Economic production quantity (EPQ) model in ‘pull’ managed single-machine multi-item production systems

Abstract

For decades researchers have been facing the issue of adapting the economic production quantity (EPQ) to the case of multi-item production contexts characterised by a single (shared) resource with finite capacity. The economic lot scheduling problem (ELSP), which is still of interest to researchers, has addressed this issue. A recent attempt by Rossi et al. (Omega 71:106–113, 2017) addressed the problem while avoiding scheduling. Notwithstanding their relevance, these approaches present limitations in adapting the EPQ model to multi-product ‘pull’ production systems. The present work attempts to overcome these limitations through the development of a methodology based on the equation proposed by Mallya (1992) and restricting items production frequencies to define feasible solutions while avoiding scheduling. The feasibility and performance of the proposed model are evaluated through its application to well-known benchmarking instances (Bomberger’s, Eilon’s and Mallya’s problems) and a large set of test problems.

Appendices

Appendix A.

The equations that explain the proposed model, from Eq. (6) to Eq. (9), are reported in the following.

Given the relation $f_i={\hat{f}}/n_i$, from Eq. (2), the batch quantity of any item ($Q_i$) is that which ensures the satisfaction of demand ($D_i$) over the period of time between its subsequent processing ($\frac{T}{f_i}$), which depends on its processing frequency ($f_i$), and is modelled by Eq. (A.1a).

As the time interval in which all items are produced at least once (T) is the same for all items processed by the single machine, $Q_i$ and $Q_j$ can be related as in Eq. (A.1b).

The ratio between the common time interval (T) and the maximum frequency (${\hat{f}}$) is large enough to accommodate the production of all items once, Eq. (A.1c). This constraint avoids run-out, while allowing different consumption times.

Combining Eq. (A.1a) and Eq. (A.1c), $Q_i$ can be expressed as in Eq. (A.1d).

Exploiting Eq. (A.1c), it is possible to calculate the time interval (T) from items setup times ($s_i$) and demand and production rate ratio parameters, ratio of maximum frequency (${\hat{f}}$) divided $f_i$, ($n_i$), and maximum frequency (${\hat{f}}$) variables. From Eq. (A.1e), the denominator sets the feasibility constraint for the pull inventory management system, expressed by Eq. (7).

$$\begin{aligned} Q_i&= \frac{T \cdot n_i \cdot D_i}{{\hat{f}} }&\forall i \in I \end{aligned}$$

(A.1a)

$$\begin{aligned} Q_j&= \frac{Q_i \cdot n_j}{D_i \cdot n_i} \cdot D_j&\forall i, j \in I \end{aligned}$$

(A.1b)

$$\begin{aligned} \frac{T}{{\hat{f}}}&=\sum _{ {i\in I}} \Bigl ( \frac{Q_i}{P_i} + s_i \Bigr ) \end{aligned}$$

(A.1c)

$$\begin{aligned} Q_i&= \frac{D_i \cdot \sum _{ {j\in I}}}{s}_j{\frac{1}{n_i} - r_i - \sum _{ {j \in I | j \ne i}} \Bigl (r_j \cdot \frac{n_j}{n_i}\Bigr ) } \quad&\forall i \in I \end{aligned}$$

(A.1d)

$$\begin{aligned} T&= {\hat{f}} \cdot \frac{\sum _{ {i\in I}} s_i}{1 - \sum _{{i\in I}} r_i \cdot n_i} \quad&\forall i \in I \end{aligned}$$

(A.1e)

Appendix B.

1.1 Appendix B.1. Eilon’s Problem (1962)

This test reference is used since together with Bomberger’s 10-items problem. As in the original version $\sum r_i > 1$, we use the modified instances provided in Grznar and Riggle (1997). In the considered instance, six items are considered and different utilisation levels, $\sum r_i$, characterise the instances. In particular, demand set 1 (D1) corresponds to 0.22 utilisation, demand set 2 (D2) corresponds to 0.44 utilisation, demand set 3 (D3) corresponds to 0.67 utilisation, and demand set 4 (D4) corresponds to 0.90 utilisation. The data set corresponding to the instances is presented in Table 5.

