Integrated Procurement-Production Inventory Model with Two-Stage Production

Inventory is one of the main factors that become a problem in the production process [1]. Inventory is usually considered an asset in the company [2]. Since economic order quantity (EOQ) supply control was introduced in 1912, many researchers have developed models for solving various supply problems [3] [4] [5]. Conventionally, the economic lot size of the raw material is determined separately from the finished product. This decision leads to the resulting sub-optimization solution [6]. The demand for raw materials depends on the production plan for the finished product [7]. Because of this, Goyal [8] was the first researcher to develop the Economic Order Quantity (EOQ) formula into an Integrated Procurement-Production (IPP) system. In this IPP model, the ARTICLE INFO ABSTRACT

The production process requests as much 1 semi-finished material. The semi-finished material is processed at the 2 production rate. Because it is assumed that one unit of the finished product requires one unit of half-finished material, so 1 is equal to 2 . From the second stage, as many 2 finished products are produced. In the two-stage manufacturing process, raw materials are sent m times. In Fig. 2 it is explained that the factory places orders for raw materials by the ( 0 ) number of orders and the 0 0 ⁄ order cycle. The production rate of the first process is 1 . Since it is assumed that one unit of semi-finished material requires one unit of raw material, so 0 is equal to 1 . Therefore, the ordering cycle is the same as 0 1 ⁄ . Furthermore, the inventory of semi-finished materials, where the raw materials are processed into WIP materials during 1 ( 1 = 1 1 ⁄ , in which 1 is the quantity with the production rate ( 1 )). During 2 , the number of WIP will run out continuously. on the WIP system is the same as ( 1 − 2) ) * 1 1 ⁄ . In the finished product inventory at the factory, WIP is processed at ( 2 ) production rate within production limits ( 2 ) to meet demand during production (T), or 2 = 2 * that 2 is equal to 2 * 2 ⁄ . The finished product is shipped in batches (by 2 transfer batch size) of (n) times during production. Therefore, the finished product order cycle in retailer is equal to 2 2 ⁄ The Integrated Procurement Production (IPP) model aims to provide a policy for supplying raw materials, in-process products, and finished products. Fauza, et al. [29]. The development of the IPP system model needs to determine the assumptions and notations used. The assumptions of this supply model are: 1. The production rate and the level of demand are constant and deterministic where 1 > 2 > 2 2. Shortage and backorder are not allowed 3. Ignoring waiting time transportation 4. There is no loss of quality 5. One finished product requires one WIP unit, and one WIP unit requires one raw material The notation used in this problem is 1 : production rate of semi-finished products production (units/month) 2 : production rate of finished products production (unit/month) 0 : the number of raw material demands (units/month) 1 : the number of demands for semi-finished products (units/month) in which : the total cost of finished products inventory system (IDR/time unit) JTC : the total cost of the integrated inventory system (IDR/time unit) JTR : the total revenue/income of integrated inventory system (IDR/time unit) JTP : the total profit of the integrated inventory system (IDR/time unit) m : frequency of raw materials demands (times/demand) T : time during the production cycle (one time) n : frequency of finished products shipping (times/time unit)

Model Development
This section describes the development of an IPP system model to maximize profit. The development of a mathematical model of the problem can be formulated as follows: Equation (1) -(3) performs an average inventory starting from ( 0 ) raw materials, ( 1 ) in-process products, and ( 2 ) finished products. Equation (4) calculates the cost system of total raw material inventory, which consists of raw material purchase costs, transportation costs of raw material delivery, raw material storage costs, and the cost of decreasing raw material quality. Equation (5) is the calculation of the total cost of inprocess products. The cost of finished product inventory is done by adding up WIP costs, installation costs for finished processing products, storage costs for finished products, transportation costs of finished product delivery, and finished product purchases as in equation (6). Meanwhile, equation (7) is the calculation of the total income in an integrated inventory system.
From equations (1) - (7), the total profit of the integrated inventory system can be calculated by using the following mathematical model: Constraint P ≥ D (9) T > 0 (10) m,n > 0 (integer) (11) Equation (8) is the objective function in the model development of the IPP system. This objective function aims to determine the maximum total profit. Equation (9) -(11) is a delimiting function.

Dragonfly Algorithm
This study proposes the Dragonfly Algorithm method to help solve problems. Dragonfly Algorithm is an algorithm that is inspired by the behavior of dragonflies. Dragonflies are luxurious insects. There are nearly 3000 different species of these insects worldwide [28] [30] [31] [32]. The main inspiration of the DA algorithm comes from clustered behavior in static and dynamic manners. These two clustering behaviors are very similar to the two main phases of optimization using meta-heuristics: exploration and exploitation [30]. The notations needed in Dragonfly Algorithm include:

Si
: the avoidance of static collisions from one individual to another Ai : indicates the suitability of an individual's velocity for another in the environment Ci : the tendency of the individual towards the center of the environment mass Fi : attraction to food sources Gi : disturbance of enemy emersion : calculation of Xt, and ∆ +1 ( ) : the dragonfly flying direction around the search room using a random path Dragonflies form sub-swarm and fly over different areas in static swarms, which is the main objective of the exploration phase. However, dragonflies fly in larger groups and together in one direction, advantageous in the exploitation phase. These two phases are mathematically implemented in the following sections: Wherein X is the individual's current position, Xj shows the position of the individual neighbor j-th, and N is the number of individual neighbors.
Wherein Xj indicates the velocity of the individual neighbor j-th.
Wherein X is the individual's current position, N is the number of environments, and Xj represents the individual's position to the j neighbor.
Wherein X is the individual's current position, and X + indicates the position of the food source.
Wherein X is the current position of the individual, and X-indicates the enemy position.
Wherein s denotes the weight of the separation, Si shows the individual separation to i-th, a is the alignment weight, A is the alignment of the individual to i-th, c represents the weight of the cohesion, Ci is the cohesion of the individual to i-th, f is the food factor, Fi is the individual's food source to i-th, g is the enemy factor, Gi is the individual's enemy position to i-th, w is the inertia weight, and t is the iteration counter.
Wherein t is the current iteration. Where t is the current iteration, and d is the dimension of the position vector.

