Economic production quantity model with trade ... - Inderscience Online

Int. J. Procurement Management, Vol. 7, No. 5, 2014

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Economic production quantity model with trade credit financing and price-discount offer for non-decreasing time varying demand pattern Puspita Mahata Department of Commerce, Srikrishna College, P.O. – Bagula, Dist. – Nadia, 741205, West Bengal, India E-mail: [email protected]

Gour Chandra Mahata* Department of Mathematics, Sitananda College, P.O. and P.S. – Nandigram, Dist. – Purba Medinipur, 721631, West Bengal, India E-mail: [email protected] Abstract: As a policy when some suppliers offer trade credit periods and price discounts to the retailers in order to increase the demand of their product, the retailers have to face different types of offer of discounts and credits for which they have to take a decision which is the best offer for them for profit. One thing is important to the retailers that they try to buy good quality items at a reasonable price and if possible, they try to invest returns obtained by selling those items in such a manner that their business is not hampered. In this paper, we intend to develop an economic production quantity (EPQ)-based model with non-decreasing time varying (quadratic) demand pattern in order to investigate the inventory system as a profit maximisation problem when delay in payment and price discount are permitted by the suppliers to the retailers. Mathematical theorems are developed analytically to determine optimal replenishment policies and a lot of managerial phenomena are obtained through numerical examples. Keywords: inventory; trade credit; time-quadratic demand. Reference to this paper should be made as follows: Mahata, P. and Mahata, G.C. (2014) ‘Economic production quantity model with trade credit financing and price-discount offer for non-decreasing time varying demand pattern’, Int. J. Procurement Management, Vol. 7, No. 5, pp.563–581. Biographical notes: Puspita Mahata is an Assistant Professor in the Department of Commerce, Srikrishna College, Bagula, Nadia, West Bengal, India. She received her BCom and MCom in 1997 and 1999 respectively from Vidyasagar University, Midnapore, West Bengal, India. Her research interest focuses on inventory control/management and supply chain management. Gour Chandra Mahata is an Assistant Professor in the Department of Mathematics, Sitananda College, Nandigram, Purba Medinipur, West Bengal, India. He received his BSc and MSc in 1998 and 2000 respectively, from Copyright © 2014 Inderscience Enterprises Ltd.

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P. Mahata and G.C. Mahata Inventory Control/Management, Operations Research in 2007 from Indian Institute of Technology, Kharagpur, India. His research interest focus on inventory control/management, fuzzy inventory control, supply chain management, operations research and optimisation theory.

1

Introduction

One of the challenging problems for the researchers and the scientists belonging to the research and development (R&D) section of the different marketing sector is to improve the capability of the respective sector by introducing new planning and proposals for their companies. They use different techniques like optimum production technology (OPT), flexible manufacturing system (FMS), just-in-time (JIT), etc. Generally, under the traditional economic order quantity (EOQ) model it is assumed that the retailer must pay for the items upon receiving them. In practice, the supplier hopes to stimulate his products and so he frequently offers retailers a time period, namely, the trade credit period, for payment of the amount owed. Before the end of the trade credit period, the retailer can sell the goods and accumulate revenues and earn interests. In a competitive market, the supplier offers different delay periods with different price discounts to encourage the retailer to order more quantities. On the other hand, a higher interest is charged if the payment is not settled by the end of the trade credit period. Therefore, it makes economic sense for the retailer to delay the settlement of the replenishment account up to the last moment of the permissible period allowed by the supplier. The permissible delay period for payment produces two benefits to the supplier: 1

it attracts new buyers who consider it a type of price reduction

2

it may be applied as an alternative to a discounted price because it neither illicit price cutting from competitors, nor does it introduce permanent price reductions.

