33 Inventory Models: Non- Instantaneous Replenishment

Vikas Singla

33.0 Objectives

33.1 Introduction

33.2 Non-Instantaneous Replenishment Model: Economic Production Quantity (EPQ)

33.2.1 Assumptions of EPQ

33.2.2 Calculation of EPQ

33.3 Practical Example

33.4 A Case of EPQ

33.5 Summary

33.6 Glossary

33.7 References/ Suggested Readings

33.8 Short Answer Questions

33.9 Model Questions

33.0 OBJECTIVES

This chapter would help students to understand:

Application of Non-Instantaneous Replenishment model
Assumptions of Economic Production Quantity (EPQ) model
Calculation of optimum batch size and minimum total inventory cost

33.1 INTRODUCTION

A fast food restaurant sells a fixed quantity of particular kind of burger to customers. A supplier of buns supplies number of buns every day depending on that fixed number of burgers. The burgers are provided in batches depending on demand. On the other hand same restaurant continuously produces French fries to sell with each order. The production is continuous as instead of burgers being ordered in batches these French fries are ordered continuously. Thus, the major difference between two production processes is that one product is produced and supplied in batches whereas another is produced and consumed simultaneously as it is used continuously because of its high or continuous demand. Economic order quantity (EOQ) model was used to answer questions of how much and when to order in first scenario whereas Economic Production Quantity (EPQ) inventory model would be used to answer same questions of inventory in second scenario.

33.2 NON-INSTANTANEOUS REPLENISHMENT MODEL: ECONOMIC PRODUCTION QUANTITY (EPQ)

Understanding of EPQ model becomes necessary in order to manage inventory of products which are produced and used simultaneously at a constant rate. For instance, in a continuous production line involving dressing of doll twp processes of putting up sock and then shoe are done in a sequence. In such a process one doll enters the system, its sock being put up and then goes to process of putting up a shoe. At the start of process only one doll moves along the process. After finishing of process at one workstation it moves forwards and is preceded by another doll. Thus, products enter and leave one-by-one in sequence rather than being provided in one full batch as was done in EOQ model. This is the reason this model is also termed as non-instantaneous replenishment as when a specific number of units whether one or more are produced they are provided for usage. Out of the produced units if few are used then remaining becomes inventory.

33.2.1 Assumptions of EPQ

The demand for the item is uniform and constant (d)
The production of an item start immediately as stock level reaches zero at constant production rate (p).
Units are supplied to inventory at constant rate. Instead of inventory being built with constant size of Q units after a particular time period it gets built at constant rate in continuous fashion. For instance in EOQ model a batch size of 50 was supplied after fixed time period when daily demand was of 10 units. But in EPQ model only 10 units would be supplied per day to fulfill demand of 50 units in 5 days.
Model is applicable only in situations where production rate is either equal to or greater than usage rate. Uniform production rate ‘p’ is greater than uniform demand rate ‘d’. For instance, if demand rate is of 100 units daily then production process should be able to produce either 100 or lesser number of units per day.
Any gap between production and usage rate is constant. For instance, if production rate per day is 100 units and daily demand is 10 units for each day then inventory would be build at 100-10=90 units per day till production level is reached.
After production run number of units in inventory are consumed or used at constant rate. For example if at the end of production run of 3 days inventory buildup is of 100 units and daily demand is of 10 units then only consumption will be carried out at constant rate of 10 units daily.
The demand for the item is certain, constant and continuous
Lead time is fixed
Holding cost (H) per unit per unit time is constant and does not change for different order quantity
Ordering cost (O) per order is constant and does not vary with number of orders
No stock outs are permitted.

33.2.2 Calculation of EPQ

In the model during the production run, demand diminishes the inventory at constant rate while production adds to inventory at constant rate. As discussed it has been assumed that production rate would always be either equal to or more than usage rate so there would be continuous build up of inventory till the production run. When the production run is completed units from buildup inventory gets used up till zero level is reached. The entire process is then repeated again. This cycle is illustrated in Fig.1.

