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

1 INTRODUCTION

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

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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.

```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
```