CC BY
29
TRADE CREDIT FINANCING SCHEME ON RETAILER'S ORDERING QUANTITY FOR IMPERFECT QUALITY ITEM WITH LEARNING EFFECTS AND STOCKING STRATEGIES
A. R. Nigwal1, U. K. Khedlekar2, N. Gupta3 and L. Sharma4
•
1,4Department of Mathematics, Ujjain Engineering College, Ujjain, 456010, Madhya Pradesh India, 2,3Department of Mathematics and Statistics, Dr. Harisingh Gour Vishwavidyalaya, Sagar, 470003, Madhya Pradesh, India, (A Central University) 1arnw@rediffmail.com
Abstract
Today's life is a age of modern life, and in the modern life for any kind of business setup, customer service, pricing, stocking strategies and trade credit financing schemes are effective, essential and survival parameters to grow the business. In this paper, we have developed, an economical order quantity model for imperfect quality product by considering retailer's stock sensitive demand of product under trade credit financing policy. Further in this paper we have studied the Learning effect on screening process on every batch of imperfect quality product. Under the trade credit financing scheme, we have considered that, the supplier proposes to the retailer, a fixed credit time period for payment and retailer also offers to his customers to a fixed credit time period of payment.
Finally an appropriate total profit function per unit time has been derived under the various trade credit financing periods of payment including various expenditure and other related parameters. A sensitivity analysis has been done to verify the optimum results and also a numerical example has been given to verify the model's outputs.
Keywords: Learning Effect, Stocking, Imperfect quality items, Trade credit policy. Screening
process
1. Introduction
Generally, it has been analyzed, that a large number of consumer goods displayed in a shelf at shopping center or supermarket are connected with on sale items to induce more sales and profits. For any kind of an item, increment of shelf space induces more consumers to buy it. This possible because of its visibility, prominence or variety. Conversely, low stocks of certain paved goods might raise the feeling that they are not good and fresh. Therefore, demand is often based on inventory-level. In the last some decade, a considerable literature has been written in the operational research area on how inventory-level-dependent demand affects inventory control policies. Wu et al. [1] developed a inventory model for determining the optimal ordering quantity for non-instantaneous deteriorating products considering with stock-level dependent demand. They suggested that this may more beneficial for those situations in which backlogging parameter increases and decreases the order quantity. In (2005), Teng and Chang [2] Giri, and Bardhan, [3] formulated two layer supply chain coordination policy for deteriorating items with price stock level depended market demand of single product under revenue sharing contract. They concluded that the centralized system is the best strategy instead of decentralized system. developed EPQ models for deteriorating items considering with selling price and stock level dependent demand. They proposed appropriate decision for managerial activities. Ray and Chaudhuri,
[4], developed an EOQ model assuming a deterministic stock-dependent demand, incorporating with shortage, inflation and time discounting. They show through the numerical example that inventory backlogging is beneficial from both (retailer and supplier) organizational as well as economic viewpoints; Giri, and Chaudhuri, [5] designed a model considering with deterministic stock-dependent demand rate of perishable products incorporating time dependent nonlinear holding cost of the products. Parlar and Wang, [6] designed a quantity discounting decisions model for supplier and buyer relationship in which they start with Stackelberg equilibrium of the problem. They concluded quantity discount policy can be very effective in obtaining the maximum profit increase that the supplier and the retailer can possibly obtain together in certain cases. A deterministic inventory model is developed by Pal et al., [7] assuming that the demand rate is stock-dependent and the items deterioration rate is constant. They highlighted the various problems related stocking of goods and further they optimized the profit function with respect to decision variable time and economical order quantity. Muth, and Spremann [8] provided a classical square root formula on the class of economical lot sizing problem considering with learning effects on the production process. Salameh et al. [9] developed a economical production inventory (EMQ) model which was formulated under the learning curve effect on the finite production rate. Cheng [10] Formulated an economical manufacturing quantity (EOM) under the influence of learning process. The order size is considered as to be large enough to allow the manufacturing learning phenomenon to manifest itself. The set-up cost is also assumed to reduce as a result of learning over the life of the product. Salameh and Jaber [11] developed a traditional (EOQ/EPQ) model for imperfect quality items using the (EOQ/EPQ) formulas. They also assumed that at the end of 100 percent screening work the poor-quality items are sold as a single batch with lower price.
Jaber and Guiffrida [12] Provided an (EPQ) model with rework for imperfect quality items using (WLC) wright learning curve. For this they proposed two different cases, first one is learning process adopted in production, no learning process in reworks and second one is learning process adopted in production and rework both. Eroglu and Ozdemir [13] developed an economical order quantity (EOQ) model in which they considered that each ordered lot contains some defective items incorporating with shortages at retailers end. They analyzed, how to affects optimal solution by increasing rate of percentage of defective items. They also assumed that, after 100 percent screening of each lot, the good and defective items are separated into two collection of imperfect quality and scrap items.
