Comparison of optimal buffer allocation in flow lines under installation buffer, echelon buffer, and CONWIP policies

Abstract

We compare the optimal buffer allocation of a manufacturing flow line operating under three different production control policies: installation buffer (IB), echelon buffer (EB), and CONWIP (CW). IB is the conventional policy where each machine may store the parts that it produces only in its immediate downstream buffer if the next machine is occupied. EB is a more flexible policy where each machine may store the parts that it produces in any of its downstream buffers. CW is a special case of EB where the capacities of all buffers, except the last one, are zero. The optimization problem that we consider is to maximize the average gross profit (AGP) minus the average cost (AC), subject to a minimum average throughput constraint. AGP is defined as the average throughput of the line weighted by the gross marginal profit (selling price minus production cost per part), and AC is the sum of the average WIP plus total buffer capacity plus transfer rate of parts to remote buffers, weighted by the inventory holding cost rate, the cost of storage space, and the marginal cost of transferring parts to remote buffers, respectively. Numerical results show that the optimal EB policy generally outperforms the optimal IB and CW policies. They also show that as the production rates of the machines decrease, the relative advantage in performance of the EB policy over the other two policies increases. When the cost of transferring parts to remote buffers increases, the dominance of the EB policy over the IB policy decreases while the dominance of the EB policy over CW increases.

Appendices

Appendix 1: Numerical tests for the 8-machine line verifying the concavity of the net average profit under the EB policy

Tables 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 and 22 present the buffer capacity designs under the EB policy that resulted in each iteration of the two-phase optimization algorithm, for scenarios $L_{1} ,T_{3} ,$ and $O_{1}$–$O_{11}$ of the 8-machine line. Each row shows the iteration number, the buffer capacity design at that iteration, and the eigenvalues corresponding to the Hessian matrix of partial derivatives of the average net profit with respect to the buffer capacities at that design. The buffer capacity design in the last row of each table is the optimal design shown in Table 8. In all cases, all the eigenvalues of the Hessian matrix are negative, indicating that the average net profit at the buffer capacity designs examined is concave.

Table 12 Numerical results per iteration for scenario $L_{1} ,T_{3} ,O_{1}$ of the 8-machine line, under the EB policy

Table 13 Numerical results per iteration for scenario $L_{1} ,T_{3} ,O_{2}$ of the 8-machine line, under the EB policy

Table 14 Numerical results per iteration for scenario $L_{1} ,T_{3} ,O_{3}$ of the 8-machine line, under the EB policy

Table 15 Numerical results per iteration for scenario $L_{1} ,T_{3} ,O_{4}$ of the 8-machine line, under the EB policy

Table 16 Numerical results per iteration for scenario $L_{1} ,T_{3} ,O_{5}$ of the 8-machine line, under the EB policy

Table 17 Numerical results per iteration for scenario $L_{1} ,T_{3} ,O_{6}$ of the 8-machine line, under the EB policy

Table 18 Numerical results per iteration for scenario $L_{1} ,T_{3} ,O_{7}$ of the 8-machine line, under the EB policy

Table 19 Numerical results per iteration for scenario $L_{1} ,T_{3} ,O_{8}$ of the 8-machine line, under the EB policy

Table 20 Numerical results per iteration for scenario $L_{1} ,T_{3} ,O_{9}$ of the 8-machine line, under the EB policy

Table 21 Numerical results per iteration for scenario $L_{1} ,T_{3} ,O_{10}$ of the 8-machine line, under the EB policy

Table 22 Numerical results per iteration for scenario $L_{1} ,T_{3} ,O_{11}$ of the 8-machine line, under the EB policy

Appendix 2: Additional numerical results for the 20-machine line with randomly chosen input parameters

Table 23 shows the randomly generated parameter values for instances 41–80 of the 20-machine line, and Table 24 and shows the optimization results for all these instances.

Table 23 Values of production rates $p_{n} ,n = 1, \ldots ,N$, and basic parameters $I_{c} ,I_{h} ,I_{r} ,I_{b} ,I_{\nu }$ and $I_{t}$ for instances 41–80 of the 20-machine line

Table 24 Numerical results for instances 41–80 of the 20-machine line

Appendix 3: Numerical results for the 8-machine line with base production probabilities 0.4 and 0.8

Table 25 shows eight additional scenarios for the production rates of the 8-machine flow line example considered in Sect. 6.2. These scenarios are denoted $L_{5} , \ldots ,L_{12}$ and are similar to scenarios $L_{1} , \ldots ,L_{4}$ shown in Table 5, except that they use a different base production probability than 0.6, which is used in scenarios $L_{1} , \ldots ,L_{4}$. More specifically, scenarios $L_{5} , \ldots ,L_{8}$ use base production probability 0.4, while scenarios $L_{9} , \ldots ,L_{12}$ use base production probability 0.8.  As in Table 5, the production probabilities of the slowest machine in scenarios L₆-L₈ and L₁₀-L₁₂ are shown bold.

Table 25 Production rate scenarios with base production probability 0.4 and 0.8 for the 8-machine line

We solved problem (1)-(2) for all combinations of scenarios L₅–L₁₂, T₁–T₃, and O₁–O₁₁, i.e., for a total of $8 \times 3 \times 11 = 264$ instances. Tables 26, 27, 28, 29, 30, 31, 32 and 33 show the optimization results for all these instances.

Table 26 Numerical results for scenario $L_{5}$ of the 8-machine line

Table 27 Numerical results for scenario L₆ of the 8-machine line

Table 28 Numerical results for scenario $L_{7}$ of the 8-machine line

Table 29 Numerical results for scenario L₈ of the 8-machine line

Table 30 Numerical results for scenario L₉ of the 8-machine line

Table 31 Numerical results for scenario L₁₀ of the 8-machine line

Table 32 Numerical results for scenario L₁₁ of the 8-machine line

Table 33 Numerical results for scenario L₁₂ of the 8-machine line