Optimal replenishment level - DETAILED INVENTORY CONTROL MODEL

8. DETAILED INVENTORY CONTROL MODEL

8.5 Optimal replenishment level

In the theoretical model, the optimal replenishment level is computed. First, the computation for the adjusted replenishment level of model 1 and 3 is shown, after that the fixed replenishment level for model 2 and 4 is explained.

Model 1 and 3

The replenishment level is calculated with the use of historic ratios. The sum of the historic ratios for the variants with purchase item j in it is the probability of that purchase item, referred to as 𝑝_𝑗^𝑖. Model 1 and 3 assume a variable replenishment level and Binomial distributed demand in PIs. The realization of the demand in components is 𝑑𝑖 and the stochastic demand in PIs is:

𝑋_𝑗(𝑡)~𝐵𝑖𝑛(𝑑_𝑖(𝑡), 𝑝_𝑗^𝑖)

The replenishment level of t+L can be computed with the cumulative distribution function:

𝐹 (𝑆̃_𝑗(𝑡 + 𝐿)) = 𝑃 (𝑋_𝑗(𝑡 + 𝐿) ≤ 𝑆̃_𝑗(𝑡 + 𝐿)) 𝑃 (𝑋_𝑗(𝑡 + 𝐿) ≤ 𝑆̃_𝑗(𝑡 + 𝐿)) ≥ 𝛼_𝑗^𝑜𝑏𝑗

𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑆̃(𝑡 + 𝐿) ≥ 𝐹_𝑗 ⁻¹(∝_𝑗^𝑜𝑏𝑗) Model 2 and 4

Model 2 and 4 assume the demand in components is stochastic. Therefore, a fixed and time-independent replenishment level is computed. The probability distribution function of the demand in components is approximated by calculating the mean and standard deviation of the demand given in the SPO Future demand and then fitting a theoretical discrete probability distribution. The fitting procedure developed by Adan et al. (1995) is used for this. The best fit for the demand in projects and exits was the Poisson distribution with mean 𝜆𝑖 (Appendix L). Consequently the demand in purchase items is:

𝐷_𝑖~𝑃𝑜𝑖𝑠(𝜆_𝑖) 𝑋_𝑗~𝐵𝑖𝑛(𝐷_𝑖, 𝑝_𝑗^𝑖)

A truncated distribution function is used, as the domain of the probability function is restricted with an upper boundary of the demand in components:

𝑢_𝑖= 𝑢𝑝𝑝𝑒𝑟 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑒𝑚𝑎𝑛𝑑 𝑖𝑛 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑖 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑃(𝐷_𝑖≤ 𝑢_𝑖) > 0.995 𝐹(𝑆̃_𝑗) = 𝑃(𝑋_𝑗≤ 𝑆̃_𝑗)

= 𝑃(𝑋𝑗≤ 𝑆̃_𝑗|𝐷_𝑖 = 0) ∗ 𝑃(𝐷𝑖= 0) + 𝑃(𝑋𝑗 ≤ 𝑆̃_𝑗|𝐷_𝑖 = 1)

∗ 𝑃(𝐷_𝑖 = 1)+. . +𝑃(𝑋_𝑗≤ 𝑆̃_𝑗|𝐷_𝑖 = 𝑢_𝑖) ∗ 𝑃(𝐷_𝑖= 𝑢_𝑖) 𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝑆̃ ≥ 𝐹_𝑗 ⁻¹(∝_𝑗^𝑜𝑏𝑗)

45 8.6 Order of events

The order of events at week t for item j in the simulation:

1. The on hand inventory, backorders, and inventory position of week t-1 are the starting position;

2. The order placed at t-L arrives²;

3. The backorders of t-1 are, as far as possible, satisfied;

4. The actual demand for item j of week t is fulfilled as far as possible, and unmet demand is backordered;

5. The actual demand of week t+4 is known;

6. The replenishment level of week t+L is known;

7. The base-stock level at [t,t+L] is calculated;

8. The order for week t+L is calculated and placed;

9. The on hand inventory at the end of week t is calculated;

10. The inventory position at the end of week t is calculated;

11. The back orders at the end of week t are calculated.

8.7 Mathematical Model

First, model 1 is discussed in full, lateron the differences for model 2, 3, and 4 are defined. The mathematical model is discussed in the same order, as the order of events.

