34 Inventory Models: Quantity Discounts

Vikas Singla

34.0 Objectives

34.1 Introduction

34.2 Quantity Discounts Model

34.2.1 Issues in Estimating Best Lot Size

34.2.2 Procedure used to find the best lot size

34.2.3 Derivation of Quantity Discount model

34.3 Practical Examples

34.4 A Case on Quantity Discount Model Of EOQ

34.5 Summary

34.6 Glossary

34.7 References/ Suggested Readings

34.8 Short Answer Questions

34.0 OBJECTIVES

This chapter would help students to understand:

Application of Quantity discount model of inventory management Assumptions of Quantity Discount model

Calculation of optimum batch size and minimum total inventory cost

34.1 INTRODUCTION

A fruit vendor sells a particular fruit at Rs.50 per kg. Thus two kilogram of fruit is sold at Rs.100. But at the end of the day the vendor due to perishable nature of product and intention to clear the stock is willing to sell two kilograms of same fruit for Rs.80. If a customer is interested in buying more than two kgs vendor is willing to reduce price further. The question for customer is how much he/she should buy and at what cost. Buying more would might result in fall in total cost but it also increases holding and maintenance cost of larger bulk of products. These similar questions of how much and at what cost have been answered in other inventory models. But in those cases a particular size of quantity was ordered at constant price. There was no fluctuation in prices of product with change in size of order. But in practical situations especially in case of perishable products or products which are produced and sold in high quantities discounts are offered by the manufacturer depending on size of order. In these cases a special inventory model called as Quantity Discounts is applied to answer the questions of ordering optimum quantity at lowest cost.

34.2 QUANTITY DISCOUNTS MODEL

Quantity discount model examines the effect of price incentives given in case of larger quantities of products ordered. Thus, in such model only those problems are discussed where price fluctuates with regard to size of order. Larger the size of order more would be the discount offered. Larger size orders would help in reducing ordering cost as frequency of orders would be less but increases holding cost as larger number of units would be stocked. The model tries to provide a quantitative solution to balance the tradeoff between holding and ordering cost. Following illustration would help in understanding the working of model.

For example, a supplier is selling first Q=1000 units at a price of Rs.10 per unit and then offers three discount plans which were:

Discount 1	Q=0-1000 units	Rs.10.00
Discount 2	Q=1001-2000	Rs.9.50
Discount 3	Q=more than 2000	Rs.9.00

As it can be seen from above data that price of product is not fixed and with varying quantity of product ordered price keeps changing. With change in price of product total cost gets affected. The total cost curve for three different discount levels have been shown in Fig. 1.

The figure explains following points:

• First 1000 units are being offered at Rs.10 per unit and yields first U shaped total cost curve indicated by total cost curve for discount 1.
• The cost curve drops down with increase in quantity and reduction in price per unit as shown by first price break point at 1000 units. The total cost U shaped curve for this situation is shown by total cost curve for discount 2.
• Lastly, at second price break point occurring at 2000 total cost curve falls further as indicated by total cost curve for discount 3.

The result is a total cost curve, with steps at the price breaks.

34.2.1 Issues in estimating best lot size

The figure also illustrates EOQ level indicating estimation of optimum batch size at minimum possible cost.

The EOQs shown in the figure do not necessarily produce the best lot size for two reasons:

1. EOQ is not feasible: The estimated lot size at particular price point which would result in minimum cost might not correspond with quantity offered by the supplier. For instance, point a in the figure indicates that by buying at price of Rs.9.50 per unit number of units to be ordered per batch would be less than 1000. But this quantity at Rs.9.50 is not feasible for the supplier. So order size has to be increased from a to minimum of 1000 units to avail discount.

2. Total cost is not feasible: Suppose estimated batch size calculated at particular price point is feasible for supplier. For instance EOQ comes out to be at 1750 units at Rs.9.50 which is feasible as it falls in the range offered by the supplier. But its total cost is higher than the price break point of for instance 2000 units. Then even in the case of EOQ being feasible batch size with lowest total cost would be ordered.

