Inflationary Implications on an Inventory with Expiration Date, Capital
Constraint and Uncertain Lead Time in a Multi-Echelon Supply Chain
Richa Jain*
Operations Management and Decision Sciences, Asia Pacific Institute of Management, New Delhi, India E-mail: dr.richajain@gmail.com
*Corresponding author S.R. Singh
Department of Mathematics, D.N. (P.G.) College, Meerut, Uttar Pradesh, India E-mail: shivrajpundir@gmail.com.
Abstract
The present study considers a multi-echelon supply system comprising of a supplier, a retailer and the end customer. The inventory is deteriorating and has a certain expiration date beyond which there is no demand for the item. The practicalities of real business world are embedded in the setup as the supplier’s lead time has been considered, which is a random variable. The effect of inflation has been accounted for to provide economic feasibility to the model. There is an upper limit to the expenses that the supplier can bear, providing him with a capital constraint. An optimal solution has been arrived at, and the sensitivity of the solution shows that the model is very stable.
Keywords: Inventory, inflation, expiration date, lead time, capital constraint.
1. Introduction
The concept of deteriorating inventory is not new and has been considerably explored by the researchers. Any item or product, after being produced or manufactured experiences a decline in its attributes and characteristic properties. This may be due to time, whether, storage conditions, pilferage, or any other aspect. But due to its non-ignorable importance in practical business world, this fact has been studied at large by various researchers and scientists in different forms.
There is hardly any country in the capitalist world today, which is not afflicted by the specter of inflation. It is on account of this that the phenomenon of inflation has widely attracted the attention of the economists from all over the world, but despite this fact, there is no generally accepted definition of inflation. Normally, we can define
inflation as too much currency in relation to the physical volume of business being done. Put simply, inflation is the general increase of prices and subsequent fall in purchasing power of money. The cycle of price inflation starts when production costs increase for the same level of production which results in an increase in the product price. This leads to a reduction of purchasing power which in turn, leads to higher wage demands, and therefore, higher production costs. This is precisely the reason that scientists from the research world are busy interpreting the effects of inflation over inventory systems.
The supply chain models in inventory are a comparatively new foray for researchers. There has been especially very limited research for a supply chain in inflationary environment. Supply chain is the linked set of resources and processes that begins with the sourcing of raw material and extends through the delivery of end items to the final customer. It includes vendors, manufacturing facilities, logistics providers, internal distribution centers, distributors, wholesalers and all other entities that lead up to final customer acceptance. The extended supply chain for a given company may also include secondary vendors to their immediate vendors, and the customers of their immediate customers. Every business establishment works through a complex supply chain. That is why it became imperative for researchers to deal with this particular branch of study.
The uncertainty in the lead time of a supplier is such a phenomenon which has deep roots in reality. Almost every supplier faces this problem at some time or the other during his business deals. Even then, this is a realm which has not been sufficiently explored by researchers.
In the present study, we have strived to combine all the above mentioned factors into a single problem. We shall undertake to explore a three echelon supply chain, comprising of a supplier, a retailer and end customers. The whole environment of business dealings has been assumed to be inflationary, which conforms to the practical market situation. The supplier experiences the problem of lead time at his end. So even though the retailer has placed the order, he has to wait for the supplier to deliver. The lead time of the supplier is a probability density function of his managing cost. The more the supplier is ready to spend as managing expenses, smaller will be the lead time and vice versa. However, there is a constraint upon the amount the supplier can afford to
spend, hence there is capital constraint also involved. The product starts deteriorating as soon as they reach the retailer, wherefore their demand also decreases with decay. There is even an expiration date beyond which the product cannot be used. Hence, the retailer plans out his cycle to finish off his inventory before the product reaches its expiration date. As a matter of fact, the retailer takes the length of the lifetime of the product to be his planning horizon and does not wish to retain inventory after that date.
Also, since the lead time is originating at the supplier’s end, hence the retailer makes it a point to recover his losses of shortage and lost sale from the supplier himself. In this way, the supplier is even more desperate to reduce his lead time as much as possible.
The whole combination is very unique and very much practical. The setup has been explored numerically as well, an optimal solution has been reached at. The sensitivity of that solution has also been checked with respect to various system parameters. The final outcome shows that the model is not only economically feasible, but stable also.
