Models and Algorithms for Throughput Improvement Problem of Serial Production Lines via Downtime Reduction

Abstract Throughput is one of the key performance indicators for manufacturing systems, and its improvement remains an interesting topic in both industrial and academic fields. One way to achieve improvement is to reduce the downtime of unreliable machines. Along this direction, it is natural to pose questions about the optimal allocation of improvement effort to a set of machines and failure modes. This article develops mixed-integer linear programming models to improve system throughput by reducing downtime in the case of multi-stage serial lines. The models take samples of processing time, uptime and downtime as input, generated from random distributions or collected from real system. To improve computational efficiency while guaranteeing the exact optimality of the solution, algorithms based on Benders Decomposition and discrete-event relationships of serial lines are proposed. Numerical cases show that the solution approach can significantly improve efficiency. The proposed modeling and algorithm is applied to throughput improvement of various systems, including a long line and a multi-failure system, and also to the downtime bottleneck detection problem. Comparison with state-of-the-art approaches shows the effectiveness of the approach. Supplementary materials are available for this article. Go to the publisher’s online edition of IISE Transactions.

The binary variables y g j,k for g = 1, ..., G j,k represent if x j,k ∈ (x g−1 j,k , x g j,k ], and y 0 j,k represents if x j,k is positive. The real value variables λ g j,k show the propotion of the distance between x j,k and x g j,k over the length of the interval if x j,k ∈ (x g−1 j,k , x g j,k ] or x j,k ∈ (x g j,k , x g+1 j,k ].

Appendix II: Continuous improvement algorithm
CI is used for comparison in Section 6.2.1. The procedure is shown in Algorithm 3. The input includes the budget B * and the stopping condition. For MP1, stopping condition is B = B * . For MP2, the budget B * is equal to +∞, and the stopping condition is T H ≥ T H 0 (1 + ∆T H * ). The input parameters r, s 0 , s min are related to the step length of improvement, s, at each iteration.
The step length s is initially set to be equal to s 0 , and each time the selected machine shifted to a different position from the last iteration, the step length is reduced to r(s−s min )+s min , where r ≤ 1 and s min is the defined minimal value of s. The procedure includes simulating the system (step 1), selecting the machine to be improved ( Step 2), and improving the selected machine with a certain amount s (step 3 and 4). Lines 18-19 assure that the budget can be fully used in MP1. The procedure stops when the stopping condition is true, or no machine is selected to improve at step 2. The machine to be improved is selected as the machine whose downtime is with the highest sensitivity and whose reduction amount does not reach the upperbound U j,k .
The way to calculate the sensitivity of each machine d j,k in line 5 is from Chan and Schruben (2008), with the dual optimal solution from Algorithm 2.
In Section 6.2.1, CI algorithms with eight sets of different values of s 0 , s min , r are implemented, and their values can be found in Table 2.  Get dual solutionū i,j with Algorithm 2 in the article. k(xj,k,ri,j,k,q) ∂xj,kū i,j |. y (j,k)l ← 1. U (j,k)l ← 0.

20:
l ← l + 1.  Tables 3 to 6 indicate buffer space b j , mean and Coefficient of Variation (CV) of cycle time (P), mean of uptime (T up ) and mean of downtime (T down ) of the systems in the numerical experimen. In all experiments, processing time is generated from truncated normal distribution on interval (0, 2CT), and time to failure is generated from Weibull distribution with shape parameter k = 2. Downtime distributions are varied in the different sections. In sections 6.1 and 6.2, repair time follows right-skewed triangular distribution. The relationship among lower limit a, upper limit b and mode c is b − c = 2(c − a), and the value of c − a is reported in the tables. In section 6.3, repair time follows exponential distribution.
MP1-S-Dual: The weights on arcs are equal to the time delays, and all the nodes e s i,j and e d i,j hold flows. The 'start' node is a source, and the 'end' node is a sink absorbing ϑ N unit flow.
Proof. Figure 4(b) presents the same element as in Figure 4(a), with dual variables in MP1-S-Dual labeled on the arcs, instead of the time delay in MP1-S. For the node e s i,j , the balance of its flow, i.e., the difference between input flow and output flow, is equal to s i,j + v i,j − u i,j − w i+bj−1,j−1 , the same as the lefthand side of constraint (7). Thus, (7) can be interpreted as   Lines 10-14 in Algorithm 1 assure that constraints (2) and (5) are satisfied. Lines 5-9 in Algorithm 1 assure that constraints (3) and (4) are satisfied. Therefore, Algorithm 1 provides feasible solution e s i,j , e d i,j of MP1-S. Line 17 asssures that constraint (1) is satisfied. Lines 6-7 in Algorithm 2 imply that u i,j + w i,j = (bu i,j + bw i,j )(s i,j+1 + v i+1,j ). Each event e d i,j is with only one input flow, i.e., bu i,j + bw i,j = 1, so u i,j + w i,j = s i,j+1 + v i+1,j , i.e., constraints (8) hold. Similarly, lines 4-5 in Algorithm 2 assure that constraints (7) hold. Line 1 assures that constraints (9) to (10) are satisfied. Therefore, Algorithm 2 provides feasible solution of MP1-S-Dual.
In the spanning tree represented by bu i,j , bv i,j , bw i,j , bs i,j , there is one and only one path from the start node to the end node. We denote the path as P.
Lines 6,8,11,13 in Algorithm 1 imply that each event occurring time e s i,j or e d i,j is calculated by adding a time delay, either 0 or t i,j , to its triggering event occurring time. Following a triggering path, the occurring time of the last event on the path can be calculated as the sum of all the time delay on the path. The occurring time of event e d N,M is equal to the sum of all the time delays on P. Thus, providing the solution in Algorithm 1, the objective function of MP1-S is equal to accumulated time delay along P N .
Lines 4-7 in Algorithm 2 imply that the input flow of one node is equal to its output flow.
Input flow of one node e s i,j or e d i,j is equal to sum of the flow absorbing by the leaves of its spanning tree. Since the end node is the only sink in the graph absorbing 1 N unit flow, the only arcs carrying non-zero flow are the arcs on P. Providing the solution in Algorithm 2, since the weights on the arcs are equal to the time delays, the objective function of MP1-S-Dual is equal