Retailer's Optimal Replenishment Policy With Trade Credit Under Inflationary and Fuzzy Environment With Different Demand Pattern

Traditionally inventory modeler assumed that retailer's capitals are sufficient and must pay for the products as soon as the products are received. However, this is not always true in the actual business world as every businessman strives to increase his profit, his goodwill and his retailer base. In today's business transactions, it is more and more common to see that supplier always offers a specified period (say, 30 days) to the retailer to settle the account. In literature this period has been termed as trade credit period. The trade credit financing produces two benefits to the supplier: 1. It should attract new customers who consider it to be a type of price discount. 2. It should cause a reduction in sales outstanding. Some established customers pay more promptly in order to take advantage of trade credit more frequently. In India, gas stations adopted a pricing policy that charged less money per gallon to the customer who paid by cash, instead of credit card. Likewise, store owners around the world usually charge a customer 5% more if the customer pays by a credit card, instead of cash. As a result, the customer must decide which alternative to take when the supplier provides not only a cash discount but also a permissible delay. Hence, trade credit can play a major role in inventory control policy for both the retailer and the supplier. This idea caught attention of inventory practitioners and results are that various articles dealing with trade credits have appeared in various inventory journals. In business world, time is a phenomenon, which affects everything around it. Inflation is a concept closely related to time. Inflation is that state of disequilibrium, in which an expansion of purchasing power tends to cause, or is the effect of an increase in the price level. During inflation, there is too much currency in relation to the physical volume of the business being done. Economists generally agree that high rates of inflation and hyperinflation are caused by an excessive growth of the money supply. Low or moderate inflation may be attributed to fluctuations in real demand for goods and services, or changes in available supplies such as during scarcities, as well as to growth in the money supply. However, the consensus view is that a long sustained period of inflation is caused by money supply growing faster than the rate of economic growth. Due to high inflation and consequent sharp deckle in the purchasing power of money in the developing countries like India, Argentina, Brazil, Bangladesh etc., the financial situation has been completely changed. So, it is sufficient reason in itself to compel the researchers over the world to study the effect of inflation on their analysis. It was very obvious fact that given some time, every item can create a niche for itself in the customer's mind, hence increasing its demand with the passage of time. Demand of goods may vary with price or even with the instantaneous level of inventory

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trade credit. It is a common belief that large piles of goods displayed in a supermarket will lead the customer to buy more. According to Levin et al. (1972), one of the functions of inventories is that of a motivator, as indicated in the statement that 'At times, the presence of inventory has a motivational effect on the people around it.' In a competitive market, price of the goods plays an important factor to a customer. Silver and Peterson (2012) noted that sales 'at the retail level tend to be proportional to inventory displaced. Pal et al. (1993) observed that sales of some items at the retail level were directly related to the' amount of inventory displayed. Urban and Baker (1997) generalized the EOQ model in which the demand is a multivariate function of price, time and level of inventory. Urban (2005) conducted a comprehensive review of that literature, distinguishing between type 1 models in which the demand rate of an item was a function of the initial inventory level and type 2 models in which it was dependent on the instantaneous inventory level. Teng and Chang (2005) extended an Economic Production Quantity (EPQ) model for perishable items, considering the demand rate as the sum of two terms: first term was inversely proportional to the price and second term was directly proportional to the stock-level of inventory displayed. Sana and Chaudhuri (2008) presented the retailer's profit-maximizing strategy when confronted with supplier's trade offer of credit and price-discount on the purchase of merchandise. They had taken different demand pattern in that analysis. First in literature, credit-linked demand function had been coined by Jaggi et al. (2008) in the analysis of inventory control systems. Jaggi and Kausar (2010) developed a supply chain model by taking credit-linked demand function and determined the optimal replenishment time and credit period for the retailer. They assumed that items deteriorate when physically present in stocks. Most of the inventory practitioners while discussing trade credit policy in their models implicitly assumed that the retailer settle the account when whole of its inventory depleted to zero. But, in most business transactions, this assumption is unrealistic. As each retailer wants to settle the accounts as soon as possible in order to pay minimum interest to the supplier. It is also observed that most of the inventory practitioner ignore the fact that credit period offered by the supplier to the retailer has a positive impact on the demand. There are various factors which can stimulate the demand of the product such as by reducing selling price and through the display of stock. It was also observed from literature that most of the researchers take fixed interest rate while developing models with trade credit policy. But this is not realistic as there is always impreciseness present in interest rate. Inventory modelers have not contributed much in that direction. So it is the need of the hour to discuss this in detail. Through this chapter the researcher has tried to soon as retailer has sufficient amount and tries to maximize his profit. All analysis has been performed in inflationary and fuzzy environment. In this chapter, two different inventory models have been considered. From literature it was observed that credit period or selling price have positive impact on demand of the product. In this section to stimulate the demand it is assumed that demand depends on either credit period or selling price, fixed by the retailer. Hence, demand function D(.) can be any one of the following forms: a. Credit Linked Demand Function: Demand rate is a function of retailer's credit-period offered by the supplier (M). The demand function can be represented as a differential difference equation: S : maximum demand r : rate of saturation of demand (which can be estimated using the past data) under the assumption that the marginal effect of credit period on sales is proportional to the unrealized potential of the market demand without delay. The solution of the above difference equation, under the condition that at M=0, D(0) = s(initial demand), keeping other attributes like price, quantity, etc. at constant level, is given by

