -Model 1) less than (OIL -Model 2), where holding cost for (OIL -Model 1) equals to12188000 and for (OIL -Model 2) equals to 40575500 dinars.3. The percentage of purchasing cost 89% , Transportation cost 10% and holding cost 1% from the total cost of (OIL -Model 1). 4. The percentage of purchasing cost 87% , Transportation cost 10% and holding cost 3% from the total cost of (OIL -Model 2). 5.The percentage of purchasing cost of cotton is the biggest value, more 99% of purchasing cost for two models.6.The proposed methodology can be applied to another industrial company, especially organizations which work in a dynamic environment more than Wasit company.

Figure 3 .
Figure 3. Bill of materials for products of textile factory.

Figure 4 .
Figure 4. Comparing runs in the MATLAB programming for the first model.

Figure 5 .
Figure 5. Dividing total cost for the first model.

Figure 6 .
Figure 6.Comparing runs in the MATLAB programming for the second model.

Figure 7 .
Figure 7. Dividing total cost for the second model.


Araújo M. and Alencar D., 2015, IntegratedModel for Supplier Selection and Performance Evaluation, South African Journal of Industrial Engineering August Vol.26. Baz, M.A., 2004, A Genetic Algorithm for Facility Layout Problems of Different Manufacturing Environments, Computer and industrial engineering , vol.47, No.23, pp.233-246. Bredenkamp F., 2005, The Development of a Genetic Just-In-Time Supply Chain Optimization Software Tool, Master thesis, Unversity of Stellenbosch, Industrial Engineering. Burger F., 2003, The Impact of Warehousing and Transportation on Supply Chain Effectiveness, Msc.Thesis, Rand Afrikaans University.

Table 1 .
Materials required from cotton ( ton) to the year 2016.

Table 2 .
Materials required from colors ( kg) to the year 2016.

Table 3 .
The relation between price costs (thousand dinars) per unit and material order quantity.

Table 6 .
Best order quantities for first model.

Table 7 .
Inventory levels for the first model.

Table 8 .
Holding costs ( thousand dinars) for the first model.

Table 9 .
Purchasing costs ( thousand dinars ) for the first model.

Table 10 .
Transportation costs ( thousand dinars) for the first model.

Table 11 .
Best order quantities for second model.

Table 12 .
Inventory levels for the second model.

Table 13 .
Holding cost ( thousand dinars ) for the second model.

Table 14 .
Purchasing costs ( thousand dinars) for the second model.

Table 15 .
Transportation costs ( thousand dinars) for the second model.