Application Oftimal Control Theory for Inventory

Application Oftimal Control Theory for Inventory Production System Pardi Affandi1, Faisal2, Yuni Yulida3 1,2,3

Staf Pengajar Prodi Matematika, Fakultas Matematika dan Ilmu Pengetahuan Alam UNLAM, Banjarbaru e-mail: [email protected]

Abstract. In this paper, we are using optimal control approach to determine the optimal production rate in production Inventory System, first build the mathematical models. Some references theories apply theory to control the inventory problem is

Sprzeuzkouiski (1967), Hwang, Fan and Erickson (1967), Pekelman (1974), Bensoussan, Affandi P. (2011). The classical problem in the inventory problem is how to manage changes in consumer demand in a finished product. One such problem can be modeled and solved using optimal control techniques. Optimization problem that arises is how to balance between the interests of the smooth production and inventory storage price. Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method to lose control policy. This method largely inspired by the work of Lev Pontryagin and his colleagues in the Soviet Union and Richard Bellman in the United States. Explicit optimal control is obtained for the two general inventory levels depend inventory production. inventory is used more specifically limited to the production of inventory problems.

Key words : Inventory production Problem, Control Theory.

1. Introduction Many problems in the life involving systems theory, optimal control theory and some applications. One is the inventory, the problem is how to manage changes consumer demand in a finished product. So that the company should make good planning in order to produce goods in accordance with the number of requests. The finished goods should fit in a place before it was booked by consumers. This has led to the emergence of inventory that certainly will add to the cost in the form of storage costs such as the cost of physically storing goods or costs arising out of the company’s capital tied up in the form of goods. This problem can be modeled using mathematical optimal control techniques, that is optimal control. The classic problem in inventory related to changes in demand for products and the other problem is how to balance between the interests of the smooth production at a price of inventory storage. So that production can run smoothly while the cost of inventory storage can be arranged according to consumer demand. To finish this one using optimal control techniques. Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method to lose control policy. This method inspired by the work of Lev Pontryagin and his colleagues in the Soviet Union and Richard Bellman in the United States. One of the major work is Pontragyn’S Lev Pontryagin maximum principle or Pontryagin maximum principle. This principle has a pressure point on the necessary condition (Necessary condition) for optimality. Besides having to pay the actual inventory because there is the advantage of having the first inventory can quickly meet consumer demand both in the warehouse as a place to store inventory during periods of little demand and can serve requests back during high demand periods. Then one way to overcome the problems of inventory is the optimal control, differential equation models beforehand simplified. Various problems that involve many systems theory, optimal control and some applications. One of them is the inventory problem, the problem is how to manage the change in consumer demand in a finished product. Besides having to pay the actual inventory because there is the advantage of having the first inventory can quickly meet consumer demand both in the

warehouse as a place to store inventory during periods of little demand and can serve requests back during high demand periods. Optimization problem that arises is how to balance between the interests of the smooth production at a price of inventory storage. .

2. Theoretical Aspect 2.1 The set of convex and convex function The concept of a convex function underlie some part in the discussion section. The following definitions and theorems related to the set and convex function. Definition 2.1.1 (Mangasarian) The set such that for

called convex set if for to any .

and

Definition 2.1.2 (K.V. Mital) Suppose with A is set convex set . A function θ (x) is called a convex function in A if and only if for any two points x1,x2 ϵ A and every λ ϵ [0,1], apply Theorem

2.1.3

(Mangasarian) differenciable in for each

Given .

a If

set

open convex

. in

Function then

.

2.2 Nonhomogen Linear Differential Equation and Solution Definition 2.1.4 (Ross, S.L 1984) Differential equations are equations containing derivatives of one or more dependent variables for one or more independent variables. Definition 2.1.5 (Ross, S.L 1984) Order linear differential equations-n, with the dependent variable y, and the independent variable x, can be expressed as follows (2.1) With

not equal to zero. If F is equal to zero then the equation reduces to (2.2)

called homogeneous differential equation. For F(x) ≠ 0, referred to as non homogeneous differential equation. Theorem 2.1.6 (Ross, S.L 1984) Provided a solution to nonhomogeneous linear differential equation (2.1) which does not contain any constants. If general solution of linear differential equations homogeneous (2.2) so any solution of differential equations (2.1) can be expressed as for an election constants appropriate.

2.3 Differential Equation System Theorem 2.1.7 The transition matrix is a homogeneous linear system x = Ax solution e A(t − s ) . System of differential equations in the form , the solution can be completed in order to obtain . 2.4 Inventory model Specifically inventory includes raw materials, goods in work process and finished goods. To limit the scope of understanding of inventory, then in this paper inventories limited to the production of inventory in the form of finished goods. Inventory production is a stock manufactured goods used to satisfy and meet customer demand. So that limits this inventory has a narrower meaning than in the previously discussed inventory. It that no waste production, the need for careful planning and precise. Many made a useful way of production planning for system planning and scheduling needs of raw materials for production, and inventory in the company should also be known for production planning so that waste does not occur. In this paper, we discuss about the inventory of finished goods production. This deterioration is certainly cause loss expense for the company. So the company has the problem of how to optimize the amount of production of goods so as to meet customer demand at the same time that these items. 2.5 Optimal control Definisi 2.1.8 (M.Athans 1966) Optimal control problem for the system with the target S, the objective function , the set of admissible control U, and the initial state x0 at time t0 is decisive control u ϵ U that maximizes objective function . Any control u * which provides a solution to the problem of optimal control called optimal control. In the following discussion, the problems given in the case of optimal control with state end and the end time is known. In other words, the target set S shaped S = { }x{ } in the form with specially element in Rn and element at (T1,T2). Given the state system by the end and the end time unknown

with x(t) vector state sized , u(t) input vector sized , f a vector valued function. Initially given state is X0 and initially time is t0. Target set S form with known value and . Optimal control problem is to find the admissible control u(t) with the initial value and the final value that maximizes the objective function To solve the problems mentioned above optimal control, first determined necessary condition for optimal control are met.

