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An economic order quantity model with shortages, price break and inflation.

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 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 lead-time, or in both [2]. Generally, the determination of the optimal policy of an 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.

[FIGURE 1 OMITTED]

The following notations are used in the model development.
[C.sub.o]    Initial Purchasing

               cost/Unit
[C.sub.1]    Set up cost
[C.sub.2]    Holding cost/
               Unit/Unit time
[C.sub.3]    Shortage cost/
               Unit/Unit time
C(t)         Cost at time t
C(0,L)       Ordering Cost
               over the
               period (0, L)
D            Demand rate

MT           Megaton
R            Rate of return
               on inventory
               investment
k            Effective
               inflation rate
l            Number
               of orders
L            Planning
               Horizon
S            Maximum
               inventory level
q            Lot size
I            Inventory
               cycle
[t.sub.1]    Time interval
               before shortage
[t.sub.2]    Shortage period
T            Ordering interval
C01          Purchase cost
               per unit at q1
C02          Purchase cost
               per unit at q2
q1           Quantity at first
               Price break
q2           Quantity at Second
               Price break
ym           Quantity at which
               TIC(L,T) is at
               minimum
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) (1)

The expression for each cost component is derived as follow:

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:

[C.sub.1](T) = [C.sub.1][[??].sup.kT] (2)

Assuming of batch supplies then

L = lT

Over the period (0, L) then

[C.sub.1](0,L) = [C.sub.1] + [C.sub.1](T) + [C.sub.1](2T) + .......... ....... + [C.sub.1]((l - 1)T) (3)

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

The geometric series can then be expressed as

[C.sub.1](0, L) = [C.sub.1] [[e.sup.kL]-1 / [e.sup.kT]-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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

over the period (0,L) we then have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

but

[C.sub.2](nT, nT + w) = [RC.sub.2](T)w

that is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Then the holding cost over the period (0,L) is expressed as:

[C.sub.2](0,L)= [C.sub.0]DR[(T-[t.sub.2]).sup.2] / 2 [[e.sup.kL]-1 / [e.sup.kT]-1] (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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (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.sub.0][e.sup.kt] (12)

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

[C.sub.1](0, L) = DT([C.sub.0] + [C.sub.0](T) + [C.sub.0](2T) + [C.sub.0]((l-1)T)

That is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

This yield

[C.sub.0](0,L)= [DTC.sub.0] [[e.sup.kL]-1 / [e.sup.kT]-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.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (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 [less than or equal to] q1: In this case, the first price break is introduced. Corresponding price offered for this is Co1. TIC(ym) = TIC([y.sup.*]) if it is the least cost and also feasible at any value less than q1.

q1<ym<q2: This indicates the existence of further price break with quantity between q1 and q2. The range in the quantities creates a limitation for the possible solution.

q2 [less than or equal to] ym: The second quantity that provide incentive for buyer at a much more reduced price C02. TIC(ym) = TIC(y*) if at a quantity equal to or greater than q2, there exist a feasible solution that is less than other feasible solutions [16].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

[FIGURE 2 OMITTED]

3. APPLICATION

In order to demonstrate the application of the proposed model, data for an imported item, in respect of a cement manufacturing company in Nigeria was obtained. Questionnaires, interviews and company record were utilized to generate the relevant data. Basic data for which the model was applied were stored in spreadsheet format and graphed using EXCEL 2007. Data used include: Times were varied with minimal interval ([DELTA]T= 0.01year) in order to locate point of inflection on the concave graph obtained. The optimal reorder period ([T.sup.*]) was obtained at minimum total inventory cost. Optimal reorder quantity is then: [Q.sup.*] = [T.sup.*]D
16

Unit cost ([C.sub.0]) (at [less than
  or equal to] ym)                      = N 120, 000 / MT
Ordering Cost ([C.sub.1])               = N 100, 000 / MT
Stock out cost ([C.sub.3])              = N 50, 000 / MT
Annual Demand (D)                       = 250MT
Effective Inflation rate (k)            = 25%
Interest rate (R)                       = 40%
Time shortage is allowed (t2)           = 1 month
Planning horizon (L)                    = 1 year
Quantity for which no discount is
  allowed (q)                           = [less than or equal to] 75MT
Quantity at first Price Break (q1)      = 76-105MT
Quantity at Second Price Break (q2)     = >105
Price Break at q1 (Co1)                 = N105,000/MT
Price Break at q2 (Co2)                 = N95,000/MT


4. DISCUSSION

Table1 presents the inventory stock level for different unit prices for varying quantities. The stock level obtained where checked if feasible within the limit allowed for price per unit to be discounted. The feasible price break quantity exists between 76 and 105MT having price per unit of N105,000 and minimum inventory cost of N3013761.27. The associated order interval is 0.32 (3.125orders).

Figure 3 shows that the best purchase quantity for which the deepest discount is found on q1 curve with corresponding value of 80MT.

A comparison with current inventory cost of N35,664,453.00 shows a decrease of 15.5%. This is indicative of the significant benefits derivable from the use of the model by the organization.

Figure 4 shows that price per unit varies with purchase quantity at different levels of price break Figure 5 however shows that the inventory cost is quite sensitive to available price level and that the higher the quantity the lower the total inventory cost.