2. Literature Review
The concept of deteriorating inventory is not new and has been considerably explored since the advent of Ghare and Schrader (1963). Since then, lots of researchers have studied this concept. Later Nahamias (1982) also undertook to study the perishability of items. Raafat (1991) put forward an extensive survey on continuous deterioration of inventory models which was extended by Goyal and Giri (2001). Yang and Wee (2002) explored a supply chain model with single vendor and multiple buyers for a deteriorating inventory. Balkhi and Benkherouf (2004) undertook to study inventory models with stock and time dependent demand for a deteriorating product. Lately, Jaggi and Verma (2010) have developed a two-warehouse inventory model for deteriorating items.
Buzacott (1975) was the first scientist to have addressed the problem of inflation. Later on many scientists followed his league, like Misra (1979), Chandra and Bahner (1985), Bose et. al. (1995), Wee and Law (1999). Chung and Lin (2001) studied an EOQ model for a deteriorating inventory subjected to inflation. Hou (2006) derived an inventory model for deteriorating items with stock-dependent consumption rate and
shortages under inflation and time discounting. Lo et. al. (2007) developed an integrated production and inventory model from the perspectives of both the manufacturer and the retailer under inflation. Jaggi and Khanna (2009) addressed the retailer's procurement policy, where the decision was influenced by the inflation and time value of money under a permissible delay in payments. Lately Singh and Jain (2009b) have deliberated the effects of inflation on a supply chain with reserve money and supplier credits.
The idea of joint total cost of the supplier and the customer was first introduced by Goyal (1977). Later, Cohen and Lee (1988) determined material requirement for all materials at every stage in a supply chain. Pake and Cohen (1993) extended the above study to consider for stochastic sub systems. Gyana and Bhabha (1999) explored a single manufacturing system for procurement of raw materials with a multi-ordering policy that minimizes the total inventory costs of both the raw materials and the finished goods. Sarker et.al (2000) explored a supply chain model for determining an optimal ordering policy under inflation and allowable shortages. Chien and Lin (2004) investigated the optimal order interval and discount price such that the joint total cost is minimized during a finite planning horizon. Ahmed et. al (2008) have recently coordinated a two level supply chain in which they considered production interruptions for restoring of the quality of the production process. SrikanthaDath et. al. (2008) studied the relationship between the factors of SCM and the measures of performance is investigated in a moderated environment in India. More lately Singh and Jain (2009a) have studied the effects of inflation on a supply chain with reserve money and supplier credits. Also, Kim and Goyal (2009) investigated how the transportation cost affects the joint lot-sizing policy in a multiple-supplier-single-buyer supply chain.
Coming to lead time problems, Liao and Shyu (1991) showed that lead time can be controlled through crashing. Later on, Ouyang et. al. (1999) investigated uncertain lead time and studied the effect of cost reduction in a continuous review inventory model. Even then, this is a realm which has not been sufficiently explored by researchers.
3. Assumptions and Notations
The following assumptions and notations have been adopted for the proposed model to be discussed.
1. Deterioration starts as soon as items are received by the retailer.
2. There is no repair or replenishment of deteriorating items during the period under consideration.
3. There is no deterioration at the supplier’s end due to proper storage conditions. 4. Single item inventory is considered.
5. The demand rate is a decreasing function of time.
6. Lead time of the supplier has been considered, which is a random variable. 7. Inflation and time value of money is considered.
8. Shortages are allowed and partially backlogged. The following notations are used in our study:
Ir(t) Inventory level of the retailer at any time t.
D Demand rate (units/unit time), D=e−λt where λ is a constant. θ Deterioration rate of the inventory, where θ is a constant.
r Constant representing the difference between the discount rate and inflation rate. y Lead time of the supplier which depends on the supplier’s managing cost.
B Constant rate of partial backlogging for the retailer. c Purchasing cost per unit item for the retailer $/unit item. s Selling price per unit item for the retailer $/unit item. c1 Inventory holding cost for the retailer $/unit item/year.
c2 Shortage cost per unit backordered per unit item for the retailer $/unit item/year.
cL Lost sale cost for the retailer $/unit item/year.
cp Production cost for the supplier $/unit item/year.
