INVENTORY MODEL UNDER TRADE CREDIT WHEN PAYMENT TIME IS PROBABILISTIC
Dr. Mohini Bhat, Associate Professor,
Department of Computer Applications
T. John College , Bangalore
In practice, the supplier may simultaneously offer the retailer a permissible delay in payments to attract new customers and increases his/her sales and also a cash discount to motivate immediate payment and reduce credit expenses. However, not all the time retailer is able to pay within the fixed period. Here we take into account the possibility of all situations like making the payment before and after the trade credit limit. This is incorporated in the model through probability distribution functions. Since all cash outflows related to inventory control that happen at different points of time have different values, we use discount cash flow approach to set up an optimal ordering policies to the problem. The model is illustrated through various numerical examples.
Keywords: Inventory model, discounted cash flow, trade credit, probability distribution.
The standard economic order quantity (EOQ) model assumes that the retailer must be paid for the items as soon as the items were accepted. However, in real-life positions, the supplier hopes to motivate his products, he will offer the retailer a delay period, which is the trade credit period, in paying for the amount of purchasing cost. In addition, the supplier offers a cash discount to encourage the retailer to pay for his purchases as early as possible. The retailer can obtain the cash discount when the payment is made before the cash discount period presented by the supplier. Otherwise the retailer will pay full payment within the trade credit period. Thus the supplier often makes use of this trade credit policy to promote his /her commodities, also use the cash discount policy to invite retailer to pay the full payment of the amount of purchasing cost to cut down the collection period. The credit term that contains cash discount is very practical in real life business situations as an incentive for an earlier payment.
Several papers discussing this topic have shown in the literature that look into inventory problems under varying conditions. Some of the prominent papers are discussed here. Goyal (1985) derived an Economic order quantity model under the condition of permissible delay in payments. However in real situations ” time” is a important key concept and plays an important role in inventory models. Certain types of commodities deteriorate in the course of time and hence are unbalanced. To provide more practical features of the real inventory systems, Aggarwal and Jaggi (1995) and Hwang and Shinn (1997) extended Goyal’s (1985) model for deteriorating items. Jamal et al. (1997), Sarker et al.(2000) and Chang et.al.(2002) further extended Aggarwal and Jaggi (1995) model to allow for shortages and makes it more applicable in real world. Liao et al.(2000) developed an inventory model for stock dependent expenditure rate when delay in payment is allowed. Huang and Chung (2003) extended Goyal’s (1995) model to cash discount period suggested by the supplier. Cash discount rate is decided by the behavioral and operating Characteristics of buyers and sellers. Buyers with lesser credit quality are offered superior cash discounts. superior cash discounts are also connected with buyers who typically pay cash and who accumulate inventory to take advantage of the higher cash discount. Many related articles regarding inventory models with cash discount and trade credits are found in Jinn – Tsair, Teng , (2006), Yung-Fu Huang , Kuang- Hua Hsu, (2008) , Kun-Jen Chung, Jui-Jung Liao (2009) and their references.
Most of the earlier inventory management studies have not considered the discounted cash flow concept and hence the importance of money value is not considered. One of the inventory models that is discussed in the modern literature of inventory model is considering the time value of money. As the transactions that occur at different points of time will have different values and that cannot be compared with one another, so the face value of amounts paid at different time points cannot be considered as such. Certain authors discussed inventory models taking DCF concept. K.H Chung (1989) presented the discounted cash flows (DCF) concept for the analysis of an optimal inventory policy in the presence of trade credit. Kim & Chung (1990) identified the need to discover the inventory problems using the net present value approach or discounted cash flow concept. (DCF). Teng (2006) implemented discount cash flow (DCF) concept to establish the optimal order policies when suppliers offer both a cash discount and permits delay in payment. Chung & Liao (2006) incorporated all these concepts of a DCF approach and trade credits linked to ordering quantity and developed new model for deteriorating items.
In the literature of inventory model, so far we discussed trade credit policy with ‘two – part’ strategy, cash discount and delayed payment. Thus at most two intervals are considered to make the payment. However in intense situations the retailer is not able to pay within the trade credit period. Though it may not happen quite commonly but is not improbable. Such situations are to be captured in the model. Hence in this present paper, we propose one more payment interval with a penalty rate which occurs for a customer with certain probability.