On the other hand, the policy of granting credit terms adds an additional cost to the supplier as well as an additional dimension of default risk. Researchers in the past have established their inventory lot-size models under trade credit financing by assuming that the demand rate is constant. However, from a product life cycle perspective, it is only in the maturity stage that demand is near constant. During the growth stage of a product life cycle (especially for high-tech products), the demand function increases with time. One can usually observe in the electronic market that the sales of items increase rapidly in the introduction and growth phase of the life cycle because there are few competitors in market. To obtain robust and generalised results, we advise here that the demand rate should be represented by a continuous quadratic function of time in the growth stage of a product life cycle. As a result, the fundamental theoretical results obtained here are suitable for both the growth and maturity stages of a product life cycle. In the present paper, we attempt to develop an inventory model for credit period and price discount offer with quadratic time-varying deterministic demand, where the model starts with a constant replenishment/supply rate k up to time t = t1. During [0, t1], inventory piles up by adjusting the demand. The accumulated inventory level at time t1 depletes gradually to meet the demand and the level reaches at zero level at time T

Economic production quantity model with trade credit financing

565

(t1 ≤ T). The agreement between the supplier and the retailer is such that total purchasing cost of whole amount (kt1) would be paid within the time M (M > t1) with purchasing cost at discount rate σ. The different delay periods with different discount rates on the purchasing cost are permitted by the supplier to the retailer. In the credit period, the retailer can earn interest by selling the product whereas interest of purchasing cost is charged against the delay of excess time of credit of payment period by retailer to the supplier.

2

Literature review

The first classical EOQ formula, which is also known square-root formula, was developed in 1915 by F.W. Harris. In this formula, demand of items was assumed to be constant over time. This formula was also derived apparently independently by Wilson (1934). The discrete case of this basic EOQ model was first discussed by Wagner and Whitin (1958). After the pioneering attempt by Harris, several other researchers extended the EOQ model considering variable demand rate. To determine EOQ, the general case of a time varying demand rate was first discussed by Silver and Meal (1969). For mathematical convenience, Donaldson’s (1977) analytical solution considered finite time horizon and linear time dependent demand. His model’s requirements were a substantial amount of computational effort to obtain the optimal time of replenishment. Following Donaldson (1977), significant contribution in this direction came out from researchers like Silver (1979), Buchanan (1980), Silver and Peterson (1985), Goyal et al. (1986), etc. All these models were developed on the assumption that there was no shortage. Deb and Chaudhuri (1987) were the first to introduce shortage into inventory with a linear increasing time-varying demand which is an extension of Donaldson (1977) model with shortage. These models were again extended by many researchers like Goswami and Chaudhuri (1991), Giri et al. (2000), Hariga (1996), Goyal et al. (1992) and others. Hariga and Bankherouf (1994) discussed some optimal and heuristic replenishment models for deteriorating items with an exponentially time-varying demand. Wee (1995) studied an EOQ model allowing shortages where demand declines exponentially over time. Sana and Chaudhuri (2008) discussed a production inventory model with quadratic demand pattern which was an extension of Khanra and Chaudhuri (2003). It is notable that Hsieh et al. (2009) generalised the demand rate for an EOQ model with upstream and downstream trade credits to an increasing function of time, and proved that a unique global minimum cost per unit of time exists. Recently, Teng et al. (2012) developed an EOQ model with trade credit financing for linear non-decreasing demand function of time. In a competitive market, price of goods plays an important factor to a customer. Generally, a reduced price encourages a customer to buy more. In this point of view, Abad (1988), Kim and Hwang (1988), Hwang et al. (1990) and Burwell et al. (1991) developed the traditional quantity discount models. Urban and Baker (1997) generalised the EOQ model in which the demand is a multivariate function of price, time and level of inventory. Datta and Pal (2001) analysed a multiperiod EOQ model with stock dependent and price-sensitive demand rate. Teng and Chang (2005) extended an EPQ model for perishable items, considering the demand rate as the sum of two terms: first term is inversely proportional to the price and second term is directly proportional to the stock-level of inventory displayed. Cárdenas-Barrón (2009a, 2009b) presented optimal