Fig.1.1 shows the demand function for different values of 'r' and M. It shows that credit period have

achieve faster if the value of 'r' is high. b. Demand Depends on Selling Price of Retailer: The market demand rate for the product is a downward sloping function of the selling price. In this case, demand is of the form D(.) = a(i1)-b where a>0 is a scaling factor and b>l is a price elasticity coefficient.

Fig.1.2 shows that demand changes significantly with respect to the markup price (i1). Demand gradually decreases as the value of 'i1' increases for fixed value of 'b'. 1.2.1 Assumptions: In this sect' n, mathematical model is developed under the following assumptions. 1. Supplier provides a credit period to the retailer in payment. Supplier offers different discount rate of purchasing price at different credit periods. 2. Demand rate is function of credit period or selling price. 3. The replenishment rate is infinite. 4. Inflation and time value of money is considered. 5. A constant fraction of the on-hand inventory deteriorates per Unit time. 6. Lead-time is negligible. 7. The inventory system involves a single type of items. 8. Time horizon is infinite. section. T : Cycle time D(.) : Demand rate Q : Initial inventory level k : Constant representing the difference between discount rate (k1) and inflation rate (k2). A(t) : Ordering cost for an order placed at time t, i.e., A(t)=A0e-kt, where A0 is the ordering cost at time zero i : Inventory holding cost per unit per year excluding interest charges  : A constant fraction of on-hand inventory which deteriorates per unit time le : Annual interest that can be earned per unit Ip : Annual interest charge payable per unit (Ip>Ie) M : Credit period for settling accounts C(t) : Unit purchasing cost per item at time t, i.e., C(t) = C0e-kt, where C0 is the purchasing cost at time zero and depends on the credit period (M) P : Payment time for retailer CD : Total cost of deterioration per cycle CH : Total holding cost per cycle Dr : Amount of materials deteriorated during a cycle time (T) S(t) : Selling price per unit item, i.e., S(t)e-kt, where S0 (=i1C0) is the selling price at time zero.

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occur when the materials are physically present in the inventory at time t (0≤t≤T). Let I(t) be the inventory level at any time t. Inventory level, I(t), during the time period (0≤t≤T) is given by [Fig. 5.3] The purchasing cost at different delay periods are

C0 =

Where Cr = maximum retail price per unit, Mj (i=1,2,3) decision point in settling the account to the supplier at which supplier offers % discount to the retailer. M3 is the maximum delay period after which the supplier will not agree to give trade offer of credit and price-discount for sale of the items to the retailer. Consequently, the supplier decides Co when M>M3, i.e., retailer never purchases at an infinite cost. (j=1,2,3) are the constant discount rates decided by the supplier. Solution of equation (5.1) using the boundary condition is I(t) = ..........(5.2) Consequently, initial inventory after replenishment becomes