At the starting position in week t, the on hand inventory of t-1, backorders of t-1, inventory position of t-1, and all orders that still had to be received in t-1 are known:

𝐼_𝑗^𝑂𝐻(𝑡 − 1), 𝐵𝑂_𝑗(𝑡 − 1), 𝐼𝑃_𝑗(𝑡 − 1),

𝐼𝑂_𝑗(𝑡 − 𝐿), 𝐼𝑂_𝑗(𝑡 − 𝐿 + 1), 𝐼𝑂_𝑗(𝑡 − 𝐿 + 2), 𝐼𝑂_𝑗(𝑡 − 𝐿 + 3), 𝐼𝑂_𝑗(𝑡 − 1)

Next, the order placed at t-L (𝐼𝑂𝑗(𝑡 − 𝐿)) arrives and, if possible, backorders of t-1 are satisfied. Actual demand (𝑥𝑗(𝑡)) at t is fulfilled and unfulfilled demand is back ordered. Moreover, the realization of demand in t+4 (𝑥𝑗(𝑡 + 4)) becomes known. The replenishment level of week t+4 is calculated by the theoretical model, and this replenishment level is used to calculate the base-stock level over [t,t+L]:

𝑆_𝑗(𝑡, 𝑡 + 𝐿) = 𝑥_𝑗(𝑡 + 1) + 𝑥_𝑗(𝑡 + 2) + 𝑥_𝑗(𝑡 + 3) + 𝑥_𝑗(𝑡 + 4) + 𝑆̃(𝑡 + 𝐿) _𝑗

Next, the order at t can be calculated and placed at the second tier supplier.

𝐼𝑂_𝑗(𝑡) = (𝑆_𝑗(𝑡, 𝑡 + 𝐿) − 𝐼𝑃_𝑗(𝑡 − 1) + 𝑥_𝑗(𝑡))⁺

After that, the on hand inventory, back-orders, and inventory position are calculated.

𝐼_𝑗^𝑂𝐻(𝑡) = (𝐼_𝑗^𝑂𝐻(𝑡 − 1) + 𝐼𝑂_𝑗(𝑡 − 𝐿) − 𝐵𝑂_𝑗(𝑡 − 1) − 𝑥_𝑗(𝑡))⁺

2In the models with lead-time uncertainty, the moment at which an order should arrive the actual lead-time of the order is known. The actual lead-time for the order ordered at week t-L can be L or L+1. If the actual lead-time is L+1, the order is received in the next week. Moreover, if the actual lead-time of the order ordered at week t-L-1 was L+t-L-1, it is received in this week.

46 replenishment level is done in the theoretical model, resulting in different input for the simulation, but no difference in calculations.

Model 3

The third model assumes a deterministic demand in components. Hence, the computations for the replenishment level and base-stock level are similar to that of model 1. The lead-time is stochastic, and there is a possibility to deliver orders at t+L+1 instead of t+L. This is known at the moment an order should be received. The Binomial distribution is used to simulate the actual lead-time. An extra input parameter is introduced in this model:

𝑃(𝐿(𝑡) = 𝐿) = 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑜𝑓 𝑎𝑐𝑡𝑢𝑎𝑙 𝑙𝑒𝑎𝑑 𝑡𝑖𝑚𝑒 𝑏𝑒𝑖𝑛𝑔 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑙𝑒𝑎𝑑 𝑡𝑖𝑚𝑒 Model 1 is equal to a specific setting of model 3 with P(L(t)=L)=1. The calculation for the on hand inventory and back-orders changes into:

𝐼_𝑗^𝑂𝐻(𝑡) = {(𝐼_𝑗^𝑂𝐻(𝑡 − 1) + 𝐼𝑂_𝑗(𝑡 − 𝐿) − 𝐵𝑂_𝑗(𝑡 − 1) − 𝑋_𝑗(𝑡) + 𝐼𝑂_𝑗^{𝑑𝑒𝑙𝑎𝑦𝑒𝑑}(𝑡 − 1))⁺

The calculation for the inventory position remains the same.

