With the purpose of easing the comparison efforts for readers, this section uses the same example as in Chiu et al. (2013) to demonstrate the proposed research result. Reconsider a production plan of producing five different items on a single machine in sequence under a common cycle time policy. Annual demand rates *λ*
_{
i
} for these five items are 3000, 3200, 3400, 3600, and 3800, respectively. The production rates *P*
_{1i} are 58,000; 59,000; 60,000; 61,000 and 62,000, respectively. During individual production process, there are random nonconforming rates *x*
_{
i
} for each item and they follow Uniform distribution over the intervals of [0, 0.05], [0, 010], [0, 0.15], [0, 020], and [0, 0.25], respectively. All nonconforming items produced go through a rework process at the rates *P*
_{2i} of 1800, 2000, 2200, 2400, and 2600 items per year, respectively; with additional unit rework costs *C*
_{Ri
} of $50, $55, $60, $65, and $70, respectively. During the reworking, there are failure-in-rework rates *φ*
_{i} of 0.05, 0.10, 0.15, 0.20, and 0.25, respectively. Additional values of system parameters are given as follows: *K*
_{
i
} = the setup costs are $3800, $3900, $4000, $4100, and $4200, respectively, *C*
_{
i
} = production costs per item are $80, $90, $100, $110, and $120, respectively, *C*
_{Si
} = disposal costs per item are $20, $25, $30, $35, and $40, respectively, *K*
_{1i
} = fixed costs per delivery are $1800, $1900, $2000, $2100, and $2200, respectively, *C*
_{Ti
} = unit transportation costs are $0.1, $0.2, $0.3, $0.4, and $0.5, respectively, *n* = number of shipments per cycle, it is assumed to be a constant 3 (i.e., *n* + 1 = 4), *h*
_{
i
} = unit holding costs are $10, $15, $20, $25, and $30, respectively, *h*
_{1i
} = holding costs per reworked item are $30, $35, $40, $45, and $50, respectively.

The optimal common cycle time *T** = 0.7279 (years) can be obtained by applying Eq. (22). Total expected system costs E[*TCU*(*T**)] = $2,013,956 can also be obtained from computation of Eq. (15). Variation of mean defective rate and mean failure-in-rework rate effects on the expected system cost E[*TCU*(*T*)] is illustrated in Fig. 4. It is noted that as mean defective rate increases the E[*TCU*(*T*)] increases significantly, and as mean failure-in-rework rate increases the system cost E[*TCU*(*T*)] increases slightly.

As stated earlier, the proposed model aims at reducing vendor’s inventory holding cost for each product *i* during the production cycle. As a result from this numerical example, the percentage of overall holding cost reduction is 24.7 % (i.e., from $109,476 (Chiu et al. 2013) down to $82,431). Figure 5 demonstrates the percentage of holding cost reduction for five different products, respectively as compared to that of Chiu et al.’s work (where *n*-delivery policy is adopted).

In summary, the proposed study realizes a significant system cost savings of $56,358 (i.e., $2,070,314 − $2,013,956) or 16.09 % of other system interrelated costs (i.e., E[*TCU*(*T*)] − *λC*, which is the expected system cost excludes variable manufacturing cost).