..........(5.3)

Since the total demand during T is D(.)T, the amount of materials which deteriorates during one cycle is

DT = Q – D(.)T= ...........(5.4)

Now, the various costs associated to inventory are calculated as follows: Income from selling the product is Interest earned after P is CR = A(0) + A(T) + A(2T) + ............................................+ A((m-l)T)+..................... = Total deterioration cost is Holding cost is = i The interest payable per cycle for the inventory not being sold after the due date is given by = Theorem-1-1: Initially retailer orders Q quantity and thus owes C0Q to the supplier. The following two cases may now arise: Case-1: If then loan amount will be paid at M=P. Case-2: If then loan amount will be off at P such that Thus, total profit of retailer (TP) is defined as

TP =

- or

TP =

...........(5.5)

TP = i1Fl + i(-F2) + IP(-F3) + ilIe(F4) + ilIP(F5) + IPilIe(F6) + (–F7) ............(5.6) where

= = = = = = =

In order to find the optimal solution of the proposed inventory model in the presence of uncertainty, opportunity cost, interest earned and interest paid are represented by triangular fuzzy numbers. The triangular fuzzy profit is calculated by using arithmetic operations based on the Function Principle. The Function Principle defines an efficient way to perform fuzzy arithmetic operations on fuzzy numbers (Chen, 1985). Now, let us consider i, i1, le, and Ip as imprecise parameters and expressed by triangular fuzzy numbers, respectively then i = () = (i-) =

() = (-) () = (-)

A normal triangular fuzzy number, for example, (a-, a, a+) and >0, has the following membership function: In this study, Chen's (1985) Function Principle for the calculation of the fuzzy profit function is used. According to this, the fuzzy profit function TP=(TP1,TP2,TP3) becomes a triangular fuzzy number where TP1 = (il-3)F1–(i+2)(F2)-

(+)()+()()() + () () () + () () () () – ()

TP2 = –i () - () + + ()

+ () – ()

TP3 = () – (i–) () - (–) ()+

() () ()+ () () () + (+) (+) () () -

(), j =1,2,3 and = (1,1,1) The membership function of total profit is given by The α-cut, TP(α), of consists of points x such that TP(α) = . Since the total profit is a triangular fuzzy number, so -cut of is TP(α)= [TPL(α), TPU(α)], α∈[0,1], where TPL(α) = TP1+(TP2– TP1) and TPU() = TP3–(TP3– TP2)α. In order to

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signed distance. Using this method, equivalent crisp profit expression is TPS ≡ d (T.) = (TP1 + 2TP1+TP3)/4 ............(5.7)

In this case, demand function is considered of the form D(.) = α + β where Q is initial stock-level and α, β and γ are positive constant.

Fig.1.4 presents the demand pattern with respect to different value of ' γ ' It is observed that as the value of 'γ' increases demand also increases.

With the same notations and assumptions proceed up to equation (5.5) and using the relation D(.)=α + β profit expression is as follows: TP = il

+ +

+ or TP =

+ + +

+ + or + + +

…...(5.8)

Where ,

Now, let us consider i, and , as imprecise parameters and expressed by triangular fuzzy numbers, respectively. Proceeding as previous section and after defuzzification by signed distance method, profit expression is

TPS = (TP'1+2TP'2+TP'3)/4 ...............(1.9)

Where TP'1 = (il-3)–(i+2)()-

(+)()+()()() + () () ()- () () () ()–()

TP'2 = –i ()- ()++ ()+

( – ()

TP3 = () –(i–) ()-(–)

()+() () () + () () () + () () () () – ()

In this model, the retailer tries to find out the payment time to supplier in order to maximize their profit and minimize interest which has to be paid to the supplier. The proposed model has been illustrated with the help of following data which is taken from the literature in appropriate units: Example-1: Taking D(.)1000(constant), A0=200, Cr=20, k=0.2, i = 0.12 per year, = 1.2 per year, =0.13 per year, , θ=0.2, =0.8, =0.7,

0.7, = 20/365, = 25/365, = 30/365, =

15%, = 5%, = 0%. Calculation is performed through MATLAB 7.1 and the results thus obtained are listed in Table 1.1 corresponding to M=20, 25, and 30 days.

