An Economic Order Quantity Model with Shortages, Price Break and ...

Int. J. Emerg. Sci., 1(3), 465-477, September 2011 ISSN: 2222-4254 © IJES

An Economic Order Quantity Model with Shortages, Price Break and Inflation

Onawumi. AS, Oluleye. OE. and Adebiyi. KA. Department of Mechanical Engineering Ladoke Akintola University of Technology Ogbomoso, Nigeria. Department of Industrial and Production Engineering University of Ibadan, Ibadan. Department of Mechanical Engineering Ladoke Akintola University of Technology Ogbomoso, Nigeria. [email protected], [email protected], [email protected]

Abstract. The effect of inflation has become a persistent characteristic and more significant problem of many developing economies especially in the third world countries. While making effort to achieve optimal quantity of product to be produced or purchased using the simplest and on the shelf classical EOQ model, the non-inclusion of conflicting economic realities as shortage, price break and inflation has rendered its result quite uneconomical and hence the purpose for this study. Mathematical expression was developed for each of the cost components the sum of which become the total inventory model over the period (0, L) (TIC(0, L)). Significant savings with increase in quantity was achieved based on deference in the varying price regime. With the assumptions considered and subject to the availability of reliable inventory cost element, the developed model is found to produce a feasible, and economic inventory stock-level with the numerical example of a material supply of a manufacturing company in Nigeria. Keywords: Inflation rate, Inventory Management, Interest rate, Price break, Shortage Cost.

1 INTRODUCTION The classical economic order quantity (EOQ) model seeks to find the balance between ordering cost and carrying cost with a view of obtaining the most economic quantity to procure by the distributor. Literature reveals that sometimes up to 60percent of the annual production budget is spent on material and other inventories [5],[11]. An effective customer friendly and efficient supply system could be achieved by operating on economic order supply level (EOSL). Limiting conditions inherent in the traditional EOQ model give opportunity for several other inventory models which are published in some Journals [7],[13]. In this paper economically realistic situation where the availability of quantity discount which results in price break is explored in the face of double digit inflation rate and possible shortage of needed supply was considered and modeled. Incessant economic recession 465

Onawumi. AS, Oluleye. OE. and Adebiyi. KA.

worsened with the alarming inflation rate is a common trend in many third world countries. This is an issue of economic concern that necessarily requires major attention on management of inventory of products [12]. One of the advantages often explored to cushion the burden of net inventory cost and to enjoy substantial savings is the benefit from procuring large enough quantity that reduces the unit price of the item. Likewise in the absence of capacity restriction, the replenishment cost per unit is reduced as the quantity is increased. While procured quantity cannot be uncontrolled due to the adverse effect it could have on holding cost which ought to be held minimum, optimal inventory cost is only possible as all considered costs are balanced up [17]. It is evident that any quantity above the EOQ is a loss to the buyer who invariably is forced to reduce quantity procured to a level close to the calculated optimal quantity. Seller on the hand could loose sales as a result of this customer’s decision. To encourage buyer to increase his buying interest without experiencing any loss seller could provide price incentives to purchase large which in effect reduce price per unit. This could shift the EOQ in favor of both the seller as well as the buyer [12]. Quantity price structure is then considered as a benefit made available by manufacturer/seller as a matter of marketing policy to encourage retailer who buys larger quantity than actual order. The varying peculiarities of the supply inventory categories as well as divergent operating factors affecting inventory has contributed to the lack of particular inventory model that has general applications to the entire variants inventory situations. Consequently a variety of inventory models have emerged which address specific inventory problems [6], [9],[16]. In like manner the recent problem of economic meltdown also possesses much task on investment managers in the area monitoring the effects of inflation together with interest rate (return on capital) in relation to inventory problems. Usually, the problem is that of balancing the costs of less–than–adequate inventory (Under-stocking) and that of cost of more-than-adequate inventory (Over–stocking). The goal is to have adequate items at all times at minimal cost [7],[14][15]. Solution methods used for solving these problems are basically analytical techniques and the sophisticated application of mathematical programming. However, the mathematical complexity of the resulting models increases as we move away from the assumption of deterministic to probabilistic non-stationary demand [10],[14]. Silver [13] reviewed many classifications of the inventory problem, highlighting the limitations while also advocating the bridging of the gap between theory and practice. Buzacott, [3] noticed that the assumption of the classical EOQ formula that all relevant costs and prices cannot be valid in an economy that is plagued with double digit annual inflation rate and therefore developed an inventory model with inflation factor included. He suggested obtaining the optimal order quantity by a process of iteration. However closer examination reveals that the solution method is indeed an approximation; because of the assumption inherent in his use of a quadratic approximation. This paper considered an approach that seeks to balance the advantages of lower prices for purchased items and fewer orders and the disadvantages of the increased inventory holding cost. The amounts generally includes expected demand during lead time and perhaps an extra cushion of stock, which serve to reduce the risk of experiencing a stock-out during lead-time especially in the environment when variability is present in the demand, in the leadtime, or in both [2]. Generally, the determination of the optimal policy of an 466

