A cyclic production scheme for the synchronized and integrated two-level lot-sizing and scheduling problem with no-wait restrictions and stochastic demand

Abstract

In this paper, we propose a production scheme for a two-step packaging system as part of a make-and-pack production process including parallel production units in all stages. In the first stage of the packaging system, flavored liquids are filled into cans of different sizes which are immediately palletized in the second stage, i.e., work-in-progress inventories do not exist. However, each filling unit can feed more than one palletizer at a time. Final products can be stored in a warehouse with limited capacity. Among others, the proposed scheme consists of a periodic production sequence, also referred to as a cycle, for each production unit and a control strategy that keeps cycle lengths close to a target length. In addition, an approach to specifying the parameters of the scheme is developed. This approach accounts for sequence-dependent setup times, downtimes of production units, capacitated storage and uncertain demand for final products that is satisfied from stock or backlogged. We evaluate our approach conducting computational experiments that are based on real-world and random data.

Appendices

System properties and assumptions

See Table 4.

Table 4 Overview of the production system

Make-and-pack systems discussed in the literature

Table 5 Overview of the literature

Table 5 gives an overview of contributions on make-and-pack systems in the literature and the characteristics of the production planning problem considered in the paper at hand. For each of the relevant papers, it outlines what characteristics each of these contributions takes into account.

As mentioned in Sect. 1, demand is assumed to be deterministic in contributions that consider parallel packaging units. Vice versa, contributions that investigate production environments with a stochastic demand for final products focus on a serial supply chain. In most cases, stages are decoupled by a storage location facilitating synchronization of production and packaging processes. Furthermore, demand and supply of final products is decoupled by a storage location for final products. Sequence-dependent setup times are also part of most of the examined production environments whereas unit downtimes are only considered by Venditti et al. (2010). Remember that Briskorn et al. (2016), Rappold and Yoho (2014) and Vaughan (2009) focus on a single production unit.

A hierarchical framework comprising both a planning phase (production scheme) and an execution phase (detailed scheduling according to the scheme) combined with cyclic patterns is proposed only in a minority of contributions. Among these contributions, Günther et al. (2006), Rappold and Yoho (2014) and Vaughan (2009) consider a pure (P) rotation sequence, whereas Mehrotra et al. (2011), Soman et al. (2007), Bourland and Yano (1994) and Briskorn et al. (2016) propose a general (G) production sequence.

Nomenclature

$a_{(i,j)}$ :

Number of units of intermediate i necessary to produce one unit of SKU (i, j)

$\alpha _{(i,j)}$ :

Minimum service level of SKU (i, j)

$\alpha _{(i,j)}^r$ :

Service level of SKU (i, j) observed in simulation run r

C :

Total available storage space for SKUs

$C^{+}$ :

Median of the SR values of the additional storage space requirement found by simulation

$C^{+^r}$ :

Additional storage space requirement observed in simulation run r

$CL_z$ :

Target cycle length of subsystem z

${\overline{CL}}_z$ :

Average cycle length of subsystem z divided by the number of sections $NC_z$ in subsystem z

$\epsilon$ :

Parameter that defines the maximum permissible deviation of the cycle length from the target cycle length

$\eta ^m_{(m,(i,j))}$ :

Median of the SR values of the average production cutback at the mth PR of SKU (i, j) found by simulation

$\eta ^r_{(m,(i,j))}$ :

Amount by which the mth PR of SKU (i, j) is cut short on average in simulation run r

$F_g$ :

Frequency of group g

$F_{(i,j)}$ :

Frequency of SKU (i, j)

FU :

Set of filling units

${\overline{FU}}$ :

Set of correlated filling units

$FU_i$ :

Set of filling units that can handle a PR of intermediate i

${\tilde{FU}}_g$ :

Set of filling units that can handle the PR of an intermediate that is related group g

${\hat{FU}}_z$ :

Set of filling units in subsystem z

G :

Set of groups of SKUs

$h_{(i,j)}$ :

Inventory holding costs of SKU (i, j) per pallet and hour

I :

Set of intermediates

$IJ_g$ :

Set of SKUs forming group g

$IJ_{i,s}$ :

Set of SKUs that are based on intermediate i with a PR of i being assignable to the set s of suitable filling units

$I{\hat{J}}_z$ :

Set of SKUs that are an element of subsystem z

IC :

Median of the SR values of average inventory holding costs found by simulation

$IC^{*}$ :

Average inventory holding costs observed for the best candidate scheme found

$IC^r$ :

Average inventory holding costs observed in simulation run r

J :

Set of SKUs

$J^{f}$ :

Set that contains SKU $(i,j) \in I{\hat{J}}_z$ if $NC_z\ /\ F_{(i,j)} = f$ with $z=1,\ldots ,Z$

$\lambda _{(i,j)}$ :

Demand rate of SKU (i, j)

$LB_{(m,(i,j))}$ :

Minimum lot size for the mth PR of SKU (i, j)

M :

Big number

$NC_z$ :

Number of sections in subsystem z

$\omega _{(i,j)}$ :

Standard deviation of demand of SKU (i, j)

$\Omega _{(i,j)}$ :

Parameter that defines the maximum permissible deviation from the average lot size for a PR of SKU (i, j)

P :

Simulated period

$\pi _{(i,j),u}$ :

Maximum packaging rate of SKU (i, j) on packaging unit u

$\pi _{i,u}$ :

Maximum filling rate of intermediate i on filling unit u

PU :

Set of packaging units

$PU_{(i,j)}$ :

Set of packaging units that can handle a PR of SKU (i, j)

$PU_{(i,j),u}$ :

Set of packaging units that can handle a PR of SKU (i, j) and can be fed from filling unit u

$PU_u$ :

Set of packaging units that can be fed from filling unit u

$q_{(m,(i,j))}$ :

Lot size for the mth PR of SKU (i, j)

${\overline{q}}_{(m,(i,j))}$ :

Expected lot size for the mth PR of SKU (i, j)

$\rho _{g,u}$ :

Utilization rate of group g if the corresponding intermediate is produced on filling unit u

$\rho _{(i,j),u}$ :

Utilization rate of SKU (i, j) on packaging unit u

$RP_{(m,(i,j))}$ :

Risk period for the mth PR of SKU (i, j)

s :

Set of suitable filling units

$S_{(m,(i,j))}$ :

Base-stock level for the mth PR of SKU (i, j)

sf :

Safety factor

$SI_u$ :

Sequence of PRs on filling unit u

${\overline{\sigma }}_{CL_z}$ :

Standard deviation of the cycle length of subsystem z divided by the number of sections $NC_z$

SR :

Number of simulation runs

$st_{(m,(i,j))}$ :

Safety stock level assumed to be available in $t_{(m,(i,j))}$

$t_{(m,(i,j))}$ :

Expected time between the beginning of the mth and the beginning of the next PR of (i, j)

$TS_u$ :

Total setup time on unit u

$\theta _u$ :

Downtime rate of unit u

$UB_{(m,(i,j))}$ :

Maximum lot size for the mth PR of SKU (i, j)

x :

Base value for determination of lot size boundaries

Z :

Number of subsystems