3. Discussion Many company manufacturing using production inventory system are being made to regulate changes in consumer demand in a finished product. So the consumer and customer demand can be met, but also the production of goods is deterioration. The slump in production inventory may occur due to spoilage or damage to goods inventory of production for a certain period. Usually the production inventory is often experienced this deterioration is the production of food and drink. Because food and drinks that are too long to be in warehousing, the effect is that it can decay or damage, this is called degenerate. So the company must pay a slump. For that the company must make a plan so that products finished goods in the warehouse did not fit in deterioration before the booked by consumers and can manage inventories so as not to delay requests made by consumers. This has caused some problems, in addition to warehousing problems that certainly will add to the cost of storage also makes good planning so that the items are in storage until the deterioration. So the optimization problem that arises is how to balance between the level of production of goods in order to meet the demands of customers and consumers, but did not issue a production inventory storage costs and huge slump. 3.1 Model and Results Production Inventory Model In the early part first introduced the notations used are as follows I(t) : inventory levels at time t, P(t) : average production at time t, S(t) : average demand at time t, T : plan within a certain length of time, I0 : initial inventory level, : average goals deteriorating, : level inventory goals, : average production goals, c : positive unit production costs h : holding cost coefisien c : positive unit production costs λ : constant nonnegative diskont cost Interpretation of the inventory level I is to save stock production so the company can secure the demand with existing inventory items. For example , you can secure sales for the two months the demand of 100 units until the request can be served. Very similar to the amount of the production level can be interpreted so that more efficient production quantities as desired. With these notations we get the condition of the state model, the stock flow equation is diffrensialnya (t) = P(t)-S(t), I(0)= Io While inventory in time t will be increased with the passage of production and reduced by the value of sales as a function of demand. Then the objective function of the model are:

Interpretation of the objective function is we want to keep the inventory is closed to enable fit for purpose and also maintain the production value of p is closed to fit the purpose . Squared values stating that the final determination of the level of response goals are not covered in the first goal and P. Solutions with the principle of Maximum known relationship λ adjoint function with equation (t) = P(t)-S(t), I(0)= Io and we can write down the value of advanced functions Hamilthon as: In equation we can use negative (meaning not given piece (undiskon). So the minimum of J in equation is equivalent to a maximum of functionality - J. Then using the application Pontryagin maximum principle, the equation at diffrented and right segments equal to zero, it will be obtained:

This form will be obtained from the decision

So that the value of the right side is not negative. In most cases P limit is non-negative, so that from the shape is control optimal

In the rest of this discussion and so we will assume is so large that (6.4) will always provide value not negative of production value. Assuming a very large and I0 is quite small, we will get: and we will be able subsitusikan (6.4) to (6.1) so that would be obtained I(0)= Io 6.6 Adjoint equation with variable easily be obtained λ(T) = 0

6.7

From equation 6.6 is the initial value problem and the final value of 6.7 is the problem, so that both provide two point boundary value problem, we can use the method of simultaneous equations, which work only in a few cases, including this issue. This method uses the knowledge and tricks for solving simultaneous difrensial with differential and substitute up to one variable in elimination.

After 6.6 at diferensial especially by his use of the t, with the creation of the equation here, then using (6.7) for mngeliminasi and 6.6 to eliminate λ from the equation will be obtained = + = We make like 6.8 Where constant α give with

Equation 6.8 can be solved by a standard method, so that would be obtained as the following equation yields 6.8 with two real roots. and Note that

6.10 .

Then solution 6.8 6.11 Where Q (t) is the integral solution specifically from 6.8. Otherwise da additional requirements involving and From here will always assume that Q (t) as an integral customized solutions. Although the 6:11 has two constants and only one boundary conditions, to get one other boundary conditions we differensialkan 6:11 and we subsitusikan result into equation 6.6 so the solution is obtained in the form of: 6.12 Note that we get the boundary conditions in λ so that we can obtain a constant and for the next job we define two constants, namely:

Now the next step determine the boundary conditions in 6:11 and 6:12 the following:

and

6.13 6.14 solutions for

6.15 6.16 If we enter the negative and negligible and then we will get and

positive values, then when T is sufficiently large that it is 6.17 6.18

Note that for the value T which is great, is closed to zero and is closed to zero. However reason to restrain the exponential nature 6.18 that was multiplied in 6.12 that when a small t is small, become large and important when t is closed to T. Thus the value and press and with 6.4, 6.11 and 6.12 we now write down the expression for the value of I, P and λ. We will observe the expression of three parts, namely the starting of correction that is useful for value when t is small, exspression turnpike is significant for all value t and ending exspression useful when t is closed to T. Starting correction

Turnpike exspression

Ending correction

Note if b1 = 0 which means that I0 = Q (0), then here did not start correction. At the time of I0 = Q (0) is the start of inventory caused the solution to be faster. And the same way if b2 = 0, then the correction end vanished between the formulation and solution are in rapid time to end solution.

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