Fig. 3. Pseudo-code Dragonfly Algorithm
In this two-stage IPP problem, the number of dimensions used is 3. This number is based on the decision variables m, n, and T. m and n variables are discrete variables. While the T decision variable is a continuous value. This study proposes a continuous conversion of numbers to discrete, as illustrated in Fig. 4. For example, the conversion provision is that the value of m and n is an integer with a range of 1 to 100. At the same time, T is a continuous number with a range of 0 to 1. The m and n variables are rounded up if a decimal value is 0≥0.5. Otherwise, it will be rounded down.

Experimental Data and Procedures
In this section, data collection and experimental procedures are described. A case study was conducted in a manufacturing company in Indonesia. There are several data parameters used in the study. The required data parameters consist of 1 , 2 , 0 , The objective function of the IPP is to maximize the total system profit (Equation 8). This experiment uses a population and iteration number of 10, 50, and 100. The dimension meant here is the decision variables used in the IPP. There are three dimensions used, namely the frequency of raw materials demands (m), the frequency of the finished product delivery (n), and the time of production cycle (T). The upper and lower limits are used as the value limits for the decision variables. The optimal solution is also compared to the GA algorithm used, namely crossover rate of 0.7 and migration of 0.2 for every 20 generations, referring to the research of Nagib, et al. [33].
This research uses demand, production, and costs which can be seen in 2. The setting of the Dragonfly Algorithm (DA) parameter value consists of; Population : 100 Iteration : 100 The lower limit : [0 0 0] The upper limit : [10 10 1] Experiments were carried out using the MATLAB R2014a software. After knowing the value of the decision variable that produces the best profit, an analysis of m, n, and T's values is carried out on the resulting profit. Analysis of the decision variables m and n was tried with ten different values. As for the T value, a sensitivity analysis was also carried out with a value range of 1 to 0.1844. The sensitivity analysis procedure is as follows: When one of the decision variables is changed, the other decision variables are set to the optimal decision variable value. This sensitivity analysis is used to determine the effect of the decision variable value on profit.

Effect of Iteration and Population on Profit
The Dragonfly Algorithm (DA) method is used to process the data obtained in Table  2 using the MATLAB R2014a software. The results of the software calculation experiments are presented in Table 2. The experimental results in Table 2 shows that the profit increases when the population used is getting bigger. Conversely, when the population is used smaller, the profit generated will also be smaller. The biggest profit obtained was 569,380,000 IDR on 10, 50, and 100 iterations with a population of 100. Table 3 also shows decision (m), (n), and (T) variables. The decision variable in 50 iterations 100 population, 100 population 50 iterations, and 100 iterations and 100 population show the same results because the profit obtained is maximum.
The total profit calculation results using the Dragonfly Algorithm are equal to 569,380,000 IDR in the integrated inventory system. By using the company's existing solution, the company's profit is 560,740,000 IDR. Based on the results of the proposed solution, the efficiency is 1.52%. It shows that the total profit with Dragonfly Algorithm is more efficient.

Effects of Changes in m, n, and T on Total Costs
The effect of changes in m, n, and T on total costs is presented in Table 3 to Table  5. Sensitivity analysis is used to determine the effect of changes in (m), (n), (T) decision variables on TP, JTR, TC0, TC1, TC2. From the results of the analysis in Table 3, the selected m variable is 1. When m is increased, the total profit (TP) will decrease, JTR is constant, TC0 increases, TC1 and TC2 are constant. The m variable value affects the cost of TC0. This result is in accordance with the research of Fauza, et al. [19] From the results of the analysis in Table 4, the selected n variable is 2. When n is increased, the total profit (TP) will decrease, JTR, TC0, and TC1 remain constant, and TC2 will increase. These results indicate that the n variable value affects TC2 costs. This result is in accordance with the research findings of Fauza, et al. [19]. From the analysis results in Table 5, the optimal T variable is 0.8844. When T is lowered and increased from the optimal value of T, the total TP profit decreases.  Table 4. The effect of changes n on total costs (IDR)  n  TP  JTR  TC0  TC1  TC2  1  568,550,000  1,063,800,000 421,910,000 31,559,000 41,784,000  2  569,380,000  1,063,800,000 421,910,000 31,559,000 40,951,000  3  569,210,

Conclusion
This study proposes a two-stage production of the IPP model to maximize profit. This study also proposes DA as a procedure to solve problems. The research results on the effect of DA parameters on profit show that the greater the iteration and population used, the more maximum the results will be obtained. The sensitivity analysis shows that the m, n, and T variables influence total profit. Some of the limitations of this study are the demand that is assumed to be constant and deterministic. In addition, backorders and shortages are not allowed. For further research, the stochastic characteristics of demand need to be examined in this study. The model needs to consider backorders and shortages.