With the understanding of above mentioned assumptions of EPQ following procedure is adopted for formulating the quantitative formula for calculating number of units to be ordered per batch and total cost. As discussed total cost would be a function of only ordering and holding cost. Notations used in the model are:

D: Annual demand

H: Holding cost per unit

S: Ordering or set-up cost per order

Q: Batch size

p: number of units produced daily

t: production time

d: daily demand

_{Fig.1 is illustrated by taking following scenario.}

Suppose a batch size Q=300 units are required. Daily p=100 units are produced out of which 10 units (d) are consumed each day.
Thus production time would be 3 days (t) in which total of 300 units would be produced.
So on day 1, 100 units are produced and 10 are used. Remaining 100-10 = 90 units are stored in inventory as shown by A.
Similarly on day 2, another lot of 100 units are produced and 10 are consumed. Remaining 90 units become part of inventory making a total of 180 units as stock after day 2 as shown by B.
Lastly, on day 3 last batch of 100 units is produced and 10 are consumed. A total of inventory of 270 units as shown by C is formed. This whole process is shown in the shaded region indicating process of both production and consumption.
It needs to emphasize that max inventory level denoted by Imax is of 270 units and not 300 units.
After completion of production run the stock of 270 units is being used at a constant rate of 10 units daily shown by downward moving line. This shaded area indicates only consumption with no production.
If there would not have been any daily consumption and only production of 100 units then stock buildup would be of 300 units as shown by dotted line and indicated by D.
When this stock is fully consumed then cycle is repeated again.

Understanding of process as modeled in Fig.1 calculation of holding and ordering cost is done as under. The calculation process is discussed for two cases. In the first case production and consumption time period in a given year are different. In second case formula is modified when production occurs for each demand day.

Case 1: when production is done till production run

Holding cost:

Case 2: when product is produced every day till completion of ordering cycle

In the above deduced formula it was assumed that production and consumption time periods are different. In the first phase production and consumption happened simultaneously for first three days and then only consumption happened for remaining cycle time. But in case if a product is produced daily for entire cycle time. At same time usage of product produced also happens daily then formula can be modified to: Suppose, a production facility operates for 250 days per year. If daily demand for the product is d then annual demand can be computed as:

D = 250d or d = D/250 where D is annual demand

Similarly if product is produced daily for 250 days then annual production denoted by P would be:

P = 250p or p = P/250

Putting these values in the lot size and total cost formula computed in case 1 we get:

TC = H * (Q/2) * (P-D)/P + (D/Q) * S

Q = √(2DS/H) * √(P/(P-D)

The formula is almost similar with changes in daily production and daily demand to annual production and annual demand.

33.3 PRACTICAL EXAMPLE

Example for Case 1: A plant manager must determine the lot size of a product that has a steady demand of 30 units per day. The production rate is 190 units per day, annual demand is 10,500 units, set up cost for a production run is 200.00 and holding cost is 0.21 per unit. Plant operates for 350 days per year. With this information calculate:

a) Economic production lot size

b) Total annual cost

c) Time between orders (TBO) or cycle length

d) Production time per lot

Solution:

Annual demand, D = 10,500 units

Daily demand, d = 30 units

Production per day p = 190 units

Set up cost S = 200.00

Holding cost H = 0.21 per order

(a) Economic production lot size

Q = √(2DS/H) * √(p/(p-d)

Q = √(2*10500*200/0.21) * √(190/(190-30)

= approx. 4874 units

(b) Total annual cost

TC = H * (Q/2) * (p-d)/p + (D/Q) * S

TC = 0.21 * (4874/2) * (190-30)/190 + (10500/4874) * 200

= approx. 862.00

(c) Time between orders (TBO) or cycle length

TBO = Q/D

= (4874/10500) * 350 days per year

= approx. 162 days

Thus each cycle runs for approximately 162 days

(d) Production time per lot

t = Q/p

= 4874/190

= approx. 26 days

Thus, per cycle production and consumption simultaneously takes place for 26 days.

Example for Case 2: Given annual demand D = 6400 units, ordering cost = 100.00 and holding cost = 2.00 per unit per year. Compute the minimum cost production lot size for each of the following production rates:

(a) 8,000 units per year

(b) 10,000 units per year

(d) 100,000 units per year

Also, compute Economic order quantity (EOQ) and compare the results.