Mainly this paper is proposed to include and develop the model that includes the possibility that the retailer is not able to pay within the trade credit period, but can pay later along with interest as a form of penalty. Model is developed by including the possibility of later payment which happens with certain probability and the delay duration for the payment after trade credit could be assumed to follow appropriate probability density function. Here the general tendency of making payment will be usually last day of the policy rather than within the trade credit period, because of time value of money. Under these conditions, we model retailers inventory system as a cost minimization problem to determine the retailers optimum inventory cycle time and optimal order quantity. Numerical examples and sensitivity analysis are presented to illustrate the proposed model. Finally, summary and conclusion are made.
D = Demand rate per year.
A = Ordering cost per replenishment
C = Purchasing cost per item
h = Unit stock holding cost per item per year .excluding interest charges
? = Cash discount rate (0 < ? < 1)
M = Trade credit period
T = Cycle time in years
r = Interest rate for net present value
r1= Discounted interest rate for payment made earlier to M but not at t =0.
rp = Penalty rate, where rp >r
T* = The optimal cycle time
Q* = The optimal order quantity = DT*
Demand rate “D” is known and constant.
Shortages are not allowed.
Planning horizon is infinite.
Replenishment happens instantaneously on ordering, which means, lead time is zero.
Suppliers offer cash discount if retailer makes payment at the beginning . If he makes payment at trade credit period M, then regular price is applied, whereas, if he makes the payment after the trade period M, then supplier charges the penalty rate rp for the amount of purchasing cost.
Generally, retailer may not able to follow the consistent pattern of payment, that is, same pattern of payment schedule is not adhered because of uncertainty of cash in hand. However, the retailer’s payment pattern can be modeled through a probability distribution though the payments made are at different time points. According to the previous payment habit we assume that he makes payment in the beginning of the trade credit period and be eligible for the discount price with probability p1, the probability that he makes payment at trade credit by paying regular price is p2 and he makes payment after the trade credit period with probability p3, where p3=1- p1- p2. Let g(.) denote the conditional density function of the random duration of the payment which is made after the trade credit period.
Hence the cumulative probability function of the payment made is obtained by,
Where p1 + p2 + p3 =1
The present value of the total cost is based on the following elements:
Present value of the ordering cost
Present value of the inventory carrying Cost
Present value of the purchasing cost.
PV1 (T) = Present value of all future cash flow when payment made within trade credit period
PV2 (T) = Present value of all future cash flow when payment made at trade credit period “M”
PV3 (T) = Present value of all future cash flow when payment made after M with penalty rate rp.
Present value of the ordering cost:
Present value of the inventory carrying cost:
Present value of the purchasing cost can be discussed in three different cases as follows:
Case (i) when payment is made without any delay.
Case II: when payment is made at Trade credit period “M ”
Case III: when payment is made after Trade credit period “M “with penalty rate rp.
If payment is made at time t after the trade credit period M, with penalty rate rp then payment towards the purchasing cost is,
. Hence conditional expectation of the this payment is
Net present value of the above cost with interest rate r is,
Take y = t-M
Where represents Moment generating function of distribution function X.
Any distribution which has limit 0 to infinity can be considered to derive the cost function.
One of the appropriate distributions for the delay in payment beyond trade credit period is gamma distribution. In our discussion here, we assume this distribution and derive the cost function.
Where ? and ? are parameters of gamma distribution.
, ( by definition of gamma function)
Continuously discounted present value of the purchasing cost is
Present value of total cost
Present value of all future cash flow when payment made within trade credit period with probability p1 + Present value of all future cash flow when payment made at trade credit period “M” with probability p2 + Present value of all future cash flow when payment made after M with penalty rate rp with probability p3
To obtain optimal time T* we need to minimize PV (T) with respect to ‘T’ and we get,
Note that can be approximated as
Then equation (5) can be written as
After simplification , we get (6)
Using T* optimal cycle time the optimal order quantity Q* is obtained as Q*= DT*
We get (7) Hence, if there is no credit period, the DCF approach gives an identical solution to that of the traditional inventory analysis.
V Numerical examples
To illustrate and verify the above theoretical results, we consider few examples here.