Jaber et al. [14] extended the work of Salameh and Jaber [11] by introducing the assumption that the percentage defective items per lot decreases under the learning curve (LC) effects, which was experimentally certified and validated by actual data of automotive industry. Jaber et al. [15] investigated the quality learning curve (QLC) for the assumption that the manufacturing process is interrupted due to maintain the quality to bring the process in control again. In this article they developed two various cases, first one is learning process is adopted in production, no learning process is adopted in reworks and second one is learning process is adopted in production and reworks both. Pan [16] analyzed the effect of learning curve on setup cost for their (CRI) model. They also assumed that the controllable lead time with the mixture of backorder and partial lost sales. Lin [17] investigated the market survey and manufacturing problem for a monopolist firm for quality and cumulative sales dependent demand. They also assumed that per unit production cost reduces with the cumulative manufacturing and learning effect.
Yoo et al. [18] focused on the problem that not only imperfect production process is possible but also inspection processes are always not perfect, due to generating defects and inspection errors. For this they developed a profit-maximizing economical manufacturing (EMQ) model by incorporating imperfect quality production and two-way imperfect inspection both. Sui et al. [19] provided a model for Vendor-Managed Inventory (VMI) system in place of traditional retailer-managed inventory applying with learning curve approach in which the supplier makes decisions of inventory management for the retailer.
Khan et al. [20] extended the paper of Salameh and Jaber's [11] model by introducing a new
case where there is learning in inspection. The model is more realistic than Salameh and Jaber's [11] model in the that they considered situations of lost sales and back-orders. Wahab and. Jaber [21] developed a model for the optimal lot sizes of an item with imperfect quality which is extension of Salameh and Jaber [11] by incorporating different holding cost for good and defective items. Jaber and Khan [22] presented a model to develop a combination of performance of average processing time and process yield with respect to the number of equal batches. For the development of model, they changed the learning curve parameters in production system and rework both.
Das et al. [23] introduced a production-inventory model (EPQ) for deteriorating items in an indefinite conditions characterized by inflation and timed value of money by considering with static demand. They also considered that the planning interval of the business activity time is random in nature and follows exponential distribution function with a known mean. Khan et al. [24] extended the model of Salameh and Jaber [11] by introducing the inspection error in the time of the screening process and the probability of inspection errors is assumed to be known. Konstantaras et al. [25] developed an economical order quantity (EOQ) model for imperfect quality items considering with shortages. They also assumed that the fraction of perfect quality in each shipment increases with respect to learning effect. A new and more advance inventory model for imperfect quality items has been developed by Jaggi et al. [26] under the situations of permissible delay in payments. Shortages are also allowed and fully backlogged, which are fulfilled during screening process. In this model It has been assumed that screening rate is enough greater than the demand rate.
Teng et al. [27] also proposed an economical order quantity (EPQ) model from the retailer's' point of view to determine his/her optimal production lot size (EPQ) and trade credit financing period simultaneously. For this they assumed that (i) trade credit financing scheme encourage not only sales but also opportunity cost and default risk, and (ii) production cost reduces with respect to learning curve effect. Kumar et al. [28] proposed the effect of learning on the economical ordering policy (EPQ) for deteriorating items incorporating shortages and partially backlogging. They also assumed that due to impact learning process the ordering cost is partly constant and partly decreasing in each cycle. Further they also considered the two-level storage cost for replenishment inventor.
Givi et al. [29] introduced a Human Reliability Analysis (HRA) model that estimates the human error rate while performing a collectively job under the domination of learning-forgetting and fatigue-recovery. This model is enable to quantify the human error rate dynamically with time. Agi and Soni [30] proposed a deterministic demand inventory model for jointly pricing and inventory control of a perishable product considering both physical deterioration and freshness condition degradation. They considered the market demand of product as price and stock sensitive. They suggested that when the primary demand is high, then the retailer would be interested in greedy the benefit of this higher primary demand by increasing the retailing price and accelerating the inventory turnover by reducing a cycle time in place of pricing strategies. Sarkar, and Sumon [31] extended an inventory model for deteriorating items with stock-level dependent market demand. This model has been studied in that situations in which backlogging rate and deterioration rate are time varying with respect to time. Further a sensitivity analysis is analyzed of the model's outputs with respect to key parameters. Jayaswal et al. [32] introduced trade credit financing inventory model for imperfect quality items under the effects of learning on ordering policy. They derived average profit function per cycle time by incorporating various expenditure costs and related parameters for the retailers and the optimization process is also shown by a numerical example. Yadav et al. [33], developed two layer supply chain model to study the effect of imperfect quality items under the asymmetric information with market expenditure sensitive demand.
Soni, and Shah [34] developed an economical production quantity (EPQ) model for retailer's by considering partially constant and partially stock stock sensitive demand. Further they also consider a new progressive credit period. They concluded which credit period is more beneficial for business activity. Benyong and Feng [35] developed a two layer supply chain inventory model
with revenue sharing contract and service requirement under the unpredictability of supply and demand. They formulated the buyer's and supplier's optimal coordination and service requirement situations. They demonstrated how the service requirement's impacts the buyer's and supplier's decisions. Nigwal et aZ.[36] designed an EPQ model on retailer's order quantity using learning effect on screening process under trade credit financing scheme. Nigwal et aZ. [37] developed three stage price dependent trade credit policy for supplier, manufacturer and retailer.