μ Managing cost for the supplier for reducing lead time. A Setup cost per cycle for the retailer $/cycle.
T Replenishment cycle length and the lifetime of the commodity. T1 Time to zero inventory.
X Supplier’s maximal capital constraint.
4. Mathematical Model
The model begins operating from time t=0. The retailer places the order at t=0 but due to the lead time problem at the supplier’s end, the delivery of the order does not reach the
retailer before t=y. During this time the retailer backlogs the shortages arising due to absence of stock. These shortages are backlogged partially at a constant rate B. At time t=y stock arrives at the retailer’s end. After the arrival of stock, the retailer first clears the backlog. The rest of the stock goes into fulfilling the demand which comes to the retailer. The retailer had initially ordered stock to survive for T units of time, where T is the lifetime of the item, i.e. after T units of time, the entire leftover stock will expire and will no longer be demanded. But due to the delay at supplier’s end, the retailer is forced to backlog shortages at the beginning of the cycle itself. As a result, after the clearing of the backlog, the left over stock is not sufficient to last for another T units of time. After time t=y, the stock diminishes due to both demand and deterioration, and finally finishes off at time t=T1.
The demand for an item with an expiration date always declines steeply in the proximity of that date. Hence we have considered an exponentially declining form of demand. Graphically, the retailer’s cycle can be represented as follows:
Mathematically the system can be represented as: ' r I (t)= −BD(t) 0≤ ≤ , with t y I (0)r = 0 ' r r I (t) = -θI (t) - D(t) y≤ ≤ with t T1 I (T )r 1 = 0
Solving the above equations with the boundary conditions, we obtain:
(
t)
r B I (t)= e−λ −1 λ 0≤ ≤ t y …1 0 I0 Time Inve nt or y T1 y( ) ( )
(
)
1 T t t r e e I (t) e θ−λ θ−λ −θ − = θ − λ y≤ ≤ t T1 …2The retailer had ordered a quantity that would have been sufficient for a length of T units of time. If that amount is denoted by I0, then,
T 0 0 I =
∫
D(t)dt T 1 e− −λ = λ …3The total shortages backlogged during [0,y] sum up to, y s r 0 I = −
∫
I (t)dt y B e 1 y −λ − = + λ λ …4These inventory orders have to be delivered first in order to clear the backlogs. After clearing the backlog the inventory left with the retailer for subsequent sale is reduced by the amount backlogged, i.e. Is.
So, the inventory left after clearing the stock, is, 0 s
I(y)= − I I …5
But from equation (2), this inventory level is given by,
( ) ( )
(
)
1 T y t r e e I (y) e θ−λ θ−λ −θ − = θ − λ …6From equations (5) and (6) we get,
(
)
(
)
(
)
(
)
( ) y y y y T 1 B e 1 1 e 1 T ln e 1 e y e θ −λ θ−λ θ −λ θ − λ − = − θ − λ − + + θ − λ λ λ λ …7From here, we can easily infer, 1
T =f (y, T)
5. Various costs involved in the system
The retailer is in possession of inventory during the interval [y,T1]. Hence, the holding
cost of carrying inventory is calculated for that interval.
1 T r ( y t ) 1 1 y HC=c
∫
I (t)e− + dt 1 1 (r )T ( )T ry ry (r ) y (r ) y 1 1 c e e c e e e e (r )(r ) ( ) r r − +λ θ−λ − − − +θ − +λ = + − + λ + θ θ − λ + θ + λ …85.2. Present Worth Purchase Cost of the Retailer
The retailer has placed the order at time t=0, hence he will make the payment according to this time frame only. The lead time taken by the supplier does not affect the payment made by the retailer.