International Journal of Emerging Sciences, 1(3), 455-464, September 2011

inventory model with a stochastic demand includes the calculations of the reorder point and the order size which involve the mean rate of the demand, the demand's standard deviation, the safety factor, and the forecasted lead-time.

2.0 MODEL DEVELOPMENT Using the principles of the classical EOQ model, the following assumptions are` made: 1. Demand rate is determinable and constant. 2. Supplies are delivered in batches. 3. Replacement is instantaneous on request. 4. Inflation rate is assumed constant over a period of time. 5. Unit purchase cost and other relevant costs are affected by inflation. 6. Shortages are allowed at a cost and over a given back-ordering time frame. 7. Purchase costs per unit change with quantity with discount. The situation of a determinable demand rate in which shortages are allowed is illustrated in Figure 1. The shortages could be backordered within the limit of the backlogged of the demand. The maximum inventory level is S and occur when the inventory is replenished. The lot size is less than the order level as a result of the backorder.

Inventory Level

S

Q Q/D

2Q/ D

0

Figure 1. Lot-size model with shortages allowed

The following notations are used in the model development.

Co

Initial Purchasing cost/Unit

C1

Set up cost

C2

Holding cost/Unit/Unit time

C3

Shortage cost /Unit/Unit time

C (t )

Cost at time t

C (0, L )

Ordering Cost over the period (0, L) Demand rate

D

467

Time


MT

Megaton

R

Rate of return on inventory investment

k l

Effective inflation rate

L S q

Planning Horizon Maximum inventory level Lot size

I

Inventory cycle

t1

Time interval before shortage

t2

Shortage period

T

Ordering interval Purchase cost per unit at q1 Purchase cost per unit at q2 Quantity at first Price break Quantity at Second Price break Quantity at which TIC(L,T) is at minimum

Number of orders

C01 C02 q1 q2 ym TIC ( L , T )

Total Inventory cost over period (0,L) using ordering interval T Total inventory cost over the period (0, L), can be expressed as

TIC ( L , T ) = (Set-up costs) + (Shortage costs) + (Holding costs) + (Purchase costs) The expression for each cost component is derived as follow:

(1)

2.1 Ordering or Setup cost over the period (0, L)

A simplifying assumption is that the ordering or setup is an aggregate of a fixed cost that is independent of the amount ordered, and a variable cost that depends on the amount ordered. This is the cost of placing an order to an outside supplier or releasing a production order to a manufacturing shop. It includes amongst other cost elements Clerical/labor costs of processing orders, inspection and return of poor quality products, transport costs and handling costs. The cost estimation of this cost taking into consideration all known cost elements is quite cumbersome. The quantity ordered also known as lot size is q. C1(T) is often a nonlinear function. Considering that cost increases with time then it is expressed as:

C1(T )  C1kT

(2)

Assuming of batch supplies then

L  lT

Over the period (0, L) then

C 1 ( 0 , L )  C 1  C 1 ( T )  C 1 ( 2 T )  .......... .......  C 1 (( l  1) T )

Therefore,

468

(3)