34.2.2 Procedure used to find the best lot size

Step 1: Calculate the EOQ for each price level until a feasible EOQ is found. In the above example calculate EOQ for Rs.10, Rs.9.50 and Rs.9.00. If the calculated EOQ falls in the range of units offered by supplier then that EOQ is feasible. As discussed EOQ of 1750 units at price level of Rs.9.50 was feasible implying batch size of 1750 unit lies in the range corresponding to its price.

Step 2: Calculate total cost for each price break quantity. If the calculated EOQ is feasible for first discount level then total cost should be calculated for that quantity and for larger price break quantity at each lower price level. If calculated EOQ is feasible for second discount level then total cost should be calculated for that quantity level and higher break quantity. The quantity giving lower total cost would be considered as optimum batch size.

34.2.3 Derivation of Quantity Discount model

In quantity discount model item’s price is not fixed as assumed in EOQ derivation. Larger the batch size higher would be the discount offered. Thus total cost is a direct function of price per unit of the product. In this model total cost is combination of three costs, namely holding cost, ordering cost and product cost. Holding and ordering cost are calculated as was done in EOQ model.

Holding cost: Between time period of start and end of cycle (time period t) on an average at any particular unit

Q/2 units are being held up in inventory. If to hold one unit holding cost is H then:

Annual holding cost = Average inventory level * Holding cost per unit

                                    = (Q/2) * H

The holding cost in this model is specifically in terms of percentage as costlier the item more would be its holding cost.

Ordering cost: Suppose annual demand of an item is 1000 units and manager orders 100 units per order. Thus he/she has to place an order 10 times. Thus (D/Q) represents number of orders. If S is set up or ordering cost per order then:

Annual ordering cost = Number of orders per year * Ordering or set up cost per order

                                      = (D/Q) * S

Product cost: If ‘P’ is the cost of one unit of product and ‘D’ is the annual demand i.e. total number of units ordered per year then total product cost would be ‘PD’.

Thus

Total cost      =    Annual holding cost + Annual ordering cost + Product cost

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

Where, Q-lot size

H: Holding cost of one unit

D: Annual demand

S: Ordering or Set up cost

P: price per unit.

To find minimum possible cost the above equation is differentiated w.r.t. Q and simple minima

calculus is applied to find out batch size Q which would be

Q  =  √(2DS/H)

This formula of Q would provide manager the most optimum batch size that should be ordered with minimum total annual cost.

34.3 PRACTICAL EXAMPLES

Example 1: A supplier has introduced quantity discounts to encourage larger order quantities of an item. The price schedule is

The supplier estimates that its annual demand for the item is 936 units, its ordering cost is Rs.45 per order, and its annual holding cost is 25 percent of the item’s unit price. What quantity of the item should the company order to minimize total costs?

Solution:

Step 1: Find the first feasible EOQ, starting with the lowest price level:

EOQ (57.00) = {(2DS)/(H)}1/2

= {(2*936*45)/(0.25*57)}1/2

= 77 units

A 77 unit order actually costs 60.00 per unit, instead of the 57 per unit used in the EOQ calculation, so this EOQ is not feasible. Now,

EOQ (58.80) = {(2DS)/(H)}1/2

                      = {(2*936*45)/(0.25*58.80)}1/2

                      = 76 units

This quantity also is infeasible because a 76 unit order actually costs 60.00 per unit. So try the highest price level:

EOQ (60.00) = {(2DS)/(H)}1/2

                       = {(2*936*45)/(0.25*60.00)}1/2

                        = 75 units

This quantity is feasible, because it lies in the range corresponding to its price, P = 60.00

Step 2: The first feasible EOQ of 75 does not correspond to the lowest price level. Hence, we must compare its total cost with the price break quantities (300 and 500 units) at the lower price levels (58.80 and 57.00)

TC	= (Q/2)(H) + (D/Q)(S) + PD
TC (75)	= (75/2)(0.25*60.00) + (936/75)(45) + 60*936
	= Rs.57,284
TC (300)	= (300/2)(0.25*58.80) + (936/300)(45) + 58.80*936
	= Rs.57,382
TC (500)	= (500/2)(0.25*57.00) + (936/500)(45) + 57*936
	= 56,999

The best purchase quantity is 500 units, which qualifies for the deepest discount.

