## 3 Application to Pull-Type Production System
### Mathematical Formulation  
Consider a production system \(Q=(D,E)\) has the set of nodes \(D=\{1,2,\ldots,n\}\). There are \(K\) different final products and \(m\) machines MC\({}_{1}\), MC\({}_{2}\), \(\ldots\), MC\({}_{\rm m}\). Each arc represents the transformation from one intermediate product to another and each node in \(D\) denotes a state of the product. Every transforming process uses one of the machines. Orders of the \(K\) final products are arrived at random intervals, and processing times of the \(m\) machines are also random with specified distributions. The time horizon is defined as \(H\). The available number of raw material is equal to the expected value of the total demand for \(K\) final products. Figure 3 shows an example of three product pull-type production system. There are two stages and each process is performed by one machine of either MC\({}_{1}\) or MC\({}_{2}\). Orders for the products may arrive in batches from any of the final nodes. If finished products are ready at final nodes, orders are fulfilled immediately. If no product is ready, a set of orders is delivered to some supplying nodesaccording to the workloads on machines. The workload is defined as the time used to produce the products which are waiting in the queue. A tally is kept on the work which is already appointed but not yet finished. For example, if an order arrives to node 5, it can be either fulfilled by semi-finished products at node 3 using MC\({}_{1}\) through process \"3-5\" or by those at node 2 using MC\({}_{2}\) through process \"2-5\". According to the workloads on these two machines, the production will be allocated to the machine with light load. Ties can be broken arbitrarily. To reduce the average lead time, all intermediate products can be processed in advanced. However, if there are too many raw materials being tied up in other products, some demands will not be satisfied. Thus, the goal of production system is to minimize the expected lead time as well as keeping a tolerable service level with high probability. Because the selections of machines, supplying nodes, and demanding nodes are not unique, the following three assumptions are used to decide the priority: (i) When a process \"\(a\)-\(b\)\" enters the queue of machine MC\({}_{k}\), the machine is reserved. The machine MC\({}_{k}\) cannot begin any new arrival process until process \"\(a\)-\(b\)\" is finished, even if it has to be idle until node \(a\) becomes available. (ii) The processing time of any product in a queue is known. Thus, each machine can determine the total workload. (iii) If a demand can be fulfilled by different supplying nodes on different machines, the machine with the minimum workload is chosen. The pull-type production system can be formulated as the following PCSOP. \[\min\ E[f(\mathbf{x})] \tag{24}\]  
\[\text{subject to}\ P(g(\mathbf{x})\geq b)\geq 1-\alpha \tag{25}\]  
\[\mathbf{V}\leq\mathbf{x}\leq\mathbf{U} \tag{26}\]  
where \(\mathbf{x}=[x_{1},\ldots,x_{n}]^{\text{T}}\) is the decision vector, \(x_{i}\) is the levels of work-in-process inventory at node \(i\) (i.e., the amounts of intermediate products to be processed ahead of time), \(\mathbf{V}=[V_{1},\ldots,V_{n}]^{\text{T}}\) denotes the vector of lower bound, \(\mathbf{U}=[U_{1},\ldots,U_{n}]^{\text{T}}\) is the upper bound vector, \(E[f(\mathbf{x})]\) represents the expected lead time of work-in-process inventories \(\mathbf{x}\), \(g\) denotes the service level, \(b\) is a pre-specified acceptable service level, \(\text{P}(g(\mathbf{x})\geq b)\) denotes the probability of the acceptable risk, and \(\alpha\) denotes the risk which violates the acceptable service level. Based on the above terminologies, we can reformulate the PCSOP as follows. \[\min\ \frac{1}{L}\sum_{j=1}^{L}f_{j}(x) \tag{27}\]  
\[\text{subject to}\ \frac{1}{L}\sum_{j=1}^{L}y_{j}\geq 1-\alpha, \tag{28}\]  
Figure 3: Three product pull-type production system.