Generally, in the traditional economical order quantity (EOQ) models, it is assumed that the retailer pay to supplier as soon as the product is received. But in the practice, supplier to stimulate sales of his products,he offers to the retailer a certain permissible delay period of payment and after end of this delayed period he charges the interest. In this chapter we have considered a two stage trade credit financing periods, in which firstly supplier offers to the retailer a permissible delay period of payment and the retailer also offers to his customers a permissible delay period of payment without interest. Further, we have also assumed that every manufacturing system may manufacture some defective and good items both. The defective items may be detected by the screening process after delivery of imperfect quality items' batches.To separate the good and defective items we apply screening process on each batches of imperfect quality items on retailer's end. Furthermore we have applied learning effects on screening process. Learning curve (LC) or Experience curve (EC) was derived first by Wright [38] in 1936. It is a mathematical tool which relates the learning variables and cumulative quantity of units. In this chapter we study the impact of learning on screening process on imperfect quality items. Sigmoid function is the ideal shape of all other learning curves and in this paper we use Sigmoid function which is formulated as a(n) = ( + , where a(n) represents defective percentage rate of item
in the single batch and n represents number of batches. fi, g > 0 and a > 0 are the learning curve parameters.
Table 1: Comparative table for contribution of different authors:
Authors Learning Effects Screening Trade Credit Financing Pricing Stock level Strategies
Wright (1936) / X X X X
Muth, and Spremann (1983) / X X X X
Salameh et aZ.(1993) / X X X X
Pal et aZ.(1993) X X X / X
Parlar and Wang (1994) X X X / X
Cheng (1994) / X X X X
Ray, and Chaudhuri(1997) X X X / /
Giri, and Chaudhuri. (1998) X X X / X
Salameh and Jaber (2000) / / X X X
Jaber et aZ. (2004) / / X X X
Teng and Chang (2005) X X X / /
Wu et aZ.(2006) X X X / /
Eroglu and Ozdemir(2007) / / X X X
Jaber and Guiffrida (2008) / / X X X
Pan (2008) / X X X X
Lin (2008) / X X / X
Soni, and Shah (2008) X X / / /
Jaber et aZ. (2008) / / X X X
Yoo et aZ. / / X X X
Sui, et aZ. (2010) / / X X X
Khan et aZ. (2010) / / X X X
Wahab and. Jaber (2010) / / X X X
Jaber and Khan (2010) / X X X X
Das et aZ. (2010) / X X X X
Khan etaZ. (2011) X / X X X
Giri, and Bardhan (2012) X X X / /
Sarkar, and Sumon (2013) X X X / /
Konstantaras et aZ. (2012) / / X X X
Jaggi et aZ. (2013) X / / X X
Authors Learning Effects Screening Trade Credit Financing Pricing Stock level Strategies
Teng et al. (2013) / X X X X
Kumar et al. (2013) / X X X X
Givi et al. (2015) / X X X X
Benyong and Feng (2017) X X X / /
Jayaswal et al. (2019) / / / X X
Maher and Soni (2020) X X X / /
Nigwal et al. (2022) X X / / /
Nigwal et al. (2022) / / / / X
This paper / / / X /
2. The Mathematical Model
I. Notations and Assumptions:
: Lot size of the nth batch,
D : Demand rate of items in units per unit of time for perfect quality items, Where, D = a + bl (t),
Sc : Setup cost per order,
: Initial stock of inventory,
Cp : Purchasing cost per unit of an item,
h : Holding cost of items per unit time,
p : Retailing price per unit of perfect quality items,
v : Retailing price (On discounted rate) per unit of defective items (p > v), a(n) : Percentage rate of defective items per batch, Tn : Length of cycle for shipment per order, X : Screening rate of items per unit time (D < x), Cs : Screening cost per unit items,
Tn : Screening time of batch in planing time Tn, where, Tn = X < Tn, Ie : Interest rate per unit $ earned by retailer, Ip : Interest rate per unit $ paid by retailer, TSR : Sells revenue,
TE : Total cost, n(^n) : Retailer's total profit per unit time,
L : Length delay period of payment offered per cycle time by supplier to the retailer, M : Length delay period of payment offered per cycle time by retailer to customers, The following assumptions are assumed during the development of model:
• The supplier provides a fixed and predetermined credit period to settle the accounts to the supplier,
• For infinite supply rate, selling price p and optimal lot size $n, are decision variable,
• No scrape item will be obtain during the screening process,
• Screening procedure and demand of items occurs simultaneously (D < x).,
• It has been assumed that each lot size contains perfect and imperfect items both,
• It has been assumed that the price of the perfect quality items is greater than the imperfect quality items,
• It has been assumed that the earned interest rate is less than the payable interest rate,
• It has been that the retailer offers a permissible delay period of payment to his customers without interest to stimulate the sales,
• We has been assumed that a limited but maximum amount of stock displayed in a supermarket without leaving a negative impact on customers,
• It has been assumed that L, M € [0, Tn ], only.
• During the formulation of profit, Tn is approximated by second term, because b < 1 and a is very large, therefore b2 ~ 0.