Example 2: Following table indicates the discounts offered by the supplier for various quantity. Suppose data and cost analyses show an annual holding cost of 20%, an ordering cost of Rs.49 per order and an annual demand of 5000 units. What order quantity should be selected?

Discount 1	Q=0-999 units	Rs.5.00
Discount 2	Q=0-1000-2499	Rs.4.85
Discount 3	Q=more than 2500	Rs.4.75

Solution:

Step 1: For each discount category compute batch size i.e. Q using the EOQ formula associated with quantity discount model.

EOQ (at P=5.00)  =  {(2DS)/(H)}1/2

                              = {(2*5000*49)/(0.20*5)}1/2

                              = 700 units

This quantity is feasible, because it lies in the range corresponding to its price, P = 5.00

EOQ (at P=4.85)  =  {(2DS)/(H)}1/2

                             = {(2*5000*49)/(0.20*4.85)}1/2

                             = 711 units

A 711 unit order actually costs 5.00 per unit, instead of the 4.85 per unit used in the EOQ calculation, so this

EOQ is not feasible.
EOQ (at P=4.75) = {(2DS)/(H)}1/2
= {(2*5000*49)/(0.20*4.75}1/2
= 728 units

A 728 unit order actually costs 5.00 per unit, instead of the 4.75 per unit used in the EOQ calculation, so this EOQ is not feasible.

Step 2: Calculation of Total cost:

As discussed total cost would be calculated for the feasible quantity i.e. where the discount offered is viable and also for the quantity break points. In this example Q=700 units is feasible and Q=1000 and Q=2500 units are the break points. So, by using

TC                                         =  (Q/2)(H) + (D/Q)(S) + PD

TC (at Q=700 and P=5.00) =  (700/2)*(0.20*5) + (5000/700)*(49) + 5*5000

                                              = 350 + 350 + 25,000

                                              = Rs.25,700

TC (at Q=1000 and P=4.85)  = (1000/2)*(0.20*4.85) + (5000/1000)*(49) + 4.85*5000
= 485 + 245 + 24,250
= Rs.24,980

TC (at Q=2500 and P=4.75)  = (2500/2)*(0.20*4.75) + (5000/2500)*(49) + 4.75*5000
= 1188 + 98 + 23,750
= Rs.25,036

Thus it can be seen from calculations that at Q=1000 units and each unit being sold at a discount price of Rs.4.85 total cost is minimum. A batch size of 2500 units does offer maximum discount but larger batch size results in maximum holding cost thereby making it second best solution. The example illustrates following two very interesting characteristics of quantity discount model:

• The feasible batch size does not necessarily result in most optimum total cost.
• With increase in batch size holding cost keeps increasing and ordering cost keeps decreasing.

34.4 A CASE ON QUANTITY DISCOUNT MODEL OF EOQ

In this chapter, we shall analyze a company’s inventory control problem and apply our proposed method in solving our problem. The aim is to determine the maximum discount price that should be taken, and the quantity that should be ordered giving the quantity discount. The general practice is that most establishments do not analyze the quantity discount that companies offered them with their carrying cost. They either go for it or reject it by the discretion of people or departments in charge. These are basically inefficient as carrying cost may be high or low.

