# The effects of varying set-up costs.

Introduction

Two results of the economic order quantity (EOQ) model are presented in previous research. These are that decreasing set-up costs:

(1) lower inventory levels; and

(2) reduce total production and inventory costs.

Do these results exist in the dynamic lot size model? Zangwill suggests that stationary reduction of set-up costs will reach the same results as the EOQ model, but non-stationary reduction of set-up costs can increase inventory and raise total cost. He shows a non-stationary example which does not exist in practice. In practice, we can obtain the same results as the EOQ model by a mathematical approach and simulation, whether the set-up cost reduction is stationary or non-stationary.

The EOQ model can easily show the effects of set-up cost reduction on total set-up and holding costs, total holding costs and total set-up costs. However, this is difficult for the dynamic lot size model. Many researchers[3,4] state that stationary reduction of set-up costs can decrease total set-up and holding costs, but they do not address the degree of these effects. Moreover, they cannot discuss the case of non-stationary reduction of set-up costs. We define the same amount of set-up cost reduction to be a stationary reduction. Meanwhile, a nonstationary reduction is defined as the case in which the amount of set-up cost reduction is proportional to the original set-up cost in each period. In this article, the effects of the stationary reduction of set-up cost on the total set-up and holding costs, and the total holding costs are developed using a mathematical model. In addition, a simulation method for increasing the reduction of set-up cost is provided to describe the non-stationary case. In the dynamic lot size model, we stationarily and non-stationarily increase the amount of set-up cost reduction, and obtain the result that increasing the amount of set-up cost reduction will decrease total set-up and holding costs and total holding costs.

Let D be the product demand (assumed known) per unit of time, S the cost of setting up production, and h the cost of holding a unit of inventory per unit time. Then the EOQ formula concludes that the minimum cost per period, C, is Production

(1) C = [square root of]ShD.

Equation (1) shows that decreasing the set-up costs, S, will reduce the total cost per period, C.If the amount of set-up cost reduction is R, the total cost is reduced by [square root of]R. In the dynamic lot size (DLS) model, we cannot obtain the same results, but the same trend exists. Due to the complexity of the DLS model few researchers have attempted to develop the relationship between the total costs and set-up cost reduction. We shall discuss this using mathematical and simulational approaches.

Excellent discussions of zero inventory (ZI) can be found in Sugimori et al., Hall, Monden[7,8] and Schonberger. All previous discussions of ZI are descriptive, but set-up cost reduction is viewed as an important factor. The set-up cost is fixed in the conventional concept, but ZI differs in that it treats S as a parameter which can be optimized. Porteus has analysed a number of variations on the idea of reducing S. Karmarker proposes that reducing set-up time can significantly improve the overall efficiency of production. Zangwill has analysed a topic of "From EOQ Towards ZI" by stationary reduction of set-up cost. Lee and Seah have presented the effects of varying set-up times on the mean job tardiness and process utilization. Using a mathematical model and simulation, this paper discusses the magnitude of the set-up cost reduction on the set-up and holding costs and the total holding costs.

Zangwill's example is not practical

Zangwill presents an example that is a non-stationary reduction of set-up cost. His example shows that cutting set-up cost need not reduce inventories and total cost. This is misleading and discourages the reduction of S. Using simulation, we obtain the result that non-stationary reductions in set-up cost will decrease the total set-up and holding costs and the total holding costs. The simulation approach will be presented later. Here we may address the issue that Zangwill's example is not practical. This will be useful for our later discussions.

Zangwill's example is described as follows. A plant runs two shifts a day -- morning and night shifts. Consider two days of its operation divided into four periods. Designate period 1 as the morning shift of day one, period 2 as the night shift of day one, with periods 3 and 4 the day and night shifts, respectively, of day two. Suppose the product demand during each shift is three units. Thus [D.sub.i] = 3, i = 1, 2, 3, 4. For simplicity assume unit production cost is constant, so it can be neglected.

Scene 1

At present the plant is busy during the day and the set-up costs during the day are higher than they are at night. In particular S, = [S.sub.3] = 8, [S.sub.2] = [S.sub.4] = 5. Let X, be the production quantity at period i. It is easily seen that the optimal production schedule is [X.sub.1] = 3, [X.sub.2] = 6, [X.sub.3] = 0, [X.sub.4] = 3. The inventory levels (I) are [I.sub.1] = 0, [I.sub.2] = 3, [I.sub.3] = 0, [I.sub.4] = 0.

Scene 2

Since the set-up costs during the day are higher than they are at night, the engineering department significantly reduces the set-up cost, which becomes [S.sub.1] = [S.sub.3] = 1, [S.sub.2] = [S.sub.4] = 4. All other costs remain the same. The new optimal production schedule is seen to be X, = 6, [X.sub.2] = 0, [X.sub.3] = 6, [X.sub.4] = 0. The inventory levels are [I.sub.1] = 3, [I.sub.2] = 0, [I.sub.3] = 3, [I.sub.4] = 0.

Zangwill states that even though all set-up costs have been cut, inventory level has increased. He says that it is not necessarily true that cutting set-up cost reduces inventories.