II. The Mathematical formulation of model
The inventory level of perfect quality items at any time t, is governed by the following differential equation:
dl (t)
dt
-(a + bl(t)), 0 < t < Tn,
(1)
with the boundary conditions: I(0) = (1 — a(n))$n, and I(Tn) = 0
solution of this equation : Where A is arbitrary constant, to remove the constant using the
conditions I (0) = (1 — a(n))$n, then solutions becomes
I(t) = b(e-bt - 1) + (1 - a(n))$ne-bt at time t — Tn, the Tn can be determined by the following formula
Tn
log(1 - a(n))$n
(2)
(3)
and according to the assumptions the screening time Tn is given by the following formula
T = $n
x
(4)
The Sales revenue SR = p(1 — a(n))$n + va(n)<£n,Ordering Cost = Oc,Purchasing Cost = Pc$„, Screening Cost = Sc$n, Inventory Holding Costs = h U(1 — (ebTnTnb)) + (1—a—fc(n))^n (1 — e—bTn)
and h
tâa(n)
Now the total expenditure per cycle is given by:
TE = Sc + pc 4>n + Sc + = h
-bTn
b
- Tn) +
(1 - u(n))$ne b
-bT,
n + + (i - x(n))$n
b2
b
(5)
At a time of each replenishment a fixed and certain credit period of payment L is provided by supplier to the retailer and similarly a fixed and certain period of payment M is also provided by retailer to their customers. Where M, L€ (0, Tn) and Tn — Tn There are four different cases available for retailer and their customers.
(1) Tn > L > M (2) Tn > M > L (3) L > M > Tn (4) M > L > Tn
a
X
e
Figure 1: Inventory Level Chart
the retailer's whole profit n(<n), i=1,2,3, 4 per unit of time can be defined as:
n <) = TSRi-TEj+(Earned Interest)-(Paid Interest), where i=1, 2, 3, 4 (6)
Case 1: Tn > L > M
As assumed credit periods, firstly we consider L is greater than M as per depicted in Figure 1, the earned interest and paid interest for this case is estimated as follows:
Earned Interest by retailer:
EIr = Iep[a + b(1 - a(n))<(L - M)] (7)
Paid Interest by retailer:
PIr = IpCp[a + b(1 - a(n))Qnb(L - (1 - a(n) <] + CpIpa(n<(^ - L) (8)
a X
and total profit function per unit time may be defined as follow:
TSR1-TE1+(Earned Interest)-(Paid Interest) n1 (<n) =-t--(9)
Tn
Hence the total profit function per unit time is:
I paO2
n <) = pa+ -e*W +ba( L - M) Igp
(Cs + Cp<n + Sc<n)a - ^ - h(1 - a(n)< - k<pna(n)
(1 - a(n))<n a W/T (1 - a(n))x
(1 - a(n))<n
+ ab(L - X\ + ClrOpi (- A . (10)
V a J. (1 - a(n))\ X J
Theorem 2.1. Retailer's profit function is an optimum at retailer's ordering quantity <*, where is given by the following equation:
\
Csa - Iepa2 + CpIpa2
h(1 - a(n))2 + - CpIpb(1 - a(n))2 + ^^^ 273
a
Proof. On differentiating the equations (10) with respect to $ n , we get
dUi($n) _ Iepa2 Csa . , .. ha(n)
+ n-hû2 — h(1 — a(n)) —
#n (1 — a(n))tyn (1 — a(n))^n x(1 — a(n))
i C?v2 + CIb(1_ a(n))- cpiPa(n)a
+ (1 — «(n))rë + CpIpb(1 a(n)) (1 — a(n))x
(12)
As per optimality condition, on equating to zero the above equation (12), yields — Iepan + Csa — h(1 — a(n))n— ^^ + CpIp (V + b(1 — a(n))Y — ) = 0
On solving the above equation we obtain the equation (11) □
.2. As per optimality condition, at the point of optimality, the second derivative is always
< 0 (13)
negative if
2Iepa2 2Csa CpIp a(n)a2
(1 - a(n))$. - (1 - a(n))$n ~ (1 - a(n))3
Proof. On differentiating again the equation (12) with respect to $n, we get the second order derivative is
d2nx _ 2Iepa2 2Csa CpIpa(n)a2
dfâ (1 — a(n))$ (1 — a(n))4>n (1 — a(n))3
(14)
As per assumptions all the terms , 2Ie,p"3, , 2C3 and ^^ are always positive, and
(1-a (n))$n (1-a (n))$n (1 a(n))
therefore by numerical analysis
2Iepa2 2Csa CpIpa(n)a2 < 0
(1 — a(n))ty3 (1 — a(n))$ (1 — a(n))3
□
Case 2: Tn > M > L
As assumed credit periods, we consider M is greater than L as per depicted in the Figure 2, the earned interest and paid interest for this case is estimated as follows:
Earned Interest by retailer:
EIr — 0 (15)
Paid Interest by retailer:
PIr — IpCp[a + b(1 - a(n))$nb(L - (1 -a(n)) )$n] + CpIpa(n)$n(— - L) (16)
ax
and total profit function per unit time may be defined as follow:
^ , N TSR2-TE2+(Earned Interest)-(Paid Interest)
n2 ($n) =-t--(17)
Tn
Hence the total profit function per unit time is:
t—r /, \ , va(n)a (Cs + Cp$n + Sc$n)a 2hb , , ..