C 1 (0, L ) 

l 1

C n0

1

 KTn



C 1 ( 0 , L )  C 1  C 1  KT  C 1  2 KT  ......... C 1  ( l 1) KT



(4)

The geometric series can then be expressed as  e kL  1  C1 (0, L)  C1  kT   e  1

(5)

2.2 Holding costs over the period (0, L)

This is one of the vital costs that needs to be optimized in any logistics system. It is a well-known fact that the inventory holding costs is a part of the total logistics costs of a firm. It is also referred to as the cost of carrying an item in inventory for some given unit of time. This inventory cost component includes the lost investment income caused by having the asset tied up in inventory. Specifically holding cost is assumed to be a variable cost with cost components which include: Storage costs, Rent/depreciation, Labor, Overheads (e.g. heating, lighting, security), Money tied up (opportunity cost, loss of interest), Stock deterioration (lose money product due to deterioration whilst held), Obsolescence costs (if left product exceeds its useful life) and insurance [11],[13], [17] This is not a real cash flow, but it is an important component of the cost of inventory. During each order ordering period, the holding cost can be expressed as T t2

 DC (T,T  w)dw 2

w0

(6)

over the period (0,L) we then have l 1 T t2

C2 (0, L)    DC2 (T, T  w)dw n0 w0

(7)

but C 2 ( nT , nT  w )  RC 2 (T ) w

that is

C2 (0, L) 

DR(T  t 2 ) 2 l 1 C2 (T )  2 n 0

(8)

but C 2 ( T )  C 0  C 0 e KT  C 0 e 2 KT  .......... .......  C 0 e ( l  1 ) kT

Then the holding cost over the period (0,L) is expressed as: C DR(T  t2 )2  ekL  1 C2 (0, L)  0  kT  2  e  1

469

(9)


2.3 Shortage costs over the period (0, L)

When a customer seeks the product and finds the inventory empty, the demand can either go unfulfilled or be satisfied later when the product becomes available. The former case is called a lost sale, and the latter is called a backorder [8],[16]. The risk associated with stock out situation could place serious challenge on the firm’s customer service and goodwill. This cost includes penalty costs, machine idleness cost, operator idleness costs, loss of sales and cost of goodwill. It is also assumed to be proportional to the number of units backordered and the time the customer must wait. The constant of proportionality is p, the per unit backorder cost per unit of time. (N/unit-time). Shortage cost per period of time according to Buzacott, [1] is given by t2

 DC (T )m dm 3

n0

(10)

Thus over the period (0, L) the shortage cost becomes: l 1

C3 (0, L)  

T

 DC (nT, nT  m)dm

n o m T t1

3

But T-t1 = t2 Hence

Dt C3 (0, L)  2 2

2

 ekL  1  kT   e  1

(11)

2.4 Purchasing Costs over the period (0, L)

This is the unit cost of purchasing the product as part of an order. This include price paid on labour, material and overhead charges necessary to produce the item, [3],[14]. The product cost may be a decreasing function of the amount ordered especially in the case of purchase of large quantity which creates room for quantity discount. (N/unit). The initial purchase cost is DTC0 But cost increases over time such that C ( t )  C 0 e kt

(12) Purchasing cost over the period (0, L) is then given by

C1 (0, L)  DT(C0  C0 (T )  C0 (2T )  C0 ((l  1)T ) That is (l 1)

C0 (0, L)   DTC0e kTn

(13)

n0

470


This yield

 ekL 1 C0 (0, L)  DTC0  kT   e 1

(14)

The total inventory cost over period (0, L), TIC (0, L), was obtained by adding all the component costs developed in equations 5, 9, 11 and 14.

  ekL 1 C DR(T  t2 )2 2 TIC(0, L)  C1  0  C3Dt2  C0TD kT  2   e 1 (15) Given the objective of minimizing costs, the solution can be obtained by iteration (varying T values to obtain minimum cost). 2.5 Quantity Discount

Price breaks are introduced at q1 and q2 as shown in Figure 2. These two differ from ym ym ≤ q1: In this case, the first price break is introduced. Corresponding price offered for this is Co1. TIC(ym) = TIC(y*) if it is the least cost and also feasible at any value less than q1. q1