## 3 Application to Pull-Type Production System  
### Pull-Type Production System  
In some manufacturing environment, various products are made from the same raw material in many stages of manufacturing processes. Some intermediate states of semi-finished products may be shared by some product family members. A pull-type production system is a lean manufacturing strategy used to reduce waste in the production process [23; 24]. Since components are just replaced once after they were entirely consumed in the manufacturing process, manufacturers will only supply enough products to fulfill customer's demands. Thus, all resources of the company are used to produce components which can be sold immediately and obtain a benefit. A pull-type production system starts with the order of customers, then it works backwards and adopts visual signals to facilitate operation in each previous step of the production process. The product is pulled through the production sequence based on the demands of consumer.  
Pull-type production systems are designed to respond quickly to the demand change. The downstream machines pull the production from the upstream ones based on the demands created at their output buffers. Some kinds of message are sent to the upstream machines to indicate the demand for a particular component. When production stages do not need a specified resource, locating manufacturing equipment at essential stages and processing parts ahead of time can reduce the lead time using a low work-in-process level. After each operation, the product becomes a specific state that is closer to its final state. The closer the product is to its final shape, the shorter the lead time needs to produce it to a finished product. Nevertheless, when the product becomes closer toward its final shape, its flexibility to transform into a comprehensive final products is decreased. Suppose the demand of final products is uncertain, a good compromise between the service levels and lead times could be accomplished if inventories of semi-finished products were held in intermediate stages. If the production system is operated in a pull-type environment, this consideration will obtain a better result.  
The goal of pull-type production system is to find the optimum work-in-process inventory for minimizing the average lead time as well as: (i) satisfying uncertain demands; (ii) maintaining the inventory levels at desirable values; and (iii) keeping a tolerable service level [25]. However, the considered problem is obviously more complex since the resources are shared at several stages of production. When stages need more rare resources to be supplied with more levels of semi-finished products, appropriately allocating the work-in-process inventories is required. Because each operation must share specified resource with other activities, it is difficult for existing optimization approaches to determine an optimal solution quickly. Furthermore, the probabilistic inequality constraint of pull-type production system significantly influences the efficiency and effectiveness during the search processes. In this section, we focus on the following two subjects: (i) convert the pull-type production system to a PCSOP; and (ii) apply the OOTSA approach to determine the optimal in-process inventories for minimizing the average lead time and maintaining an acceptable service level.