n2($n) = pa + --(1 - a(n))ft--T" - h(1 - a(n))$n
h$na(n) _ I C (1 — a(n))x pC
CpIp a (n)a f ty
a lnuf(1 — a(n))tyn
(1 — a(n))tyn
+ ab [ --^Mr^ — m
(1 — a(n)) V X
^ — L . (18)
a
Figure 2: Inventory Level Chart
Theorem 2.3. Retailer's profit function is an optimum at retailer's ordering quantity where is given by the following equation:
«2
\
Csa + CpIpO2
h(1 - a(n))2 + haXr1 + CpIpb(l - a(n))2 + ^^ '
(19)
^p±puy± aKnjj t x
Proof. On differentiating the equations (18) with respect to <n, we get
dn2(<n) _ Csa - h(1 - ^(n)) - ha(n)
d<n (1 - a(n))<n X(1 - a(n))
+ ,„ CpIpal 2 - CpIpb(1 - a(n)) - CpIpa(n}a (20)
(1 - a(n))<2 vv ( ( " (1 - a(n))x { '
As per optimality condition, on equating to zero the above equation (20), yields
Csa - h(1 - a(n))2<2 - MXM + CpIp (V - b(1 - «(n)^ - ) = 0
On solving the above equation we obtain the value of given in the equation (19) □
Theorem 2
negative if
Theorem 2.4. As per optimality condition, at the point of optimality, the second derivative is always
2Csa CpIpa2
(1 - a(n))<n + (1 - a(n))3 > 0 (21)
Proof. On differentiating again the equations (20) with respect to <n, we get the second order derivative is
d2n = _ 2Csa___CpIpa2
d<2 (1 - a(n))<n (1 - a(n))3 ( )
As per article's assumptions all the terms ^-^n)),^ and (afe^yz2)3 are always positive, and therefore
2Csa CpIpa2 0
(1 - *(n))$n (1 - a(n))3
— Tn-»
Figure 3: Inventory Level Chart
Case 3: L > M > Tn
As assumed credit periods, we consider L is greater than t and M as pr depicted in the Figure 3, the earned interest and paid interest for this case is estimated as follows:
Earned Interest by retailer:
EIr = Iep[a + b(1 - a(n))$(M - L)] + vIea(n)$(L - t)
Paid Interest by retailer:
PIr
Ip Cp
a + b(1 - a(n))$n( (1 - a(n))$n - L
and total profit function per unit time may be defined as follow:
TSR3-TE3+(Earned Interest)-(Paid Interest)
n3 ($n )
Hence the total profit function per unit time is:
Tn
(23)
(24)
(25)
H3($n)
a + va(n)a + vIea(n)a ( $n\ (Cs + Cp$n + Sc$n)a 2hb 1pa (1 - a(n)) (1 - a(n)) V x) (1 - a(n))$n a
- Ip C p
a - ab L +
a2L
1 - a(n)$
+ b((1 - a(n))$
- h(1 - a(n))fn
h$na(n)a CpIpa(n)a f $n
+ —-7-\"T" I--L
(1 - (n))x (1 - (n)) x
(26)
Theorem 2.5. Retailer's profit function is an optimum at retailer's ordering quantity $*3, where $*3 is given by the following equation:
\
Csa + CpIpLa2
h(1 - a(n))2 + ^ + CpIpb(1 - a(n))2 + ^^'
(27)
Proof. On differentiating the equations (26) with respect to $ n , we get
dU3($n) dfn
Iepa2
+
Csa
(1 - a(n))$n (1 - a(n))$n
- h(1 - (n)) -
h (n)
x(1 - (n))
C p Ip a2
+ ,„ 2 + C„I„b(1 - a(n)) - CpIpa(n):
(1 - a(n))$n pp ( ( )) (1 - a(n))x
Figure 4: Inventory Level Chart
As per optimality condition, on equating to zero the above equation (28), yields
Csa + a2CpIpL - h(1 - а(п))2фП -
ha(п)фПа
X
- CpIpb(l - a(n))Yn -
v 1еааф2 X
On solving the above equation we obtain the equation (27)
Theorem
positive if
(29)
□
Theorem 2.6. As per optimality condition, at the point of optimality, the second derivative ^ф3 is always
2Csa
+
2CpIpa2 L
(1 - a(n))4>n (1 - a(n))ф]
> 0
(30)
Proof. On differentiating again the equations (28) with respect to <n, we get the second order derivative is
d2n 2Csa _ (31)
dfy
2CpIpa2 L (1 - и(п))фП (1 - a(n))fi
2C 2C I a2L
As per assumptions all the terms ,„ 3 and ,„ 3 are always positive, and therefore,
r r (1-a (n))<n (1-a (n))<n J r
2Csa
2CpIpa2 L
(1 - я.(п))ФП (1 - О.(П))Ф;
< 0.