Solution:

(a) Q = √(2DS/H) * √(P/(P-D)

= √(2*6400*100/2) * √(8000/(8000-6400)

= 800 * 2.23

= 1784 units

(b) Q = √(2DS/H) * √(P/(P-D)

= √(2*6400*100/2) * √(10000/(10000-6400)

= 800 * 1.66

= approx. 1334 units

= √(2*6400*100/2) * √(32000/(32000-6400)

= 800 * 1.12

= 896 units

(d) Q = √(2DS/H) * √(P/(P-D)

= √(2*6400*100/2) * √(100000/(100000-6400)

= 800 * 1.03

= 824 units

Calculation of EOQ:

Q = √(2DS/H)

= √(2*6400*100/2)

= 800 units

Number of units ordered per batch remains same as estimation of optimum batch size by applying EOQ model is independent of number of units produced per year or in a given time period.

This provides us two important differences between EOQ and EPQ model:

EOQ model assumes number of units produced (p) in given time period to be constant whereas in EPQ model it can vary during that time period.
The above results does indicate that with increase in number of units produced per year batch size (Q) of each order keeps decreasing implying an inverse relation between batch size and number of units produced per year in EPQ model. Whereas batch size in EOQ remains same.

33.4 A CASE OF EPQ

As discussed in earlier sections Economic Production Quantity (EPQ) model consists of two sections.

The first section is termed as production phase where in following three activities are performed:

1. Production (p) at a constant rate
2. Consumption or daily demand (d) at constant rate
3. Because of the assumption of production rate being greater than consumption rate i.e. p>d there would always be replenishment of the product.

The second section is termed as non-production phase where in following two activities are performed:

1. As required production quantity has been achieved so there is only consumption at constant rate.
2. As there is no production so there would only be depletion of quantity produced.

This has been illustrated in following example:

A bottling plant undergoes the process of filling a particular cold drink. The process involves three steps. In the first step the used and emptied bottles are cleaned. Then in second step as the bottles gets cleaned they are stored in warehouse. Lastly, bottles get into production process of filling the drink. The filling machine according to its capacity takes empty and cleaned bottles and fills it. The process is shown in Fig.2

The production phase involves all three activities whereas non-production phase involves only activities of storing and filling. The production process has the capacity of producing 4000 hectares litres per month. 25% of this production is filled into cleaned bottles. 8000 bottles are cleaned on the cleaning line per day. Further following data is given:

Annual demand D = 1,20,000 cases per year
Production rate = 1,46,000 cases per year
Annual holding cost = 20 units per case
Set up cost = 12000 units per case

By using this data the company intends to find out following inventory parameters in the endeavor of minimizing total cost.

What is the optimum LOT SIZE?
What is the maximum inventory level?
What is the minimum total annual cost?
Time period for production phase.
Time period for non-production phase.

33.5 SUMMARY

Economic Production Quantity (EPQ) model computes optimum batch size that should be ordered incurring minimum possible cost when production and usage of a particular product occurs at same time. It is different from EOQ model where replenishment was done in batches and only consumption used to happen without any concern to production. The chapter discussed assumptions and model of EPQ. Also computation

of optimum batch size was discussed under two cases. Case 1 highlighted the scenario when production happened for part of ordering cycle whereas in case 2 production and consumption occurred simultaneously for entire ordering cycle. Also, differences between EOQ and EPQ were illustrated through an example. It was found that EOQ is independent of number of units produced whereas EPQ is not.

33.6 GLOSSARY

Economic Production Quantity (EPQ): is an inventory model which computes optimum batch size for continuous processes where part of units produced is consumed simultaneously.
Constant demand rate: is an assumption made in inventory models which states that same number of units is taken from inventory every time.
Constant supply rate: is an assumption made states that inventory is build at a constant rate over a period of time.

33.7 REFERENCES

Chase, B.R., Shankar, R., Jacobs, F.R. and Aquilano, N.J., Operations & Supply Chain Management, 12th Edition, McGraw Hill.
Stevenson, W.J., Operations Management, 9th Edition, Tata McGraw Hill.
Lee J. Krajewski, Operations Management, Prentice-Hall of India, New Delhi, 8th Edition.

33.8 SHORT ANSWER QUESTIONS

1. In the EPQ model, which assumption is relaxed?

a) Lead time is constant

b) Demand is constant

c) Items are received all at once

d) Supply is certain

Answer:c

2. EPQ model is different from EOQ model w.r.t:

a) EOQ model is independent of production whereas EPQ is not

b) In EPQ model production and consumption occurs at same time whereas in EOQ model only consumption happens

c) Both are true

Answer:c

3. EPQ model is divided in two phases. Phase of production and consumption and phase of consumption only. Should both phases be of equal time period?

(a) True

(b) False

Answer:c