Model 4

Model 4 assumes a fixed replenishment level and stochastic lead-time. Therefore the calculation of the replenishment level is equal to that of model 2, and the calculations for the on hand inventory and back-orders are similar to model 3. Model 2 is again equal to a specific setting of model 4 with P(L(t)=L)=1

8.8 Average performance and costs

The output of the theoretical model consists of the average replenishment level, and the output of the simulation consists of the actual average performance and the costs.

Average replenishment level

The average replenishment level is 𝑆̃𝑗∗

with ∝𝑗≥∝_𝑗^𝑜𝑏𝑗. For model 1 and 3, this is the average of the weekly replenishment level, and for model 2 and 4, this is a fixed value computed by the theoretical model:

𝑓𝑜𝑟 𝑚𝑜𝑑𝑒𝑙 1 𝑎𝑛𝑑 3: 𝑆̃_𝑗^∗=∑^𝑇_𝑡=1𝑆̃_𝑗(𝑡) 𝑇 𝑓𝑜𝑟 𝑚𝑜𝑑𝑒𝑙 2 𝑎𝑛𝑑 4: 𝑆̃_𝑗^∗

The yearly holding costs are based on a fixed percentage of the purchase costs. The percentage includes rental costs, costs for utilities and risks. According to Durlinger (2013), the percentages that should be used are the following:

Table 8, Holding cost percentage according to Durlinger (2013)

Rental Utilities Risks Total Mechanical/Sheet

metal

18% 15% 3% 36%

Controls/Electrical 12% 10% 4% 26%

The yearly unit holding costs are therefore:

ℎ = {0.360.26

𝑖𝑓 𝑗 𝑖𝑠 𝑎 𝑚𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑖𝑡𝑒𝑚 𝑖𝑓 𝑗 𝑖𝑠 𝑎𝑛 𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐𝑎𝑙 𝑖𝑡𝑒𝑚

ℎ_𝑗 = 𝑝𝑐_𝑗∗ ℎ

Calculating the average on hand inventory and multiplying this with the holding costs per item results in the yearly holding costs based on the on hand inventory for week t. The average yearly holding cost over a number of weeks can again computed:

𝐼_𝑗^{𝑂𝐻,𝑎𝑣𝑔}=∑^𝑇_𝑡=1𝐼_𝑗^𝑂𝐻(𝑡) 𝑇 𝐶_𝑗 = ℎ_𝑗∗ 𝐼_𝑗^{𝑂𝐻,𝑎𝑣𝑔}

8.9 Validation and verification

In this sub-section, the numerical examples are presented which aim to illustrate some features of the four types of models that are established in the previous sub-sections. The optimal replenishment level is found by using two theoretical models, and a simulation is performed to compute the actual average performance and costs and to investigate the effect of lead-time variation on the average performance.

Before elaborating on the results, the model is verified and validated by the model of development process in Figure 27 (Sargent, 2013).

According to Sargent (2013) there are three entities that should be considered:

 The problem entity, being the inventory setting at Vanderlande;

 The conceptual model; being the base-stock inventory model;

 The computerized model; being the simulation executed on a computer.

Figure 27, Simplified version of the model of development process (Sargent, 2013)

8.9.1 Conceptual model validation

The conceptual model is validated by determining that theories and assumptions of the model are correct. All assumptions can be found in section 7.1 and are based on a realistic setting at Vanderlande or at theoretical assumptions. The assumptions are discussed in Appendix C.