□
Case 4: M > L > Tn
As assumed credit periods, we consider M is greater than L and t as per depicted in the Figure 4, the earned interest and paid interest for this case is estimated as follows:
Earned Interest by retailer:
EIr — Iep [a + b(1 - х(п))Фп] L + VIea(п)Фп (L - Tn)
Paid Interest by retailer:
PIr — IpCp
a + b(1 - a(n)^{ (1 - a(n))(pn - L
(32)
(33)
0
and total profit function per unit time may be defined as follow:
TSR4-TE4+(Earned Interest)-(Paid Interest)
n4 ($n ) = -T--(34)
Tn
Hence the total profit function per unit time is:
t—r /, \ , va(n)a . ,,, (Cs + Cp$n + Sc$n)a
4(fn) = pa - CpIp (b(1 - a(n))fn - aL) - (1 -pa{n))$n
a
_ а(п))ф Нфп<Х(n)a I yLa
a h(1 а(П))Фп (1 - a(n))x V
(1 - а(п))фп
+ b
+ v Iea(nK (l - - CpIPa (1 + V (35)
(1 - a(n)) \ x) p \ (1 - a(n))fn) X '
Theorem 2.7. Retailer's profit function is an optimum at retailer's ordering quantity $*ni, where $*4 is given by the following equation:
фп4
\
Csa - Iepa2L - CpIpa2L
h(1 - a(n))2 + ^ + CpIpb(1 - a(n))2 + ^
(36)
Proof. On differentiating the equations (35) with respect to $ n , we get
dU4(fn) _ Csa - h(1 - a(n)) - ha(n)a
йфп (1 - а(и))ф2 x(1 - a(n))
_ Iepa2L _CIb(1_ (n))_ (n)a IPCPL
(1 - а(п))фп CpIpb(1 a(n)) (1 - a(n))x (1 - ФШ'
As per optimality condition, on equating to zero the above equation (37), yields
h2
(37)
Csa - h(1 - a(n))2ф2 - Iepa2L - а(П^Ф (ah + Iev) - CpIp (a2 + b(1 - a(n))2ф2) = 0 (38)
л \ '
On solving the above equation we obtain the equation (36) □
Theorem 2
negative if
Theorem 2.8. As per optimality condition, at the point of optimality, the second derivative а2Д4 is always
афп
2Iepa2 L 2Csa 2CpIpa2 L
(1 - а(п))фп (1 - а(п))ф3 (1 - а(п))фп
< 0 (39)
Proof. On differentiating again the equation (37) with respect to fn, we get the second order derivative is
d2ni = 2 Iepa2 L___2Csa___2CpIpa2 L
d$n (1 - a(n))fn (1 - a(n))fn (1 - a(n))f3
As per assumptions all the terms , n-2^^ ,3 and ,2CpI,p\^L3 are always positive, and
(1 a(n))fn (1 a(n))fn (1 a(n))fn
therefore by numerical analysis
2 Iepa2 L 2Csa 2CpIpa2 L
(1 - а(п))фп (1 - а(п))ф3 (1 - а(п))фп
< 0.
3
□
3. Numerical Examples
Case-1:
We have considered the following data set of input parameters is given as: a = 160 units/unit time, $ = 1, Sc = $0.5, h = $0.8 unit/ unit time, Cp = $150 /unit, v = $45 per/unit, x = 5000 units, Cs = $100/unit, Ie = 0.003/unit time, Ip = $0.004/unit time, a(n) = 0.1599, n = 1, a = 455, b = 0.25, g = 999 L = 0.06/unit time, M = 0.05/unit time.
Following the proposed restrictions for this case we may get the optimal ordering quantity (OOQ) <n = 339 units per unit time, p = 190 and after substituting these optimum values <n, and p into the equation (10) we get the retailer's profit n(<n) = 18266, screening time Tn = 0.06 per unit time, and time interval is Tn = 0.250 in year.
Case-2:
We have considered the following data set of input parameters is given as: a = 160 units/unit time, $ = 1, Sc = $0.5, h = $0.8 unit/ unit time, Cp = $150 /unit, v = $45 per/unit, x = 5000 units, Cs = $100/unit, Ie = 0.003/unit time, Ip = $0.004/unit time, a(n) = 0.1599, n = 1, a = 455, b = 0.25, g = 999 L = 0.06/unit time, M = 0.07/unit time.
Following the proposed restrictions for this case we may get the optimal ordering quantity (O OQ) <n = 423 units per unit time, p = 190 and after substituting these optimum values <n, and p into the equation (18) we get the retailer's profit n(<n) = 17174, screening time Tn = 0.08 per unit time, and time interval Tn = 0.363 in year.
Case-3:
We have considered the following data set of input parameters is given as: a = 160 units/unit time, $ = 1, Sc = $0.5, h = $0.8 unit/ unit time, Cp = $190 /unit, v = $45 per/unit, x = 5000 units, Cs = $100/unit, Ie = 0.003/unit time, Ip = $0.004/unit time, a(n) = 0.1599, q = 1, a = 455, b = 0.25, g = 999 L = 0.09/unit time, M = 0.07/unit time.
Following the proposed restrictions for this case we may get the optimal ordering quantity (O OQ) <n = 244 units per unit time, p = 190 and after substituting these optimum values <n, and p into the equation (26) we get the retailer's profit n(<n) = 39979, screening time Tn = 0.049 per unit time, and time interval Tn = 0.219 in year.
