Bozulabilir mallar için optimal üretim planlaması

Abstract

ÖZET Bu çalışmada, zamanla bozulan malların üretim ve envanter problemleri incelenmiş ve özellikle bozulan malın üretim hızını bir planlama süresi içinde, toplam maliyet fonksiyonunun her anki değerini minimize edecek şekilde saptanması problemi ele alınmış ve optimal bir çözüm yöntemi kullanılarak Turbo Pascal dilinde bir yazılım geliştirilmiş ve ilaç sanayiine bir uygulaması yapılmıştır. Model lemede Weibull dağılım fonksiyonu kul lanı lmışt ır. v 1 1 1

OPT I HAL PRODUCTION PLANNING FOR DECAYING ITEMS SUMMARY Many models of the production and inventory problem were undertaken to determine a minimum cost schedule of production output levels in order to meet demand during a prespecified horizon. In these works, it is generally assumed that the rate of change in inventory level at a given time is equal to the difference between the production and the demand rates. However, some items produced are depleted not only by demand but also by decay, such as spoilage as in fruit, physical depletion as in highly volatile liquids, or deterioration as in electronic components or grain. In this work the problem of production planning in which the decaying items are produced over the finite period CO, T3, where T is the planning period is considered. First the types of cost models are introduced then some information about failure models is given followed by the approach used in this study. Finally, the developed software and an application to drug industry and a discussion of the results are presented. Production planning is concerned with the determination of the inventory and work force levels in order to meet the fluctuating demand. It is generally assumed that the physical resources of the firm is constant during the planning period. Since the times and amounts necessary for the best utilization of the resources are rarely the same as the ones imposed by the demand, a problem arises and production planning is concerned with finding a solution to this problem. Generally twelve months is chosen as the planning period. The most important factors taken into consideration during production planning are the costs. These may be classified as basic production cost, costs related to production rate, inventory holding costs and shortages. In order to support the management, models are utilized because these models enable the managers to visualize the unclear thoughts and allow them to examine all the variables related to the problem. These models are classified according to the assumptions they make about costs. 1 XLinear cost models are generally preferred because they can be easily solved using linear programming. Actually these models are less restrictive than they seem to be for any convex cost function can be approximated to the desired accuracy using line segments. This model is classified into two classes. 1) Constant labor model: In this model it is assumed that the work force is constant and hiring and firing to meet the demand fluctuations is not allowed. This model is not very suitable due to the fact that the number of constraints increase very fast. 2) Variable labor model: In this model, when it is feasible to hire or fire personnel in order to meet the fluctuations in demand during the planning period, labor can be taken as a decision variable. This model allows for shortages. For only one product the form of the optimal policies have been investigated and efficient algorithms have been deve 1 oped. The main advantage of the linear cost models is that they can be solved using linear programming because this method solves a problem having a large number of decision variables and constraints. Also, since it has a high performance in parametric and sensitivity analysis, suitable planning decisions can easily be made. The main disadvantage of these models is that the uncertainties in demand cannot be incorporated into the model in an open form. Another type of model is called the quadratic cost model. Since the derivative of a quadratic function is a linear function the derived decision rules generated using quadratic cost models have a linear structure and for this reason they are called linear decision rules. The first example of this type has been developed by Holt, Modigliani, Muth and Simon therefore this model is also known as HMMS model. In this model there are two decision variables work force and production, inventory is automatically specified due to the relation among these three variables. For this reason the optimum decision rules requires the determination of work force and labor that minimizes the quadratic cost function in every period. The main advantages of the quadratic cost models are that they allow the usage of a realistic cost structure, simplify the development and the solution of the linear decision rules, and when the expected demand forecasts are made in an unbiased manner linear decision rules minimize the expected cost therefore allowing the removal of uncertainties in an easy way. Their disadvantages are that the determination of the cost coefficients require complex forecast processesand the limitations imposed on the model size due to the computational complications that arise as the number of decision variables increase. Decision rules are insensitive to large errors in the cost parameters. This property is very appealing for the determination of the cost values accurately is very hard. For every item a state called failure which prevents the usage of the item for its intended purpose can be defined. This concept is a general one that is for example spoilage in fruits and cereals, physical depletion as in highly volatile liquids or deterioration as in electronic components