The sensitivity analysis on various payment time with different probability values , Purchase values and trade credit period is shown in Table 1-3, respectively
Effects of changing payment time with different probability values on the optimal solution
Demand rate per year D =1000 units; r=0.06; M=0.1year; h=$0.2/unit/year; A=$100/order ; C=50 ?=0.1; rp=0.2; ? =0.1; ? =2;
Probabilities Q* T* PV(T)
p1=0.8, p2=0.1, p3=0.1 125 0.1250 792960
p1=0.1, p2=0.1, p3=0.8 124 0.1242 852250
p1=0.1, p2=0.8, p3=0.1 124 0.1242 848010
p1=0.2, p2=0.4, p3=0.4 124 0.1241 841960
p1=1, p2=0, p3=0 125 0.1253 776630
p1=0, p2=1, p3=0 124 0.1239 855270
It is observed from above table that
Higher the value of p1 compared to p2 and p3 will results the lower values of total relevant cost
Higher the value of p2 compared to p1 and p3 will results higher the values of total relevant cost compare to case (i)
Higher the value of p3 compared to p2 and p3 will results higher the values of total relevant cost compare to case (i) and case (ii)
Effects of changing purchase cost C, on the optimal solution
Demand rate per year D =1000 units; r=0.06; M=0.1year; h=$0.2/unit/year; A=$100/order ?=0.1; rp=0.2; ? =0.1; ? =2; p1=0.8, p2=0.1, p3=0.1
Purchase cost Q* T* PV(T)
30 161 0.1614 480450
50 125 0.1250 792960
80 99 0.0989 1259800
120 81 0.0808 1881400
150 72 0.0723 2345000
200 63 0.0626 3118400
It is observed from the Table 2. that as purchase cost increases there is significant decrease in value of optimal quantity as well as the value of optimal cycle time. But there is significant increase in total relevant cost.
Effects of changing trade credit period M, on the optimal solution
Demand rate per year D =1000 units; r=0.06; h=$0.2/unit/year; A=$100/order; C=50 ;?=0.1; rp=0.2; ? =0.1; ? =2; p1=0.8, p2=0.1, p3=0.1
Trade credit period Q* T* PV(T)
0.1 125 0.1252 792960
0.15 125 0.1252 792460
0.20 125 0.1252 791960
0.25 125 0.1252 791470
0.30 125 0.1252 790970
0.35 125 0.1252 790480
It is observed from the Table 3. that as trade credit period M increases ,there is no change in optimal order quantity as well as in the value of optimal cycle time. But there is marginal decrease in total relevant cost. Which implies that credit period offered to retailers has positive impact
From the above numerical examples it is clear that there is significant difference between the total optimal costs when paying habits changes, especially after the trade credit period. Hence when a retailer is not making payment then actual cost will be much different and higher also that could be formulated without taking such situation into consideration.
Most of the inventory models with trade credit assumed that retailer pays either before the trade credit period with in cash discount or at credit period every time. Thus models allow making the payment every time at one of these two possible points. However in real market place it is common that the retailer is not able to pay consistently at the similar time point every time. Sometimes the retailer pays before the trade credit and sometimes at the trade credit period. In extreme cases he/she pays after trade credit period. In order to model this and possibly not very punctual payment habit, the model incorporates possibility of payment even after the trade credit period of course with a penalty rate that will occur with certain probability and retailer’s payment time is also considered as a random point which is modeled through a probability distribution. Further under the condition of trade credit it is beneficial to pay only at the trade credit limit point rather than before. But retailer may find it convenient to pay wherever cash is available and hence further situations arise. From this study it can be seen that if retailer is not able to adhere to same payment pattern then the total cost differs very much. Hence, assuming models without considering various probabilities will not only mislead the total cost but also the solutions obtained are suboptimal. As can be seen from above tables that the total costs differ when probabilities are different. All models discussed earlier can be taken care as special cases by assuming appropriate probabilities as zero in the present model. Hence the present model is a generalization by taking various possibilities into the present model. In addition, the computation results on the two models discussed in the paper reveal that a lower value of purchasing cost results in higher values for the optimal replenishment cycle time T* and also the optimal order quantity Q* and vice versa. For different time points the present value of total cost is also calculated.
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