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ordering policies in response to a discount offer. Cárdenas-Barrón et al. (2010) analysed optimal order size to take advantage of a one-time discount offer with allowed backorders. Recently, Cárdenas-Barrón (2012) developed an EOQ model with imperfect quality and quantity discounts. Taleizadeh et al. (2013) developed an EOQ model for perishable product with special sale and shortage. In trade-credit policy, the supplier allows a certain fixed period to pay the purchasing cost. This fixed period which is settled by the supplier is called the credit period to the retailer. During this credit period, the supplier sells his items to the retailers with different types of discounts to get profits as quick as possible during the credit period. Depending on this policy Goyal (1985) first developed the EOQ model with permissible delay in payments. Shah (1993a, 1993b) developed a probabilistic time schedule and a lot size model for an exponentially decaying inventory when delays in payments are permissible. Aggarwal and Jaggi (1995) developed the inventory model with an exponentially deteriorating rate by considering permissible delay in payments. Chu et al. (1998) and Chung (1998a, 1998b) also extended Goyal’s (1985) model by considering the case of deterioration. Liao et al. (2000) and Sarker et al. (2000, 2001) developed Goyal’s (1985) model with inflation. Jamal et al. (1997, 2000) and Chang et al. (2001, 2002) extended this model with shortage. Teng (2002) developed an EOQ model for a retailer to order small lot size in order to take more quickly the benefits of permissible delay in payments. Arcelus et al. (2003) developed an inventory model by considering the retailers maximising profit and inventory policies for vendors trade promotion offer of price/credit on the purchase of perishable items. Chung and Liao (2004) developed an EOQ model for exponential deteriorating items under permissible delay in payments depending on the ordering quantity. Huang (2007) developed an EOQ model of retailer’s inventory system to investigate the optimal retailer’s decisions under two levels of trade credit policy within the economic production quantity (EPQ) framework. Goswami et al. (2010) presented an EPQ model for deteriorating items with two level of trade credit financing. Shah et al. (2011) determine optimal pricing, shipment and payment policies for an integrated supplier-buyer deteriorating inventory model in buoyant market with two level trade credit. Singh et al. (2011) developed a two-warehouse fuzzy inventory model under the conditions of permissible delay in payment. Thangam and Uthayakumar (2011) presented a two-echelon trade credit financing in a supply chain with perishable items and two different payment methods. Mahata (2011) presented an EPQ model for deteriorating items with fuzzy cost components under retailer partial trade credit financing. Chen et al. (2013) developed retailer’s EOQ model when the supplier offers conditionally permissible delay in payments link to order quantity. In this direction, mention should be made of the works by Huang (2003), Ouyang et al. (2005), Chung et al. (2008), Hsieh et al. (2008), Teng and Goyal (2007), Jaggi and Khanna (2009), Mahata and Mahata (2011a, 2011b), Ho (2011), Mahata (2012), Chung and Cárdenas-Barrón (2013) and their references.

3

Notation and assumptions

The following notation and assumptions are used in this paper.


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3.1 Notations C0

ordering cost per order ($/order)

h

unit stock holding cost per item per year (excluding interest charges) ($/unit/year)

cp

unit purchasing cost per item ($/unit)

cm

maximum retail price per unit ($/unit)

s

unit selling price per item ($/unit)

Ie

rate of interest gaining per $ per year due to the credit balance

Ic

rate of interest per $ per year due to financing inventory

d(t) time varying demand rate k

constant replenishment rate

I1(t) on hand inventory at time t (0 ≤ t ≤ t1) I2(t) on hand inventory at time t (t1 ≤ t ≤ T) M

variable delay period in years

Mi

ith permissible delay period in years

σi

discount rate on purchasing cost at ith permissible delay period

t1

duration of the replenishment in years

t1*

the optimal duration of replenishment in years

T

the length of inventory cycle in years

T

*

the optimal length of inventory cycle in years

TP1i the average profit in $ of the system when T ≥ Mi TP2i the average profit in $ of the system when T ≤ Mi.