can all be defined as f ai lure. For the failed item the time period that passes from an initial time t=0 to the moment it becomes useless is called lifetime and is assumed to be a random variable having a probability distribution function. Experimental work have shown that the value of T cannot be obtained from any deterministic model, i.e., under identical conditions identical items fail at different and unpredictable times. For this reason the use of a probabilistic model is the most realistic approach. As the probabilistic model generally the following are used: 1) Normal failure model; It is used to model the failures due to wear. 2) Exponential failure model: In this model the lifetime is represented by an exponential distribution. The failure rate is constant, i.e., the item does not wear out in time. 3) Gamma failure model: If failure is due to some random disturbance such as sudden drops or rises in electrical voltage or chemical decomposition, then this model is used. 4) Weibull Failure model: If the system is made up of many components then this is a suitable model for failure. The leakage of dry batteries, or decay of drugs can be modelled using weibull distribution. The model considered in this work is as follows: We consider a problem of production planning in which the decaying items are produced over the finite period £0,T], where T is the planning period. Generally the rate of change in inventory level at a given time is taken as equal to the difference between the production and demand rates. However as we consider decaying items a generalized equation containing a term for the decrease in inventory level due to depletion by decay is used. Two kinds of costs following the HMMS model are considered. The production cost contains two terms; the linear term is the production cost of raw material and direct labor that is proportional to production rate. The quadratic term is the deviation of the actual production xirate form the desired or target production rate. The inventory holding cost also contains two terms. The quadratic terms in these models can be interpreted as the penalty for deviation from the target va 1 ues. We define a current value total cost function during the finite period CO,T] because we are interested in finding the production policy of the interval that minimizes this function subject to the condition imposed by the generalized equation describing the change in inventory level together with the initial value of inventory level. This model is a dynamic optimization problem with continuously distributed time-lags. The necessary and sufficient optimality conditions are derived using continuous optimal control theory. We first define the current value hamiltonian with the adjoint function. With the help of a theorem we show that the optimality conditions are not only necessary but also sufficient. Using the defined relations and optimality conditions we give an algorithm for the computational procedure. For the development of a software for production planning using the aforementioned computational procedure Turbo Pascal 5.5 was chosen as the programming language due to its advantages such as structural programming, fast compilation and execution and language features that allow the usage of high resolution graphics. In the design stage the most important subject was the selection of the methods for the numerical solution of the various equations. Since the algorithm is basically iterative a lot of numerical integrals and derivatives have to be computed until convergence is achieved. Therefore a balance between accuracy and speed had to be observed. For the often repeated integral computations less accurate but fast and for the integrals that are computed only once very accurate but slow methods were used, this approach is justified by the fact that the number of iterations for the problems considered was less than 6, i.e., the loss of accuracy does not increase the number of iterations but the overall computation time is dramatically reduced. In order to make the program more user friendly after each iteration in windows on the screen curves for demand, production rate and inventory level are drawn. xxiThe software was implemented on an PC AT compatible machine with a 80287 numerical coprocessor, and a Hercules monochrome graphics card. An appl i an eye drop was year was chose distribution fun drug. The discr continuous one u the real cost companies, gues coefficients of noticed that t consideration i drugs. Therefor the determinatio function was als cation to the drug industry was made and chosen as the item to be produced. One n as the planning period. Weibull ction was used for the lifetime of the ete demand function was approximated by a sing cubic spline interpolation. Since of the drugs is kept secret by the sed rather than actual values for the the cost equations was used. It was also he companies actually do not take into n production planning e, experimental values n of the parameters of o not avai lab 1 e the decay of the that would enable the distribution After the examination of the solutions for the cases considered the following observations were made: 1) The inventory level initially increases and then slowly decreases as it approaches the planning horizon. 2) The production condition. 3) The either of increases, either of increases. rate converges to the terminal inventory level becomes smaller as parameters of the Weibull function production rate becomes larger as parameters of the Weibull function For further work the use of a more general cost function such as a model including the cost of a regular payroll, hiring and firing and overtime can be considered as can a mu 1 1 i -product situation. xi 1 1