# Modeling of control loop in production sheduling and inventory level control process.

Abstract: Performance of manufacturing organization depends on production scheduling which is an essential part of the management of production systems. Effective scheduling can lead to performance those results in meeting the company's customer service goals, and reducing work-in-process inventories (Wiers, 1997). Objectives of production control can be graphically represented over time in diagrams: inventory, lead time, utilization, input orders and output of production process. In this paper, the fact that the input and output amount of work coming from released orders of one product are not equal in every period, requires a technique to balance input against output continuously, and to establish a control loop.

Key words: scheduling, control loop, inventory, optimization.

1. INTRODUCTION

The input and output amount of work coming from released orders are not equal in every period, so any shop floor control system has to look for a technique to balance input against output continuously, and to establish a control loop (Wiendahl, 1994). The scheduling is carried out by an employee, a computer program, or a combination of both and acts as a controller, fixing the planed status and thus leading in the control line to the actual production process. The process is monitored by feedback records of the inventory level so as to minimize influence of disruptive factor that is inconstant released orders. The aim of this paper is to define adequate mathematical model which is used to determine the influence of input parameters, release orders quantity and forecast on inventory level. The closed-loop scheduling and control cycle responsible for a continuous flow of manufacturing process and inventory level maintenance is one of internal parts of production scheduling and control process.

2. PRODUCTION PROCESS SIMULATION MODEL

To schedule production of one product, wood casement with standard dimensions and shape, which consists of three production processes, the simulation model has been made. The material flow in production process, information flow, and all process data necessary for building a simulation model is shown on Fig. 1.

[FIGURE 1 OMITTED]

The examined model, created using ProModel software tool, has all necessary properties of production process. The total lead time L/T for the product is 2.6 working days. The inventory level in the warehouse of finished products is set to value which is suitable for fulfilling the costumer orders for at least 7 days, according to the current orders quantity. The additional constraint is limit of inventory level on one hundred products in case that orders for 7 days are smaller then that inventory level quantity of one hundred finished products. The input and output curves of the work center within a period generally do not follow a straight line, and vary quite strongly at times, and leads to producing an unsteady inventory level. Because of that, simulation of the model was examined and optimized for the orders entered into simulation with external file generated by random number generator with addition of noises in generation like difference of order level for different periods of year. Orders used in simulation for optimizing the production plan are shown on Fig. 2. for one year period.

Input parameters used to establish a closed loop production control have been evaluated at the end of the scheduled week, at Wednesday morning. The production demand according to the weekly plan quantity Q can be expressed by a linear equation:

Q = [x.sub.0] + [a.sub.1] [x.sub.1] + [a.sub.2] + [x.sub.2] + [a.sub.3][x.sub.3](1)

The value [x.sub.0] is evaluated at the end of scheduled week, and gives a production plan according to the produced quantity in the last week of simulation. This is the main member of equation and gives a constant production plan in case those other parameters in equation are equal to zero.

The first parameter [x.sub.1] is the difference between quantity in orders from last simulated week and the week before. The evaluated number is then multiplied with the factor of signification [a.sub.1]. The value of this product, gives the partial influence on the production demand for next week of simulation on the way that it increases or decreases the production demand compared to last week of simulation.

The second parameter [x.sub.2] is difference between current inventory level in the final product warehouse and calculated inventory level according to the last evaluated daily order quantity multiplied by 7 days, or constraint of one hundred finished products.

[FIGURE 2 OMITTED]

The third parameter [x.sub.3] is the difference between the current production demand and forecast for next three months gathered by the costumer. Demand forecasting is frequently different from the actual production plan (Nishioka, 2003). Those three months forecast is based on the expected orders in next three months period according to the external file order list, but modified with normal distribution factor which make those data different then future orders quantity.