3.2 Assumptions 1

The deterministic time-dependent demand rate D(t) = a + bt + ct2, a, b, c > 0 at time t (> 0) is a continuous function of time in the interval (0, T), where a, b and c are constants. Here ‘a’ is initial rate of demand, ‘b’ is the rate with which the demand rate increases. The rate of change in the demand rate itself changes at a rate c. Now,

dD ( t ) dt

= b + 2ct ,

d 2 D (t ) dt 2

= 2c. Again if

dD ( t ) dt

= 0 ⇒ t = − 2bc . The rate of increase

of the demand rate D(t) is an increasing function of time. This type of demand is known as accelerated growth in demand which is seen to occur in the case of the state-of-the-art aircrafts, computers, machines and their spare parts, etc. 2

Replenishment rate is finite and always k > d(t).

3

Permissible delay in payment is considered to the retailer by the supplier. The retailer would settle the account at t = M and pay for the interest charges on items in

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P. Mahata and G.C. Mahata stock with rate Ic over the interval [M, T] as T ≥ M. Alternatively, the retailer settles the account at t = M and is not required to pay any interest charge for items in stock during the whole cycle as T ≤ M.

4

The retailer can accumulate revenue and earn interest from the beginning of the inventory cycle until the end of the trade credit period offered by the supplier. That is, the retailer can accumulate revenue and earn interest during the period from t = 0 to t = M with rate Ie under the trade credit conditions.

5

Different discount rates on purchasing cost for different delay periods are allowed.

6

Lead time is neglected.

7

Time horizon is infinite.

8

Shortages are not allowed.

Given the above, it is possible to formulate a mathematical inventory EPQ model with trade credit financing.

4

Model formulation

Based on the above assumptions, the inventory system can be considered as follows: At the beginning (e.g., at time t = 0), the cycle starts with zero stock level at supply rate k. The replenishment or supply continues up to time t1. During the time span [0, t1], inventory piles up by adjusting the demand in the market. This accumulated inventory level at time t1 gradually diminishes during the period [t1, T] due to reasons of market demand of items and ultimately falls to zero at time t = T. After the scheduling period T, the cycle repeats itself. As a business strategy, most of the suppliers offer a delay period M to the retailer to pay the total purchasing cost (cpkt1) of the items. As a result, the retailers take maximum amount of products as per their requirement since the purchasing cost is allowed with different delay periods Mi, (i = 1, 2, 3) with different discount rates σi (i = 1, 2, 3) on the MRP of the products. In this point of view, we consider the purchasing cost for different delay periods as follows: ⎧cm (1 – σ1 ) when M = M 1 ⎪ ⎪cm (1 – σ 2 ) when M = M 2 cp = ⎨ ⎪cm (1 – σ 3 ) when M = M 3 ⎪⎩∞ when M > M 3

where cm = maximum retail price per unit, Mi (i = 1, 2, 3) = decision point in settling the account to the supplier at which supplier offer σi% 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 items to the retailer. Consequently, the supplier decides cp → ∞ when M > M3, i.e., retailer never purchases at an infinite cost. Indirectly, supplier would not supply the products to the retailer while delay period M exceeds M3. σi (i = 1, 2, 3) are the constant discount rates decided by the supplier. Mi and σi are best fitted by the supplier, from his a priori knowledge of statistics.


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Now, the differential equations governing the instantaneous states of the inventory level in the interval [0, T] are given by dI1 (t ) = k – d (t ) = k – ( a + bt + ct 2 ) with I1 (0) = 0, 0 ≤ t ≤ t1 , dt

(1)

dI 2 (t ) = – d (t ) = – ( a + bt + ct 2 ) with I 2 (T ) = 0, t1 ≤ t ≤ T . dt

(2)

and

The solutions of the above differential equations (1) and (2) are respectively given by bt 2 ct 3 – , 0 ≤ t ≤ t1 , 2 3

(3)

b c I 2 (t ) = a(T – t ) + (T2 – t2 ) + (T3 – t3 ) , t1 ≤ t ≤ T . 2 3

(4)

I1 (t ) = kt – at –

In addition, using the boundary condition I1(t1) = I2(t1), we obtain kt1 – at1 –

bt12 ct13 b c = a (T – t1 ) + (T2 – t12 ) + (T3 – t13 ) . – 2 3 2 3

i.e., t1 =

bT 2 cT 3 ⎞ 1⎛ – ⎜ aT – ⎟. k⎝ 2 3 ⎠

(5)