From Table 1.1 the following observations can be made:

• From the previous literature it was found that the k and M have significant decisive effects on inventory cycle time, credit period, order quantity and on profit of retailer. On the analysis of Table 1.1, it appears correct.

• Inflation is the state of a continuous increase in price of goods and service. Hence, as obvious, an increase in the rate of inflation causes the total profit of the system going down. Such a change in the system is very appreciable.

• An increase in the inflation rate reduces the purchasing power of the retailer. So retailer can buy fewer inventories, which ultimately finish off sooner, reducing the initial level of inventory and the payment time P to clear up the account.

• The profit of the retailer shows an increasing trend as the value of M increases from 20 days to 30 days. As the credit period increases, retailer gets an opportunity to earn more interest on the accumulated capital. That is the reason for the change in value of P. Since in this situation retailers have capital to payoff in less time, this result identifies that trade credit is an effective strategy for inventory systems.

 From Table 1.1, it can be observed that T*, p*, Q* decreases whereas profit of retailer increases with an increase in the credit period (M). It shows that the longer the credit period is, shorter the replenishment period, payment time, and order quantity are, whereas the profit will increase. From managerial point of view, if the supplier provides a

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take the benefits of the credit period more frequently.

• From Table 1.2, it is observed that as the interest paid per unit item increases from 0.10 to 0.20 then inventory cycle, payment time to settle the account, order quantity and profit of the retailer decreases. From managerial point of view it implies that when the interest paid is high then retailer should order less amount of inventory.

• Through Table 1.3, the effect of deterioration rate on T, P, Q and on profit of the retailer can be analyzed. It is observed that an increase in deterioration rate results in a short cycle length, therefore payment time to settle the account, order quantity and profit of retailer decreases. By decreasing the order quantity retailer tries to decrease the deterioration quantity and increase their profit. Hence, if the retailer can effectively reduce the deterioration rate of items by improving equipment of storehouse, profit earned by the retailer will be increased.

Example-2: To analyze the effect of credit period on demand, we consider D(.) = S – (S-s)(l-r)M i.e., where demand depends on credit period. For this following values are required: S=1000, s=700, r=0.12 and rest of the value of the parameters are same as used in Example 1. The computational results for various values of M and k are as shown in Table 1.4.

• From Table 1.4 it is observed that as M increases, there is marginal decrease in cyc1 length, payment period to supplier as well as order quantity but there is significant increase in retailer's profit, which-implies that credit period offered to relation has positive impact on the unrealized demand. Therefore, the retailer must pay attention on the credit policy provided by supplier very carefully to achieve maximum profit as much possible.

• To explore the impact of trade credit and rate of saturation of demand, using the same data as shown in Example-2, optimal solutions for different values of 'r' are listed in Table 1.5. It is observed that due to the escalating values of 'r' profit of the retailer increases. Profit also increases with the increase of trade credit period offered by supplier to the retailer. Profit of the retailer is more sensible for change in credit period in comparison to Y. So, it can be concluded that profit of the retailer increases with the increase of credit period as well as the saturation rate of demand.

Example-3: Now we are going to present the analysis when demand rate is of the form D(.)=a(i1)-b For this. the following values are required: a=1 000, b=1.5 and rest of the parameters remains same as used in Example-1.

• As the selling price increases the customer demand decreases so the order quantity is also decreases whereas in this situation retailer has enough money to settle down the account of supplier in lesser time. A higher value of i1 results in a 1argr' optimal replenishment time, a smaller optimal order quantity and larger optimal profit when the supplier offers a cash discount to the retailer or a credit period to the retailer.

• From Table 1.7, it is observed that as the value of 'b' increases the profit of the retailer decreases, whereas replenishment period and payment time increases. Its managerial implication is that the demand of the product is highly affected by the selling price of the product so that profit is also fluctuating due to the change in 'b'.