0
PC=cI …9
5.3. Present Worth Setup Cost of the Retailer
The setup cost or the ordering cost of the retailer is made at the beginning of each cycle, hence,
SPC=A …10
5.4. Present Worth Shortage Cost of the Retailer
The shortages are partially backlogged by the retailer during the starting time of the cycle, hence the shortage cost incurred upon the retailer is,
y rt 2 r 0 SC=c
∫
−I (t)e− (r ) y ry 2 Bc e 1 1 e r r − +λ − − − = + λ + λ …115.5. Present Worth Lost Sale Cost of the Retailer
The shortages incurred in the initial phase of the cycle are partially lost. Due to this, the retailer has to bear an extra expense of the lost sale cost.
y rt L 0 LC=c
∫
(1 B)D(t)e dt− −(
(r ) y)
L c (1 B) 1 e r − +λ − = − + λ …125.6. Present Worth Sales Revenue of the Retailer
The revenue is obtained by the retailer when he sells off his inventory which is at time t=y and during the time from y to T1. The total revenue thus obtained is,
1 T y ry ry rt r 0 y SR =se−
∫
BD(t)dt+se−∫
D(t)e dt− 1 (r )T y (r ) y ry B(1 e ) e e se r − +λ −λ − +λ − − − = − λ + λ …135.7. Present Worth Net Profit of the Retailer
The net profit of the retailer is obtained by deducting the total costs of the retailer from the sales revenue of the retailer. But the shortage costs and the lost sales costs are levied upon the retailer due to the lead time taken by the supplier to complete the order. Hence, the retailer refuses to bear the extra burden and this cost has to be borne by the supplier, keeping the profit margin of the retailer intact.
Hence, the net profit of the retailer for one such cycle is,
r r
NP =SR −HC PC SPC− − …14
5.8. Present Worth Production Cost of the Supplier
The supplier produces the required amount of I0 units during the time interval [0,y] during which the total costs incurred are,
y rt p p 0 0 PC =c I
∫
e dt− ry p 0 1 e c I r − − = …155.9. Present Worth Managing Cost of the Supplier
This is the cost which is under the direct control of the supplier. An increase in the managing cost would cause a decrease in the lead time, hence a consequent decrease in the shortage cost as well as the lost sale cost. However, a decrease in the managing cost would proportionately increase the shortage cost and the lost sale cost.
p
MC = µ …16
Let ψ(μ) Є [0,1] be the degree of uncertainty in the lead time of the supplier. This gives that the order completion date is uniformly distributed over the interval (-ψ(μ)T, ψ(μ)T). This distributes the random variable y uniformly over the interval -ψ(μ)T and ψ(μ)T,
where we have q
( ) e− µ, q 0, 0
ψ µ = ≥ µ ≥ . This gives the probability density function of the random variable y as
( )
1( )
( )
f (y) , T y T 2 T = −ψ µ ≤ ≤ ψ µ ψ µ …175.10. Present Worth Sales Revenue of the Supplier
The revenue from the sale of the order is received by the supplier at the beginning of the cycle itself. Hence,
p 0
SR =cI …18
5.11. Present Worth Net Profit of the Supplier
The net profit of the supplier is calculated as the difference of the various costs of the supplier as well as the shortage and lost sale cost of the retailer deducted from the sales revenue of the supplier.
p p p p
NP =SR −PC −MC −SC LC− …19
Where the total cost of the supplier is TCp
p p p
TC =PC +MC +SC LC+ …20
Hence, the supplier’s expected profit NP becomes p* 0 * p p p 0 NP NP f (y)dy NP f (y)dy ∞ −∞ =
∫
+∫
…21Similarly, the supplier’s expected cost TC becomes *p 0 * p p p 0 TC TC f (y)dy TC f (y)dy ∞ −∞ =
∫
+∫
…22Now, with this theory in mind, we have the following optimization problem: Max NP p*
Max NP r
Subject to TC*p ≤ X …23
The concavity of the net profit function of the supplier is clearly demonstrated from the following graph. This graph clearly shows that optimal function not only exists but is also unique. 0 0.2 0.4 0.6 0.8 1 1.2 0 5 0 1 2 3 4 5
6 Cocavity of Net Profit of Supplier
Lead Time
6. Numerical Illustration
To test whether the model can be implemented in practicality we have taken a numerical illustration as well. We have considered the following data for the study.
A=150 B=0.75 c=15 c1=1 c2=4 cL=8 cp=6
We have used the software MATLAB 7.0.1 to arrive at the optimal solution. The concavity of the profit function very explicitly shows that the optimal solution not only exists, but is unique as well.