Table 5 Data set of Eilon’s Problem modified by Haessler and Hogue (1976)

Applying the proposed pull model to the four instances derived from Eilon’s problem, the obtained optimisation models are characterised by six integer variables and six continuous variables. Owing to the small number of variables, the proposed pull models are optimally solved by localsolver 9.5 running on a desktop computer in less than a fraction of second. The obtained sets of $Q_i$ and $ROP_I$ are reported in Table 6.

Table 6 Pull solution of the modified Eilon Problem

When the demand set corresponds to (1), the solution approach sets the decision variables n = {4,3,3, 2,3,7}. This solution provides a daily cost of $\$158.26$, which is more costly than that obtained by Grznar and Riggle (1997), via global optimal algorithm for a planned Basic Period solution. Compared to the results reported by (Grznar and Riggle 1997, pp.363), the ‘pull’ model performs better than the common cycle (based on Hanssmann (1962) and equal to $\$177.8$) and better than other planned basic period-based models (based on Haessler (1979), $\$158.5$, Madigan (1968), $\$161.3$, and Stankard and Gupta (1969), $\$164.0$)

When the demand set corresponds to (2), the ‘pull’ solution is given by the decision variables n = {3,2,2,1,1,3}. The cost of the ‘pull’ solution, $\$ 222.86$, is slightly higher than the cost of $\$221.78$ obtained by Grznar and Riggle (1997). In this case, the results reported by (Grznar and Riggle 1997, p.363) confirm the better performance of the pull model compared to the common cycle (based on Hanssmann (1962) and equal to $\$246.3$) and compared to the planned basic period-based solution based on Madigan (1968) ($\$225.5$).

When the demand set corresponds to (3), the ‘pull’ solution is given by the decision variables n = {2,1,1,1,1,2}. Again, the cost of the ‘pull’ solution, $\$222.86$, is slightly higher than the cost equal to $\$221.78$ obtained by Grznar and Riggle (1997). In addition, for this instance, planned-based frequencies would be feasible in the pull model but would not reach the same cost. In this case, the pull model performs better than just a common cycle-based model (based on Hanssmann (1962) and equal to $\$295.3$), while all other planned solutions have lower costs.

When the demand set corresponds to (4), the ‘pull’ solution is given by the decision variables n = {1,1,1,1,1,1}. The cost of the ‘pull’ solution is equal to $\$337.65$. The global optimisation algorithm for a planned basic period finds a solution characterised by cost $\$333.41$. This difference in costs also characterises the common cycle-based model. Notwithstanding equal sets of frequencies characterise the ‘pull’ and planned solutions, the latter reaches a lower cost due to planning modifications, which are restricted to the ‘pull’ model.

1.2 Appendix B.2. Mallya’s Five-Item Problem (1992)

The proposed ‘pull’ approach is finally tested on Mallya’s original problem (Mallya, 1992), which has been solved previously by several authors in the modified instance as that proposed by Moon et al. (2002); Chung and Chan (2011). This famous test problem relates to the case of a lathe operating in a light mechanical engineering workshop that produces only five products isolated from the rest of the machines. In this problem, the utilisation, that is $\sum r_i$, is high and equal to 89%. The data set of the problem is shown in Table 7. Applying the proposed pull model to Mallya’s original problem, localsolver 9.5 deals with five integer variables and five continuous variables in a fraction of a second and achieves optimality.

Table 7 Data set of the original Mallya’s Five-items Problem

The ‘pull’ solution to Mallya’s problem is given by the decision variables n = {1,1,1,1,1}, defining the set of $Q_i$ reported by Table 7. The cost of the ‘pull’ solution is $\$48.87$, higher than the cost of Mallya’s planned solution, i.e. $\$41.79$. (See Table 8)

Table 8 Pull solution of the Mallya’s Five-items Problem

Appendix C.

See Table 9.

Table 9 Test sets results