Example-4: For this example the values of the parameters are as follows: D (.)=...+.... where .....= 800, P= l .8, .....= 0.6 and the rest of the values remain same as used in Example-I.

• From Table 1.8 it is observed that there is a positive impact of y on the optimal value of T, P, Q and on the profit of retailer. This means that demand and profit of the retailer is fluctuating by changing the value of y. So there is a positive correlation between demand and the elasticity of initial inventory level of the product.

• Fig.1.5 represents the concavity nature of profit with respect to initial inventory level. Profit is maximum when Q=178 at y=0.6. From Table 5.8 it is observed that as the value of y increases from 0.2 to 0.8, the optimal initial inventory level increases from 170 to 208 and due to this variation in profit is about 30%.

• From Fig.1.6 it is observed that as the value

of ' 'increases from 0.2 to 0.8 the profit of the retailer increases from 4748.60 to 6175.00.

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initial inventory levels both increase. This shows that have positive impact on demand and on initial inventory level.

Demand boosting activities have become more and more common in today's business world. For example, Wall-Mart and Costco often try to stimulate demand for specific types of electric equipment by offering price discounts; clothiers Baleno and NET make shelf space for specific clothes items available for longer periods. McDonald's and Burger King often use coupons to attract consumers. Other demand boosting strategies include free goods, advertising and display. The demand boosting policy is very important for the supplier and has a big impact on annual profits. Our hypothesis is that consumer demand at a retail outlet depends on selling price, trade credit period as well as in-store stock of the product. In particular, demand decreases with selling price but increases with the displayed inventory of the product and credit period. The idea is that a large stock and credit period leads consumers to buy more. This phenomenon is not a general one but may be related to certain product categories, e.g., used automobiles sold from an open yard of a car dealer, or displays of large quantities of soft drinks, detergents, and canned food sometimes found in supermarkets. Keeping this phenomenon in mind, a different form of demand function has been considered in the present chapter under fuzzy environment. The different demand patterns are considered in this chapter to capture more realistic inventory situations and are as follows: a. Demand depends on credit period offered by supplier to the retailer. b. Demand is selling price sensitive. c. Demand is directly proportional to initial inventory level. In the present chapter, inventory models has been developed by incorporating some additional features like impreciseness in costs, inflation, deterioration and credit period offered by supplies to the retailer which can be associated with a number of different types of inventory. By taking impreciseness in cost parameters, decision-makers absorb all the turbulence in cost due to market fluctuation. Inflation permits a proper recognition of the financial implication.-of the opportunity cost in inventory analysis. Generally, supplier offers different price-discount on purchase of item of retailers at different delay periods. Suppliers allow maximum delay period, after which they will not take a risk of getting back money from retailers or any other loss of profit. That is why when delay period M is greater than M3, then purchasing cost is infinite, i.e., Through numerical example, it is observed that as the inflation rate increases order quantity and profit gradually decreases. Thus as inflation rate increases purchasing power of the retailer decreases. It is also observed that by increasing the value of M, retailer it as discount and orders more frequently and earns more profit. From the analysis some managerial insights are also obtained. The retailer can increase profit by ordering lower quantity when the supplier provides a credit period in payments, improving storage conditions and also takes account of inflation while taking inventory related decisions. Numerically it is found that demand of the product is fairly sensitive of the credit period, selling price and initial inventory level. There is 20% hike in profit when retailer got 20 days as credit period whereas this crops up about 60% when supplier provides credit period of 25 days. So, it is all up to the retailer how they choose credit policy to maximize their profit. When the value of 'i1' changes from 1.2 to 1.25 then profit of the retailer is escalated by 66% whereas it escalated by 125% when 'i1' is 1.30. All this shows that profit of the retailer is highly sensitive to 'i1'. The benefits of dynamic pricing techniques have long been known in the airline, restaurant, hotel, fashion goods and high-tech product industries. The findings in this paper are important to the real world. The prices of fashion-based goods, for example clothes or computers, will be marked down gradually with the passage of time. These types of products are usually characterized as having a short shelf life. Promotional activities, such as price discount and trade credit, are commonly employed to speed up the movement of the goods. The proposed models here provides an insight for the decision-maker which is useful and practical. The results are also applicable for other types of perishable goods such vegetables/fruit, baked goods, etc. which are similar in nature to the fashionable goods described above. A future study can further incorporate more realistic assumptions such as imprecise demand, allowable shortages and different deterioration rates etc. In future, this study can be further extended by developing an integrated-supplier and retailer inventory model with two level trade credits.