In practice, a major factor of set-up cost reduction is set-up time reduction. The amount of set-up time reduction should be regular, it will not be irregular. If the set-up time reduction is irregular, it will yield the results presented by Zangwill. However, the non-stationary reduction that set-up costs [S.sub.1], [S.sub.3] are reduced from 8 to 1 and [S.sub.2], [S.sub.4] are reduced from 5 to 4 does not exist in practice. Thus the Zangwill example is not reasonable. In fact the amount of set-up time reduction will be the same for each period, or the same ratio for each period. The former is a stationary case and the latter is a non-stationary case. In the stationary case of reducing set-up cost, a mathematical model is used to obtain the effects. Furthermore, a simulation method is applied to the non-stationary reduction of set-up cost. The results are similar in both cases.

List of notations

For simplicity we assume that the unit production cost is a constant, so it can be neglected. We name the total set-up and holding costs as total cost. The following notations are used. [D.sub.i]: the product demand during each period i;, [S.sub.i]: the set-up costs during each period i; [h.sub.i]: the cost of holding inventories during period i where [h.sub.i] is concave and non-negative; m: the number of set-ups; [R.sub.i]: the amount of set-up cost reduction at period i; F(t, m): the minimal total cost with m set-ups in a t-period horizon; g(t): the minimal total cost in a t-period horizon; [g.sub.m](t): the minimal total cost in a t-period horizon, in which there are m set-ups; [g.sub.m](t): the minimal total cost after reduction of [R.sub.i] in a t-period horizon, in which there are m set-ups; [sub.k][TH.sub.m]: the total holding costs with m set-ups at the kth policy; [sub.k][TS.sub.m]: the total set-up costs with m set-ups at the kth policy; [sub.k][TH.sub.m](R): the total holding costs with m set-ups after reduction of R at the kth policy; [sub.k][TS.sub.m](R): the total set-up costs with m set-ups after reduction of R at the kth policy.

Stationary reduction of set-up costs

Adopting the above notations, let us discuss the case of stationary reduction of set-up costs. Let R be the amount of set-up cost reduction for each period. Zangwill demonstrates that as R increases, the number of periods in which inventory is held does not increase. This result shows that the number of setups after reduction, [m.sub.1], is not smaller than that prior to reduction, m. That is [m.sub.1] [less than or equal to] m.

What is the new moving direction when R increases? First, we decide when the new optimal policy will be changed. Theorem 1:

If R [less than or equal to] (F(t, [m.sub.j]) - [g.sub.m](t))/([m.sub.1] -m) holds for [m.sub.1] > m, then [g.sub.m]1(t)r[greater than or equal to] [g.sub.m](t)r.

If Theorem 1 is satisfied, then the new policy will move towards [g.sub.m]1(t)r.

Theorem 2:

Suppose the amount of the set-up cost reduction is [R.sub.i] in each period i, then as [R.sub.i] increases the total cost should be decreased.

The total cost should be reduced by set-up cost reduction [R.sub.i], i = 1, ..., t. This result is important for the dynamic lot size model. Conventional models aim to achieve production economics by batch quantity, however the new concept is that reducing set-up cost can decrease the total cost and the lot size to achieve production economics.

Since there are many possible directions in which to move, we should find the solution that will yield the minimum total cost.

Theorem 3:

[m.sub.j] and [m.sub.i] are integer for [m.sub.j] > [m.sub.i]. If F(t, [m.sub.j]) > F(t, [m.sub.i]) holds, then the first new policy with higher [m.sub.j] will be at

R = min(F(t, [m.sub.j]) - [g.sub.mi](t)/([m.sub.j] - [m.sub.i]).

The above theorem addresses the moving direction owing to the reduction of R.

The results of Theorems 1, 2 and 3 can be shown as in Figure 1.

What is the new total holding costs when set-up cost is stationarily cut by R. Two cases of set-up costs in each period are discussed. These are:

(1) that the set-up costs at each period are the same; and

(2) that the set-up costs at each period are different. The following results can be developed.

Theorem 4:

Let [S.sub.1] = [S.sub.2] = ... = [S.sub.i] for i = 1, 2, ..., t. If (F(t, [m.sub.j]) - [g.sub.m](t))/([m.sub.j] - m) [greater than or equal to] R < [S.sub.i] holds, then kjTH[m.sub.j] < [sub.k][TH.sub.m] can be obtained. Theorem 4 states that the total holding costs of the new policy after reduction of R will be smaller than those of the original policy. It means that the new policy, with higher R, will decrease inventories. The result is the same as ZI policy. Figure 2 illustrates the relation between the total holding costs and setup cost reduction, R, when the new production policy is achieved. Theorem 5:

Let [S.sub.i] not be the same value for i = 1, ..., t and [m.sub.j] be larger than m.

If (F(t, [m.sub.j]) - [g.sub.m](t))/([m.sub.j] - m) [greater than or equal to]. R < [S.sub.i] and kjTS[m.sub.j] > kTSm hold,

then [sub.kj][TH.sub.mj] < [sub.k][TH.sub.m] can be obtained. Although [S.sub.i], i = 1, ..., t, is not the same value, but the result is the same as Theorem 4 when the condition of [sub.kj][TS.sub.mj] > [sub.k][TS.sub.m] exists.