Case-4:
We have considered the following data set of input parameters is given as: a = 160 units/unit time, $ = 1, Sc = $0.5, h = $0.8 unit/ unit time, Cp = $150 /unit, v = $45 per/unit, x = 5000 units, Cs = $100/unit, Ie = 0.003/unit time, Ip = $0.004/unit time, a(n) = 0.1599, n = 1, a = 455, b = 0.25, g = 999 L = 0.06/unit time, M = 0.09/unit time.
Following the proposed restrictions for this case we may get the optimal ordering quantity (O OQ) <n = 181 units per unit time, p = 190 and after substituting these optimum values <n, and p into the equation (35) we get the retailer's profit n(<n) = 17364, screening time Tn = 0.036 per unit time, and time interval Tn = 0.164 in year.
4. SEnsiTiviTy AnALysis
A perusal of Table 2, shows that if the learning ability of the workers is 1 then 16 shipments will be required for the workers to acquire the proficiency. And, if the learning ability of the workers is 1.2 then 16 shipments will be required for the workers to acquire proficiency of screening process. And again, if the learning ability of the workers is 1.4 then 13 shipments will be required for the workers to acquire proficiency of screening process. In addition, as we increase the learning
ability of the workers, the lot size, profit function, screening time and planning time increase, while the number of defective items and number of shipments decrease.The same situations are applying for table numbers 3, 4 and 5 as well. A learning efficiency of worker's not only reduces the number of shipments but also increases the profit per unit time. Table 6 shows the comparative study of various cases. Observation of tables 6, 7, 8 and 9 reveals that Case 3. gives better results per unit time and there is no considerable effect of Beta on outputs of various cases.
Table 2: Impact of Learning Rate and No. of Shipments on Outputs (Case 1)
Learning Rate ß = 1
No. of Shipment (n) 4>n % of good item Tn Tn Profit No of Shipment Required for Perfection
1 339 84.03% 0.068 0.253 9140
4 336 84.81% 0.067 0.252 9670
7 307 92.37% 0.061 0.251 14298
10 284 99.31% 0.057 0.250 17930
13 282 99.96% 0.056 0.250 18249 n=16
16 282 100.00% 0.056 0.250 18265
19 282 100.00% 0.056 0.250 18266
22 282 100.00% 0.056 0.250 18266
25 282 100.00% 0.056 0.250 18266
Learning Rate $ = 1.2
1 339 84.04% 0.068 0.253 9147
4 332 85.72% 0.066 0.252 10269
7 291 97.06% 0.058 0.251 16813
10 282 99.90% 0.056 0.250 18219
13 282 100.00% 0.056 0.250 18265 n=13
16 282 100.00% 0.056 0.250 18266
19 282 100.00% 0.056 0.250 18266
22 282 100.00% 0.056 0.250 18266
25 282 100.00% 0.056 0.250 18266
Learning Rate $ = 1.4
1 282 99.99% 0.056 0.250 18260
4 282 100.00% 0.056 0.250 18266
7 282 100.00% 0.056 0.250 18266
10 282 100.00% 0.056 0.250 18266
13 282 100.00% 0.056 0.250 18266 n=4
16 282 100.00% 0.056 0.250 18266
19 282 100.00% 0.056 0.250 18266
22 282 100.00% 0.056 0.250 18266
25 282 100.00% 0.056 0.250 18266
Table 3: Impact of Learning Rate and No. of Shipments on Outputs (Case 2)
Learning Rate ß = 1
No. of Shipment (n) % of good item Tn Tn Profit No of Shipment Required for Perfection
1 500 84.03% 0.100 0.361 8047
4 495 84.81% 0.099 0.361 8577
7 456 92.37% 0.091 0.362 13206
10 426 99.31% 0.085 0.363 16838
13 423 99.96% 0.085 0.363 17157 n=16
16 423 100.00% 0.085 0.363 17173
19 423 100.00% 0.085 0.363 17174
No. of Shipment (n) $n % of good Tn Tn Profit No of Shipment Required
item for Perfection
22 423 100.00% 0.085 0.363 17174
25 423 100.00% 0.085 0.363 17174
Learning Rate ft = 1.2
1 500 84.04% 0.100 0.361 8054
4 490 85.72% 0.098 0.361 9176
7 435 97.06% 0.087 0.362 15721
10 423 99.90% 0.085 0.363 17127
13 423 100.00% 0.085 0.363 17173 n=13
16 423 100.00% 0.085 0.363 17174
19 423 100.00% 0.085 0.363 17174
22 423 100.00% 0.085 0.363 17174
25 423 100.00% 0.085 0.363 17174
Learning Rate ft = 1.4
1 500 84.05% 0.100 0.361 8062
4 481 87.40% 0.096 0.361 10250
7 426 99.16% 0.085 0.363 16767
10 423 99.99% 0.085 0.363 17168
13 423 100.00% 0.085 0.363 17174 n=13
16 423 100.00% 0.085 0.363 17174
19 423 100.00% 0.085 0.363 17174
22 423 100.00% 0.085 0.363 17174
25 423 100.00% 0.085 0.363 17174