1. Bellman, R.E., and Zadeh, L.A. (1970) “Decision making in a fuzzy environment”, Management Science, Vol. 3 (4), pp B141-B164. 2. Tanaka, H., Okuda, T., and Asai, K. (1974). “On fuzzy mathematical programming,” Journal of Cybernotics, Vol. 3(4) pp 37-46

management contemporary policy for managing operating system”, vol. 373 McGraw, Hill New York. 4. Zimmermann, H.J. (1976) “Descriptions and optimization of fuzzy systems”, International Journal of General Systems, Vol. 2 (4) pp, 209-215 5. Zahed, L.A. (1978) “Fuzzy set as a basis for a theory of possibility”, Fuzzy Sets and Systems, Vol. 14 pp 3-28. 6. Kacpryzk, J. and Staniewski, P. (1982). “Long term inventory policy making through fuzzy decision making models”, Fuzzy Sets and Systems, Vol. 8 pp 117-132. 7. Park, K.S. (1987), “Fuzzy set theoretic interpretation of economic order quantity”, IEEE Transactionon systems, Man and Cybernetics, pp. 1082-1084. 8. Chen, S.H. (1985). “Operations on fuzzy number with function principle”, Tamkang Journal of Management Sciences, Vol. 6(I), pp 13-26 9. Gupta, R., and Vrat, P. (1986) “Inventory model for stock dependant consumption rate”, Opsearch, Vol. 23, pp 19-24. 10. Petrovic, D., Sweeney, E. (1994) “Fuzzy knowledge based approach to treating uncertainty in inventory control”, Computer Integrated Manufacturing System, Vol. 7(3), pp 147-152. 11. Petrovic, D., Petrovic, R., Vujusevic, M. (1996). “Fuzzy models for the newsboy problem”, International Journal of Production Economics, Vol. 45, pp 435-449 12. Voosevic, M., Petrovic, D., Petrvic., R. (1996). “EOQ formula when inventory cost is fuzzy”, International Journal of Production Economics, Vol. 45, pp. 499-504. 13. Taha, H.A. (1997), “Operations Research”, Prentice Hall, New Jersey, USA, pp. 753-777. 14. Ishii, H., Konno, T. (1998). A Stochastic inventory problem with fuzzy shortage cost.”, European Journal of Operational Research, Vol. 106, pp 90-94 15. Roy, T.K., and Maiti, M. (1998) “Multi objective inventory models of deterioration items with some constraints in a fuzzy environment, “Computers Operations Research, Vol. 25, pp 1085-1095. uncertain environment.” European Journal of Operational Research, Vol. 109, pp 292-309 17. Liu, B., Iwamura, I.B. (1998). “A note on chance constrained programming with fuzzy coefficients,” Fuzzy Sets and Systems, vol. 100, pp 229-233. 18. Petrovic, D., Roy, R., Petrovic, R. (1999). “Supply chain modeling using fuzzy sets”, International Journal of Production Economics, Vol. 59, pp. 443-453. 19. Chen, S.H., Hsieh, C.H. (1999) “Optimization of fuzzy backorder inventory models”. IEEE Transaction of System, pp 296-301. 20. Lee. H.M., Yao, J.S. (1999). “Economic offer quantity in fuzzy sense for inventory without backorder model”, Fuzzy Sets and Systems, Vol. 105, pp 13-31. 21. Yao, J.S., Su, J.S. (2000). “Fuzzy inventory with backorder for fuzzy total demand based on interval – valued fuzzy set”, European Journal of Operational Research, Vol. 124, pp 396-408.