The significant effect of shortage period allowed is noticed in Figure 6 which support longer shortage interval. However, Figure 7 shows that while inventory costs vary with shortage periods allowed; the ordering policy is not as sensitive. It is indeed a step function. This implies that while the shortage period may vary, the number of orders required in the planning horizon may not. In real terms however, we note that an organization would probably prefer to let the number of orders be an integer value. Given, this the optimal cost and number of orders can then be calculated by perturbation. For example, given the solution of 3.125orders/annum obtained in the case examined and compared with the cost of 3orders/annum and 4orders/annum respectively in the planning horizon, the policy with lesser cost is then chosen. In this case, 3orders/year yields a savings of 15.5%.

The combined effect of inflation and interest rate are reflected in the value of total inventory cost. While increase in interest rate has lead to significant increase in the total inventory cost, inflation rate on the other hand is responsible for the reduction in the effective value of total inventory cost. The frequency of reorder contributed to high rate of increase in holding cost and consequently to increased TIC with inflation. It important to note that inflation rate is a necessary factor in the determination of effective lending rate in a growing economy.

5. CONCLUSION

We have reviewed the importance of inventory cost minimization with a view to increasing organizational profitability and liquidity. The sensitivity of total inventory cost to price break has been demonstrated with significant cost savings. An inventory model which considers shortages, inflation factors and price break incentive was derived and applied to a case example; in order to highlight the utility of the model. The reliability of this model depends on the correctness of the inventory cost elements used which is outside the scope of this study. Special emphasis is placed the effect of bulk purchase that can enhance saving on inventory in the face of inflationary situation. However the benefits derivable from the utilization of the model appear immense.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

REFERENCES

[1.] Balakrishnan. A, Michael. SP, Euthemia. S, "Stack Them High, Let 'Em Fly." Management Science. May 2004.

[2.] Boone. T. and Ganeshan. R., "Models and Methods to Support a New Type of Inventory Performance Measure: The ESWSO", Decision Sciences, Vol. 31 (1), 2000, 243-258

[3.] Buzacott. JA, "Economic Order Quantities with Inflation" Operation Research Quarterly. 1975; Vol. 26 No. 3. 553-558.

[4.] Chun -Tao. C, "An EOQ model with deteriorating items under inflation when supplier credits linked to order quantity" International Journal of Production Economics, 2004; Volume 88, Issue 3, 18; Pages 307-316.

[5.] Ding. QP, and Kouvelis. JM, "Dynamic pricing for multiple class deterministic demand fulfillment". IIE Trans. 2007; 39(11) 997-1013.

[6.] Ding. QP, and Kouvelis. JM, "Dynamic pricing through discounts for optimizing multiple-class demand fulfillment". Oper. Res.2006; 54(1) 169-183.

[7.] Harvey. MW, "Principles of Operations Research", Prentice-Hal International Inc. London. 1987.

[8.] Khouja. M, and Sungjune P, "Optimal Lot Sizing Under Continuous Price Decrease" Omega. December 2003.

[9.] Lev. B, Weiss. HJ, and Soyster. A.L, "Optimal Ordering Policies when anticipating Parameter changes in EOQ Systems", Naval Research Logistics Quarterly, 1981; 28.(2).

[10.] Lev. TB, and Soyster. AL, "An Optimal Model with Finite Horizon and Price changes", Journal of Operations Research Society, 1979; 30(1).

[11.] Lucey. T, "Quantitative Techniques, An Instructional Manual". Third Edition, ELBS. 1988; 196-222.

[12.] Piasecki. D, "Optimizing Economic Order Quantity". IIE Solutions. January 2001.

[13.] Silver. EA, "Operations Research in Inventory Management: A Review and Critique", Operations Research, 1981; Vol. 3. No.1.

[14.] Taha. AH, "Operations Research Introduction", Macmillan Publication Company, New York. 1971.

[15.] Wang. KJ, Hui-Ming. W, Shin-Feng. G, and Shen-Lian. C, "Production and Inventory Control with Chaotic Demands" Omega. April 2005.

[16.] Wemmerlov. U, "Inventory Management and Control", Handbook of Industrial Engineering (ed. Salvendy, G.), John Wiley and Sons. 1982.

[17.] Woolsey. RE, and Ruth. M, Inventory Control (For People Who Really Have to Do It). Lionheart Publishing, March 2001.

Onawumi. A. S., Oluleye. O. E. and Adebiyi. K. A.

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.

ayoyemi2001@yahoo.com, ayodeji.oluleye@mail.ui.edu.ng, engradebiyi@yahoo.com
Table 1. Feasible Total Inventory Cost

Q (for Price Break)   [Q.sup.*]   Co(N/MT)    TIC (N)
<76                   75          120000:00   34371059.08
76-105                80          105000:00   [30137511.27.sup.*]
>105                  85           95000:00   27312824.48
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Author:Onawumi, A.S.; Oluleye, O.E.; Adebiyi, K.A.
Publication:International Journal of Emerging Sciences
Article Type:Report
Geographic Code:6NIGR
Date:Sep 1, 2011
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