Batch Size Depending on Stockout Costs (*) The carrying...

Batch Size Depending on Stockout Costs (*) The carrying costs for a certain article are 2 per unit and year. Stockout costs are 5 per unit. The average annual consumption amounts to 1000 , and the standard deviation of demand during lead time is 10 . No safety stock is intended. Normal distribution is assumed. a. How large should the batch size be, considering the optimum stockout probability? Can the fill rate target of $99 \%$ be met? What are the carrying costs per year? b. Assume a batch size of only 250 . What are the values for safety stock and fill rate corresponding to the optimum probability of stockout per order cycle? c. Now assume a safety stock of 20 units. Again, the batch size is 250 . What are the values for service level and fill rate?

Batch Size Depending on Stockout Costs (*)

The carrying costs for a certain article are 2 per unit and year. Stockout costs are 5 per unit. The average annual consumption amounts to 1000 , and the standard deviation of demand during lead time is 10 . No safety stock is intended. Normal distribution is assumed.
a. How large should the batch size be, considering the optimum stockout probability? Can the fill rate target of $99 \%$ be met? What are the carrying costs per year?
b. Assume a batch size of only 250 . What are the values for safety stock and fill rate corresponding to the optimum probability of stockout per order cycle?
c. Now assume a safety stock of 20 units. Again, the batch size is 250 . What are the values for service level and fill rate?

Key Concepts

Fill Rate

Fill rate is a performance metric in inventory management that measures the percentage of customer demand met directly from stock without backorders. Achieving a high fill rate is important for maintaining customer satisfaction, and it is closely related to the level of safety stock and the chosen service level strategy.

Safety Stock

Safety stock is the additional inventory kept to buffer against uncertainties in demand or supply, particularly during lead times. It is a critical component in inventory control policies, helping to prevent stockouts despite variations in demand, and typically influences the service level and fill rate achieved.

Demand Variability During Lead Time

Demand variability during lead time refers to the fluctuations in customer demand that occur in the period between placing an order and receiving it. This variability is commonly modeled using statistical distributions, such as the normal distribution, and is crucial for calculating the required safety stock to achieve desired service levels.

Stockout Costs

Stockout costs represent the expenses incurred when inventory levels are insufficient to meet demand, leading to lost sales, customer dissatisfaction, or disruptions in production. In inventory management, these costs need to be balanced against carrying costs to determine the optimal inventory level, thereby influencing decisions on batch size and safety stock levels.

Batch Size Optimization

Batch size optimization involves determining the optimal quantity of inventory to order or produce in one batch in order to minimize the total costs associated with ordering, holding, and stockouts. This decision takes into account various cost factors, including carrying?costs, stockout costs, and demand during lead time, and helps in balancing the trade-off between having too much inventory (which increases holding costs) and too little inventory (which increases the risk of stockouts).

Carrying Costs

Carrying costs are the expenses associated with holding or storing inventory over a period, including costs such as warehousing, insurance, and spoilage. In inventory models, minimizing carrying costs while maintaining sufficient service levels is essential to ensure that the overall cost of inventory management is kept low.

Service Level

Service level is a measure of the probability or frequency with which inventory is sufficient to meet demand during a specified period, often expressed as a percentage. It reflects the reliability of a supply system and is used to determine appropriate safety stock levels so as to meet customer expectations while controlling costs.

Transcript

00:01 Hello, this is a problem about a business selling an item at a constant rate of r units per month.

00:20 And it reorders in batches of q units.

00:26 So a reorder contains q units and the cost of reordering is a plus bq.

00:39 A plus bq.

00:45 Storage cost.

00:48 Storage is given as k dollars per item, k dollars per item, per month.

01:06 And on average, half are in storage waiting to be sold.

01:20 Okay.

01:23 Question a.

01:26 How often? how often does reordering take place? right, let's see.

01:52 How do we reorder? well, as we sell a certain number of units per month, then our reordering should keep pace with that sale.

02:12 So we're going to say how much was sold.

02:17 In let's say t months, time t given in months.

02:26 On average should be equal to the how much was reordered.

02:34 Reordered in those t months.

02:42 So we calculate the values, we calculate time, express time with the letter t.

02:48 Now let's say if we sell our units per month, then we have rt.

02:57 Rt sold in t months.

03:02 R.

03:06 Rt is sold in t months.

03:11 And how much was reordered? reordered q units.

03:21 So, each month we reorder q divided by r here.

03:29 We divide with r.

03:32 Qr.

03:41 This gives us an expression how many months pass.

03:47 This could be a fraction, not 0 .3.

03:49 We could be ordering every nine days, but basically this ratio here, q through r, where q is the reorder number of units and r is the number of units all per month.

04:06 So q through r.

04:08 Question b is the average monthly cost of reordering.

04:23 What is it? okay, so what is the cost of one reorder? the cost of one reorder is a plus bq.

04:33 And the number of months is given down here, q through r.

04:41 So we divide these two to find what the average cost.

04:46 Cost is, this is a plus bq, a plus bq through 1, divided by this means multiplied by r versus q.

05:02 We have r a plus rbq through q, which we can write as ra through q plus ra through q plus rbq plus ra through q plus rbq will cancel here...

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