Non-stationary reduction of set-up costs

The previous section adopts a mathematical approach to describing the effects of stationary reduction of set-up costs on the total cost and the total holding costs. In the non-stationary case, the total cost after set-up cost reduction also decreases according to Theorem 2. Owing to the complexity of the non-stationary reduction of set-up costs, a mathematical approach is difficult to use to find the effects of set-up cost reduction on the total holding costs. Here the simulation model is used to investigate the effects of non-stationary set-up cost reduction on the total cost and the total holding costs. Considering the practice, we assume that the amount of set-up cost reduction is proportional to the original set-up cost in each period i. Let S'c denote the set-up cost after reduction in period i. The above assumption means that the ratio of [S'.sub.i]/[S'.sub.i] is a constant for each period i. The reason for this assumption is mentioned in an earlier section.

The simulation result offers us important information that the degree of the reduction of the total cost and the total holding costs can be understood. It is useful for deciding the lot size policy.

To prove the validity of the simulation method, four different simulation cases are studied. These include:

(1) the relation of the reduction between the total cost and the set-up cost with normal distribution and 100 runs of simulation;

(2) as (1), with different simulation runs;

(3) as (1), with different planning horizons;

(4) the relation of the reduction between the total holding costs and the setup costs with normal distribution and 100 runs of simulation.

Case (1)

Let the planning horizons be 12 and the simulation runs 100. The demand [D.sub.i], the unit holding cost per period hi and the set-up cost per period [S.sub.i] are normally distributed. For simplicity the filanning horizons are assumed as a regular interval. [S'.sub.i] denotes the set-up costs after reduction and r denotes a ratio of [S'.sub.i]/[S.sub.i]. Nine ratio values of 0. 0.2, ..., 0.9 are used to compute the total cost. Difiile T per cent to be a ratio of [g.sub.m]i(t)r/[g.sub.m](t). To measure the effects of the ratio r on T per cent, a relation between T per cent and r is given in Figure 3. To understand the range of variation of the total cost after set-up cost reduction, we take maximum T per cent average T per cent and minimum T per cent during each simulation of r.

Figure 3 shows that as the [S'.sub.i]/Sr ratio decreases the [g.sub.m]i(t)/[g.sub.m](t) ratio also decreases. It indicates that as the amount of set-up cost reduction increases, the total cost is also reduced. The difference between average T per cent and r is smaller than 0.1. The difference between maximum T per cent and r is smaller than 0.2. The minimum T per cent is closer to r. These results show that we can predict the range of total cost reduction with a given r, and understand the benefits of the set-up cost reduction. This result is similar to that obtained with the EOQ model.

Case (2)

Adopting the notations mentioned in case (1), we take different simulation runs Of 20, 100, 500 and 1,000. For simplicity we use the average (avg) T per cent for each simulation run. Figure 4 shows that the result of average T per cent due to the reduction of r is independent of the simulation runs. This can demonstrate the validity of the case (1).

Case (3)

Using the notations mentioned in case (1), we take different planning horizons, of 6, 12, 18, 24 and 48. For simplicity we use the average (avg) T per cent for each planning horizon. Figure 5 shows that the result of average T per cent due to the reduction of r is independent of the planning horizons. This can prove the validity of the case (1).

Case (4)

Let TH per cent be a ratio of [sub.k][TH.sub.m](R)/[sub.k][TH.sub.m]. As for r, [D.sub.i], [h.sub.i] and [S.sub.i] is mentioned in case (1). A relation between TH per cent and r is given in Figure 6. To understand the trend of variation of the total holding costs after set-up cost reduction, we take maximum TH per cent, average TH per cent and minimum TH per cent during each simulation of r. Figure 6 shows that decreasing r will reduce the average TH per cent during simulation runs of 100. It also means that increasing the amount of set-up cost reduction will not increase the average TH per cent. This is contrary to Zangwill's example. From the viewpoint of maximum TH per cent, average TH per cent and minimum TH per cent, the total holding costs have a decreasing trend. That cutting set-up cost reduces inventories can be true.

Conclusions

This article attempts to examine the effects of set-up cost reduction on the total cost and the total holding costs. The set-up cost reduction is divided into two cases, one is stationary and the other is non-stationary. Both stationary and non-stationary cases result in the same conclusions. Some of the major findings of the present study are that:

* Increasing the amount of set-up cost reduction will decrease the total cost.

* Increasing the amount of set-up cost reduction will not increase the total holding costs. The inventory level is not increased.

* The degree of the total cost reduction due to the ratio of r can be shown as in Figures 3, 4 and 5.

* The trend of the inventory level can be developed by mathematical model or simulation.

In the dynamic lot size model the findings of (1), (2), (3) and (4) are useful for encouraging the cutting of set-up costs. Instead of being descriptive of ZI philosophy, the results of this paper give a basic theory on ZI.

[Figures 1 to 6 ILLUSTRATION OMITTED]

References

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