Table 4: Impact of Learning Rate and No. of Shipments on Outputs (Case 3)
Learning Rate ft = 1
No. of Shipment (n) $n % of good Tn Tn Profit No of Shipment Required
item for Perfection
1 288 84.03% 0.058 0.217 35175
4 285 84.81% 0.057 0.217 35454
7 263 92.37% 0.053 0.218 37890
10 246 99.31% 0.049 0.219 39802
13 244 99.96% 0.049 0.219 39970 n=16
16 244 100.00% 0.049 0.219 39979
19 244 100.00% 0.049 0.219 39979
22 244 100.00% 0.049 0.219 39979
25 244 100.00% 0.049 0.219 39979
Learning Rate ft = 1.2
1 288 84.04% 0.058 0.217 35179
4 282 85.72% 0.056 0.217 35770
7 251 97.06% 0.050 0.218 39214
10 244 99.90% 0.049 0.219 39955
13 244 100.00% 0.049 0.219 39979 n=13
16 244 100.00% 0.049 0.219 39979
19 244 100.00% 0.049 0.219 39979
22 244 100.00% 0.049 0.219 39979
25 244 100.00% 0.049 0.219 39979
Learning Rate ft = 1.4
1 288 84.05% 0.058 0.217 35183
4 277 87.40% 0.055 0.217 36335
7 246 99.16% 0.049 0.219 39765
10 244 99.99% 0.049 0.219 39976
13 244 100.00% 0.049 0.219 39979 n=13
No. of Shipment (n) % of good Tn Tn Profit No of Shipment Required
item for Perfection
16 244 100.00% 0.049 0.219 39979
19 244 100.00% 0.049 0.219 39979
22 244 100.00% 0.049 0.219 39979
25 244 100.00% 0.049 0.219 39979
Table 5: Impact of Learning Rate and No. of Shipments on Outputs (Case 4)
Learning Rate ß = 1
No. of Shipment (n) 4>n % of good Tn Tn Profit No of Shipment Required
item for Perfection
1 213 84.03% 0.043 0.163 8238
4 211 84.81% 0.042 0.163 8767
7 195 92.37% 0.039 0.164 13396
10 182 99.31% 0.036 0.164 17028
13 181 99.96% 0.036 0.164 17347 n = 16
16 181 100.00% 0.036 0.164 17364
19 181 100.00% 0.036 0.164 17364
22 181 100.00% 0.036 0.164 17364
25 181 100.00% 0.036 0.164 17364
Learning Rate ft = 1.2
1 213 84.04% 0.043 0.163 8244
4 209 85.72% 0.042 0.163 9366
7 186 97.06% 0.037 0.164 15911
10 181 99.90% 0.036 0.164 17317
13 181 100.00% 0.036 0.164 17363 n=13
16 181 100.00% 0.036 0.164 17364
19 181 100.00% 0.036 0.164 17364
22 181 100.00% 0.036 0.164 17364
25 181 100.00% 0.036 0.164 17364
Learning Rate ft = 1.4
1 213 84.05% 0.043 0.163 8252
4 205 87.40% 0.041 0.163 10440
7 182 99.16% 0.036 0.164 16957
10 181 99.99% 0.036 0.164 17358
13 181 100.00% 0.036 0.164 17364 n=13
16 181 100.00% 0.036 0.164 17364
19 181 100.00% 0.036 0.164 17364
22 181 100.00% 0.036 0.164 17364
25 181 100.00% 0.036 0.164 17364
Table 6: Comparative Table for Case 1
ft No. of ship. required for proficiency Tn Tn Profit
1 16 0.056 0.363 18265
1.2 16 0.056 0.363 18266
1.4 4 0.056 0.363 18266
Table 7: Comparative Table for Case 2
ft No. of ship. required for proficiency Tn Tn Profit
1 16 0.085 0.363 17173
1.2 13 0.085 0.363 17173
1.4 13 0.085 0.363 17174
Table 8: Comparative Table for Case 3
ß No. of ship. required for proficiency Tn Tn Profit
1 16 0.049 0.219 39979
1.2 13 0.049 0.219 39979
1.4 13 0.049 0.219 39979
Table 9: Comparative Table for Case 4
ß No. of ship. required for proficiency Tn Tn Profit
1 16 0.036 0.164 17364
1.2 13 0.036 0.164 17364
1.4 13 0.036 0.164 17364
5. Conclusion
In this article we have optimized the retailer's ordering quantity for imperfect quality items with learning effects on screening process under the trade credit financing scheme. The main focus of this study is that how affects the retailer's ordering quantity when stocking strategies is beneficial for market situations. The various interval of credit periods have been analyzed and verified through the different numerical examples. A comparative study has been done through the numerical examples. and we have concluded that Case 3. is more beneficial for this type of trade credit financing strategies. This article suggests that, those item which sale depends on stocking may earn more and more profit by increasing Tn, fn, and Tn. Article also suggests that, in the financing policy keep always Tn > M > L for better outputs. This article may be extended by incorporating the rework process on defective items. One can also extended this article by incorporating procurement cost on ordering size of items. One can also extended this article by incorporating expected quantity of defective items.
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