8.9.2 Computerized model verification

The simulated model is verified by debugging the code in VBA whenever code resulted in errors.

Furthermore, the calculations used in the mathematical model are checked by hand and only when the calculations matched the results of the model, these were used in the simulation. Consequently, it can be assured that the computer programming and implementation of the model are correct.

8.9.3 Operational validation

The validation of the model is done by first checking the behavior of the model in extreme cases. These results are presented in Appendix M. The extreme values led to expected results. Moreover, the setting of the simulation is according to the simulation method of Law (2007). The warm-up period and confidence interval are given in Appendix M. The confidence interval for the simulation is computed to check whether the number of iterations is sufficient in providing accurate output. Appendix M also elaborates on two instances with easy settings, based on real life setting of Vanderlande, which are used to validate the behavior of the models. These setting led to expected results.

8.9.4 Data validity

The input data used in the theoretical model and simulation is based on actual data of Vanderlande. The actual demand in components in week 3 to 27 of 2017 is used to fit a distribution function and determine the parameters of the function. The distribution function is used as input for the simulation.

Furthermore, the service level constraint is based on the actual average performance level Vanderlande and its suppliers strive for, and the lead-time of 5 weeks is the standard lead-time of items of the SPO.

The probability of an item j in component i is calculated with historic ratios based on actual data.

Moreover, the holding cost rate is based on literature.

8.10 Results

Two items that are part of the group with unknown characteristics and that have a lead-time longer than four weeks are considered:

 Item 1, which is part of component (B);

 Item 2, which is part of component (C).

First, the input parameters of the items are defined. Next, the output of the model is discussed.

8.10.1 Input parameters

The input parameters consist of the demand in components, demand in purchase items, standard lead-time, holding costs and purchase costs, service level constraint, and the probability of an item j required for component i. Moreover, in model 3 and 4 the stochastic lead-time is added to the model. The input parameters for a specific item in a component are similar for every type of model, in order to compare them. The input data³ is displayed in Table 9.

Table 9, Input values, item 1 and item 2

Input parameters Values for item 1 Values for item 2

𝑪𝒐𝒎𝒑𝒐𝒏𝒆𝒏𝒕 (𝐵) (𝐶)

The output that is generated by the theoretical model and simulation is the following:

 Average replenishment level;

 Average actual performance;

 Average costs.

To analyse the impact of a possibility on an actual time of one week longer than the standard lead-time, the four models are compared.

Item 1

First, the replenishment level of the models is computed. The cumulative probability 𝑃(𝐷_𝑖≤ 8) >

0.995 , therefore the upper boundary of the truncated demand is 8. The replenishment levels for the models as computed in the theoretical model are displayed in Table 10.

3It should be mentioned that the costs for both items are similar. The reason for this could be that the items are similar with for example a different angle, but therefore resulting in a similar price. Furthermore, both items are mechanical.

Table 10, Replenishment levels, item 1

Output Theoretical model Model 1 and 3 Model 2 and 4

𝐒̃_𝟏^∗ 2.36 3

The results of the simulation of model 1 and 3 are given in Table 11. The average performance of model 1 is with 99.71% well above the 98% constraint, However, a probability of 0.9 on a lead-time of 5 weeks decreases the performance with 2.14% to 97.57%. A decrease of the probability with 0.1, results in an average decrease of the performance of 3.12%, an average decrease of the on hand inventory of 0.12 items, and a decrease in the costs of on average €0.61.

Table 11, Results of model 1 and 3, item 1

Table 12 displays the results for item 1 in model 2 and 4. Model 2 meets the service level constraint, but a decrease of 0.1 on the probability of a lead-time of 5 weeks, results in an average decrease of the performance of 1.28%, an average decrease of the on hand inventory of 0.08 items, and a decrease in the costs of on average €0.42. Again, little uncertainty, P(L(t)=L)=0.9, results in a performance level under the service level constraint.