3. OPIMIZATION PROCESS

Optimization of the model has been executed by changing input factors and monitoring a result of an objective function. The objective function is the inventory level in the final product warehouse. According to the lean production principles, excessive production and high inventory level are the biggest waste in production process (Rother & Shook 2003). Inventory level higher then it is necessary for fulfilling the costumer demands, leads to a additional costs in warehouse. To decrease overall costs, the inventory has to be maintained at the lowest possible level, but high enough to realize all customer orders in next period.

Simulation has been executed with a goal to minimize number of product days in the final product warehouse, by changing factors of significations a1, a2 and a3. These factors, together with examined parameters, make changes in production week plan quantity. Optimal factors of significations are given in Table 1.

The optimal mathematical model for the next week plan for the shop floor can be expressed by equation:

Q = [x.sub.0] + 0.13[x.sub.1] + 0.08[x.sub.2] + 1.4[x.sub.3] (2)

During simulation ProModel software constantly monitors content of finished product warehouse. The content is daily evaluated from the simulation and added to accumulated value in previous period. The value of total inventory shows how many finished product days are stored in the warehouse.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

On the Fig. 3. is shown inventory level in the finished product warehouse over one year simulated period. The inventory level is unsteady, due to a very big variation in orders quantity. In case of examined production process, customer demands increases up to five times different value in period within one month. The optimal mathematical model handled this situation with rapidly increasing of production demand on shop floor. After normalizing the order quantity on the average level, the model needs additional period for stabilization of inventory level. On Fig. 4. is shown the weekly production plan generated by optimal mathematical model.

Optimal mathematical model generated by optimization of production process gives an optimal reaction of production on shop floor on varying released orders quantity. Examination of mathematical model with additional different orders lists generated by random number generator gives also the optimal results with optimal mathematical model control.

The future research will be in the direction of implementation of this or similar simple mathematical models into simulation with two or more different products processed on the group of work centers on the same shop floor. In manufacturing systems with a wide variety of products, processes, and production levels, production schedules can enable better coordination to increase productivity and minimize operating costs (Herrmann, 2006).

4. CONCLUSION

The aim of this paper was to define adequate mathematical model which is used to determine the influence of input parameters, release order quantity and forecast on inventory level in the finished product warehouse. Conclusion can be made that the mathematical model, which is used in control loop, has to be optimized to give best response on actual released orders quantities. If mathematical model has the members of equation which gives a faster, or slower response then optimal, the production plan for next week will be or insufficient, or excessive for fulfilling the costumer demands. For given input parameters from simulation model in the mathematical model for production plan, any other factors of signification values will give the larger total inventory value which makes that mathematical model for control not good as optimal one. For the production processes with different lead times the speed of response on changes in order quantity, and inventory level, will be different, and the mathematical model can be optimized by the simulation for that particular process. This approach in production scheduling can be used for encouragement of production control personnel in defining the production plans for the shop floor.

5. REFERENCES

Herrmann, W. (2006). Impr oving Production Scheduling: Integrating Organizational, Decision-Making, and Problem-Solving Perspectives, Available from: http://www.isr.umd.edu/Labs/CIM/projects/ierc2006/IERC-2006.pdf Accessed: 2007-06-14

Nishioka, Y. (2003). Collaborative Agents for Production Planning and Scheduling, Available from: http://www.pslx.org/en/doc/TR-001.pdf Accessed: 2007-06-12

Rother, M. & Shook, J. (2003). Learning to See, The lean enterprise institute, ISBN 0-9667843-0-8, Brookline

Wiendahl, H. (1994). Load-Oriented Manufacturing Control, Springer-Verlag, ISBN 0-387-19764-8, Berlin Heidelberg

Wiers, C. (1997). Human-Computer Interaction in Production Scheduling, Technische Universiteit Eindhoven, ISBN 90-386-0355-X, Eindhoven
```Table 1. Optimal factors of signification.

[a.sub.1] [a.sub.2] [a.sub.3] Total inventory [product day]
0.13 0.08 1.4 37887
```