The Economic Order Quantity (EOQ) is a pure economic model in the classical inventory control theory. The model is designed to find the order quantity so as to minimize the total average cost of replenishment under deterministic demand and some simplifying assumptions. The study focuses on inventory management when the unit purchasing cost decreases with the order quantity, Q. The main objective of the study is to model Economic Order Quantity with quantity discount to achieve optimal level of inventory. The model was analyzed against the current practices of an organization’ ordering policies to find out whether it is appropriate to go for quantity discount when offered. The Management of the organization wants to determine if it should take advantage of the discount or order basic EOQ order size offered to them by their suppliers. Suppliers of Non-Security Examination Materials, occasionally give discounts to the organization in line with its organizational policy in order to reduce a large stock of its materials which may increase its holding cost. In 2012 it offered a quantity discount pricing schedule (as shown in table below)

Quantity  (‘000)	Price (Rs. per unit)
1-49	1400
50-89	1100
90-349	900
350 and above	890

The annual carrying cost for the organization for such materials is 190, the ordering cost is 2,500, and annual demand for this particular model is estimated to be 200 units. The management wants to determine if it should take advantage of this discount or order the basic EOQ order size.

We first determine the optimal order size and total cost with the basic EOQ model.

S = 2,500

H = 190 per material

D = 200 materials per year

Where

S = the Ordering Cost

H = Carrying Cost

D = annual demand

P = Price

Q = Optimal Quantity

Q= {(2DS)/(H)}1/2

= {(2*25000*200)/(190)}1/2 = 72.5 material

Although we will use Qopt = 72.5 in the subsequent computations, realistically the order size would be 73 materials. This order size is eligible for the first discount of Rs. 1,100; therefore, this price is used to compute total cost:

TC = (Q/2)(H) + (D/Q)(S) + PD

= RS. 233,784.

Since there is a discount for a larger order size than 50 units, this total cost of must be compared with total cost with an order size of 90 and a discounted price of 900.

TC = Rs. 194, 105

Since there is a discount for a larger order size than 90 units this total cost of must be compared with total cost with an order size of 350 and a discounted price of 890

TC = 212,678.57.

The maximum discount price that gave lowest cost was 900 at a minimum quantity of 90,000. Even though this total cost is lower among all the total costs, the maximum discount price of 900 should not be taken at the required quantity range of 90,000-349,000 units should be ordered. The reason being that, if institution goes for the discount at that minimum levels it might not meet its quantity obligation to satisfy the total number of candidature. In this case or situation, the next discount price of 890 is considered at a minimum quantity of 350. We observed that there is no order size larger than 350 that would result in a lower cost.

34.5 SUMMARY

Quantity discount model is the appropriate model of finding optimum batch size which would result in least possible total cost when supplier offers differentiating price. With increase in batch size supplier might offer a batch size at discounted price. In such cases manufacturer might be lured to buy larger batch size. But this also results in higher holding cost and lower ordering cost. Thus it becomes necessary to resolve such a dilemma quantitatively by using quantity discount model. Before deriving for the most optimum batch size the chapter has also discussed two issues that should be understood in the model. Firstly, the calculated EOQ might not be feasible as though it results in lowest cost but might not correspond with the discount offered by the supplier. Secondly, total cost calculated at EOQ might not be the lowest total cost. These issues were resolved quantitatively by discussing derivation of the model and illustrated through some practical examples.

34.6 GLOSSARY

• Quantity discounts: are the lower unit costs offered by the supplier in lieu of larger quantities of product ordered by customer.
• Total cost: is the cost incurred in ordering a fixed number of units. In quantity discount model it is a combination of holding, ordering and product cost.
• Holding cost: In quantity discount model as price fluctuates with change in number of units so would be the holding cost. Thus, it is always estimated in percentage terms.

34.7 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.

34.8 SHORT ANSWER QUESTIONS

1. Consideration of ___________ of every unit differentiates a basic EOQ model from Quantity discount model.

Answer: purchase price

2. Purchase price per unit has a significant impact on ______________.

Answer: optimal batch size

3. Higher batch size would result in higher____________ cost and lower __________ cost.

Answer: holding, ordering

4. A feasible EOQ might not be best batch size because:

(a) It might not result in minimum possible total cost

(b) The price per unit offered by the supplier at that EOQ level might not be feasible.

(c) Both a and b.

Answer: c