The optimal ordering quantity is obtained as I0=15.54 units. The optimal net profit
of the retailer is obtained as 170.37$ while the expected net profit of the supplier comes out to be 226.89$. The optimal interval of the cycle comes out to be T1=27.39 days.
Obviously, the supplier would want to reduce the expected lead time to save himself from the extra burden of shortage cost and lost sale cost. In accordance with this, the lead time comes out to be y=0.0324days.
Now, to test whether the solution obtained is indeed feasible in the sense of stability, its sensitivity with respect to various system parameters has been checked.
Variation Parameter
Percentage Variation in Parameters
-10% -5% 0% +5% +10% B * p NP 226.8715 226.8811 226.8907 226.9004 226.9100 NPr 170.4632 170.4209 170.3666 170.3242 170.2695 y 0.0323 0.0323 0.0324 0.0324 0.0325 λ * p NP 240.5727 233.5820 226.8907 220.4837 214.3468 NPr 216.2795 191.8740 170.3666 151.3303 134.3751 y 0.0315 0.0319 0.0324 0.0328 0.0333 θ * p NP 226.8907 226.8907 226.8907 226.8907 226.8907 NPr 170.7461 170.5566 170.3666 170.1761 169.9850 y 0.0324 0.0324 0.0324 0.0324 0.0324 r * p NP 226.8904 226.8906 226.8907 226.8909 226.8910 NPr 160.4258 165.5547 170.3666 174.8840 179.1278 y 0.0324 0.0324 0.0324 0.0324 0.0324
There are some very interesting facts which make themselves apparent form the above study. The most striking observation from the above table is that the profit of the supplier is not affected by any parameter except λ, the demand factor. As the demand factor increases, the demand decreases, (owing to negative exponential demand rate) and hence
the profit margin of the supplier also decreases. The rest of the factors do not affect the supplier since these factors are more or less concerned with the market forces which ultimately affect the retailer only.
The profit of the retailer is almost unaffected by the changes in backlogging parameter. This is most evident, since; the retailer is retrieving his shortage and lost sales costs from the supplier. Hence this parameter does not affect him. An increase in the demand rate reflects as a decrease in the retailer’s profit margin. This can be explained on similar terms with the supplier’s profit variation. The only other parameter affecting the profit of the retailer is the inflation rate, an increase in whose value increases the net profit margin of the retailer.
The lead time shows a stark stability to any kind of variation in the parameters. This is easily explainable since lead time is not a factor which would be affected by market forces of any kind. In fact it is a shortcoming of the supplier and his management team which creates it in the first case. The managing cost spent by the supplier on reducing it is also bound by capital constraint. Hence once the optimal solution takes the lead time to its optimal value after spending the maximum possible amount on reducing it, there is not much scope for its variation.
Overall the system shows a very good stability in itself and resists changes due to market forces or customer’s preference.
7. Conclusion
Here we have studied a multi echelon supply chain with some very realistic assumptions. We studied our model in an inflationary environment. No doubt, this assumption imparts an economic viability to the whole study. Even today, most of the researches are ignoring this concept, just for the sake of simplicity of their models. But, in today’s times when inflation has become a global phenomenon, any study ignoring the concept of inflation annoy justify itself under any circumstances.
Till date, most of the researches have been done assuming a zero lead time. Needless to say, this is another unrealistic assumption which considers that an order placed is completed instantaneously. But this is only possible if the supplier is always in possession of such a large amount of inventory that he does not take any time to fulfill a customer request. And this very assumption will put a lot of inventory holding cost on the
supplier. Obviously this is neither practical nor feasible. Hence, in our study we have considered the existence of lead time. To bring it more matter-of-fact, we have considered the lead time to be a random variable. This way, there is an element of uncertainty involved which brings the study in close proximity to reality.
In addition to the above mentioned facts, we have also considered the product to be having an expiration date. Obviously, beyond this date, the demand would be decreased to a minimal. Also, we have put an upper limit at the amount the supplier can spend on fulfilling the order. This is easily explainable, since no one can be expected to spend an unconsidered amount of money on a single order.
All these facts together make this study very unique and matter-of-fact.
8. Acknowledgement
The authors are very much thankful for the constructive criticism offered by the reviewers, and wish to thank them for their valuable suggestions which helped to enrich this study.
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