Table 12, Results of model 2 and 4, item 1

A comparison of the models with adjusted and with a fixed replenishment level is illustrated in Graph 7.

A decrease in the probability on a lead-time of 5 weeks has a bigger impact in the model with an adjusted replenishment level than it has on the model with a fixed replenishment level. Model 1 performs slightly better than model 2, but model 4 performs better than model 3. The decrease of the performance of the models with an adjusted replenishment level is on average 2.98%, and the decrease of the models with a fixed replenishment level is on average 1.28%. The costs of the models with an adjusted replenishment level are always lower than the costs of the models with a fixed replenishment level. On average, the costs for holding inventory in the adjusted replenishment models are €2.09 lower than the costs for the models with a fixed replenishment level.

Three conclusions can be derived:

 If there is no lead-time uncertainty, a model with an adjusted replenishment level results in a better performance than a model with a fixed replenishment level;

 Lead-time uncertainty has a bigger negative impact on a model with an adjusted replenishment level than it has on a model with a fixed replenishment level;

 The costs of the models with an adjusted replenishment level are always lower than the costs of the models with a fixed replenishment level.

Graph 7, Average performance and cost comparison, item 1

Item 2

The tables and graph that display the results for item 2 are given in Table 26, Table 27, and Graph 9 in Appendix N, and this section elaborates on the conclusions drawn from the results.

The cumulative probability 𝑃(𝐷𝑖 ≤ 86) > 0.995 , therefore the upper boundary of the truncated demand is 86. The theoretical model resulted in an average adjusted replenishment level of 35.20 items, and a fixed replenishment level of 38 items.

Model 1 has an average performance that meets the service level constraint of 98%, whereas model 2 has an average performance that is under the service level constraint, namely 97.72%. On top of that, the cost for model 1 are €44.43 and the costs for model 2 are €61.70. Both model 3 and 4 fail to meet the service level constraint for all different probabilities on lead-time uncertainty. Again, at the moment lead-time uncertainty is added, model 4 performs better than model 3.

Lead-time uncertainty has a big impact on the average performance. A decrease in the probability on a lead-time of 5 weeks of 0.1 results in a decrease of the average performance of on average 9.82% for the models with an adjusted replenishment level, and 9.60% for the models with a fixed replenishment level. The costs of decrease with on average €4.36 for the models with an adjusted replenishment level, and €6.17 for the models with a fixed replenishment level.

If the probability on a lead-time of 5 weeks is equal to a lead-time of 6 weeks, the average performance of model 3 is 49.43%, and the average performance of model 4 is 49.70%. The difference between the average performance of the models with an adjusted and a fixed replenishment level is on average 0.19% in favor of models with a fixed level, whereas the cost difference is on average €10.50 in favor of the models with an adjusted level. The costs for model 3 are again smaller than the costs for model 4.

Three conclusions are derived:

 If there is no lead-time uncertainty, only a model with an adjusted replenishment level results in an average performance that meets the service level constraint;

 Although the difference is only 0.19%, lead-time uncertainty has a bigger negative impact on a model with an adjusted replenishment level than on a model with a fixed replenishment level;

0,00%

Costs Model 1 and 3 Costs Model 2 and 4 Performance Model 1 and 3 Performance Model 2 and 4

 The costs of the models with an adjusted replenishment level are always lower than the costs of the models with a fixed replenishment level.

Sensitivity analysis

A small sensitivity analysis on model 1 is performed to analyse the impact of the probability of an item in a component. Instead of using historic ratios, one can decide to compute the probability based on the occurrence of the item in the variants of the component. A distinction is made between the probability in the theoretical model and in the simulation:

𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑡ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝑚𝑜𝑑𝑒𝑙: 𝑝_𝑗^𝑖 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑠𝑖𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛: 𝑝̃_𝑗^𝑖

The probability in the simulation is based on the historic ratios, whereas the probability in the theoretical model is based on the occurrence (Appendix L):

𝑝₁^𝐵= 0.35, 𝑝̃₁^𝐵 =12 24= 0.5 𝑝₂^𝐶 = 0.42, 𝑝̃₂^𝐶 = 6

24= 0.25

Table 13 shows the results. If the probability based on occurrence is higher than the probability based on historic ratios, all output increases a bit. However, if the probability based on occurrence is lower than the probability based on historic ratios, this has a big negative impact on the average performance, costs and replenishment level (Table 14). Thus, using a higher probability to compute the replenishment level increases the costs for holding inventory, but diminishes the probability on a low average performance level.

Table 13, Sensitivity analysis, item 1

𝒑̃𝒋𝒊 ∝𝒋 𝑪𝒋 𝑺̃𝒋∗

0.35 99.71% €9.61 2.36

0.5 99.86% €12.02 2.62

Table 14, Sensitivity analysis, item 2

𝒑̃_𝒋^𝒊 ∝𝒋 𝑪𝒋 𝑺̃𝒋

∗

0.42 97.72% €61.70 35.15

0.25 20.83% €2.15 23.39

53 9. CONCLUSIONS AND RECOMMENDATIONS

This chapter describes the conclusion and recommendations, based on results of the redesign and simulation, and describes further research possibilities.

9.1 Conclusion

The research question defined in section 5.6 is the following:

“What is the effect of proactive inventory control for the Posisorter equipment on performance of the factories and suppliers and how can Vanderlande better control its supply?”

This question is answered in this section, by discussing each of the seven sub-questions.

9.1.1 Measurable performance indicators for the factories and suppliers

The first sub-question is related to the identification of the key KPIs for the factories and suppliers to increase material availability at the factories. Three KPI’s are defined in section 5.1:

1. On time and complete performance of the second tier suppliers;

2. Material availability, being the percentage of order lines fulfilled from stock;

3. Total costs for holding inventory.

The three KPIs consider the ETO trade-off defined by Radke and Tseng (2012) that includes a high service level, inventory budgets, and the delivery lead-time.

9.1.2 Current performance with uncontrolled stock points

The second sub-question is related to the performance of the current stock points. There is no data available about the inventory control policy of the current stock points, nor about the performance of the inventory model. Moreover, the yearly holding costs computed by Vanderlande are time independent. The performance in delivering order on time and complete to SCCE of VIM in week 2 to 37 of 2016 is 83% and of VIS in week 10 to 37 of 2016 is 65%, while the threshold for both factories is 95%.

The performance gap between the actual performance and the desired performance is therefore 12%

for VIM and 30% for VIS. These results are displayed in section 3.1.

In section 3.2, the causes are categorised. This brings to light that 35% of the delays are caused by overdue purchase parts. An analysis of the Delivery to request of the second tier suppliers, executed in section 3.4, showed that both suppliers that deliver to VIM, and suppliers that deliver to VIS are underperforming. The second tier suppliers of VIM have an average performance of 80%, and the suppliers of VIS have an average of 83.3%. The threshold for the Delivery to request performance of the second tier suppliers is 98%. The research addresses three factories and a subcontractor, but due to information unavailability only the performance of VIM and VIS and their second tier suppliers is evaluated.

To conclude, the second tier suppliers are underperforming in delivering material at or before the request date set by Vanderlande. Consequently, the material availability at the factories is low and this affects the performance of the factories.

54 9.1.3 Five local stock points and local decisions

The third sub-question investigates the impact of local decision making and local stock points. A conceptual design of a two-echelon distribution system is defined, and in section 5.3, demand data is analysed in order to estimate the demand parameter. The analysis in section 5.3 led to the conclusion

In document Eindhoven University of Technology MASTER Redesign of the control structure and inventory control at Vanderlande Gubbels, P.M. (pagina 61-0)