The elements comprising there Tailer’s profit function per cycle are listed below: a

Sales profit (SP): In this model, the production process starts at t = 0 at a rate k up to time t1 and after facing the market demand it decreases to zero at time T. Hence, the total production up to time t1 is kt1 and (s – cp) is the profit per item, therefore, the profit by selling the products is (s – cp)kt1.

b

Ordering cost (OC): The fixed cost, C0, is incurred for each order placed.

c

Holding cost (HC): The holding cost per unit per year, h, is incurred by the retailer based on the instantaneous inventory level; this cost does not include the interest paid to the supplier: The holding cost (HC) (excluding interest charges) ⎛ = h⎜ ⎝

∫

t1

0

I1 (t )dt +

∫

T

t1

⎞ I 2 (t )dt ⎟ ⎠

⎧T ⎛ b 2T 3 c 2T 5 bcT 4 2acT 3 ⎞ aT bT 2 cT 3 – = h ⎨ ⎜ a 2T + + + abT 2 + + + + ⎟+ 4 9 3 3 ⎠ 2 3 4 ⎩ 2k ⎝ a 2T abT 2 2acT 3 b 2T 3 bcT 4 c 2T 5 ⎫ – – – – – ⎬. 3k 4k 3k 9k ⎭ k k


570 d

Interest payable (IP): From assumption 3, based on the values of T and M, there are two alternative cases to be arise in relation to interest payable: T ≥ M and T ≤ M. •

Case 1: T ≥ M (see Figure 1) When the credit period M is shorter than or equal to the replenishment cycle time T, the retailer begins to pay interest for the items in stock after time M with rate Ic. Hence, the interest payable for financing inventory during [M, T] is IP = c p I c

∫

T

M

I 2 (t )dt

aM i2 bTM i bM i3 cT 2 M i cM i4 ⎞ ⎛ aT bT 2 cT 3 – aM i + – – = c p I cT ⎜ + + + + ⎟. 3 4 2T 2 6T 3 12T ⎠ ⎝ 2

•

Case 2: T ≤ M (see Figure 2) Since the cycle time T is shorter than the credit period M, there is no interest paid for financing the inventory in stock. Therefore, the interest payable in this case is zero.

Figure 1

Inventory versus time (case 1: T ≥ M)

Figure 2

Inventory versus time (case 2: T ≤ M)

Economic production quantity model with trade credit financing e

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Similarly, from assumption 4, there are also two alternative cases that can occur in relation to interest earned. •

Case 1: T ≥ M. In this case, the retailer sells the goods, accumulates sales revenue, and earns interest with rate Ie until time M. Therefore, the interest earned from time 0 to M per cycle is sI e

•

∫

⎛ aM i2 bM i3 cM i4 ⎞ ( M – t ) D (t )dt = sI e ⎜ + + ⎟. 6 12 ⎠ ⎝ 2

M

0

Case 2: T ≤ M. Since the cycle time T is shorter that the credit period M, from time 0 to T the retailer sells the goods and continues to accumulate sales revenue to earn interest sI e

∫

T

0

(T – t ) D(t )dt. From time T to M the retailer can use the

sales revenue generated in [0, T] to earn interest sIekt1(M – T). Therefore, the interest earned from time 0 to M per cycle is ⎡ sI e ⎢ ⎣

⎤ (T – t ) D (t )dt + kt1 ( M – T ) ⎥ ⎦ bTM i cT 2 M i aT bT 2 cT 3 ⎞ ⎛ – – – = sI eT ⎜ aM i + + ⎟. 2 3 2 3 4 ⎠ ⎝

∫

T

0

From the above results, the profit per unit of time can be expressed as TPi (T ) =

1 [( SP) − (OC ) − ( HC ) − ( IP) + ( IE )]. T

(6)

i.e., ⎧TP1i (T ); T ≥ M i TPi (T ) = ⎨ ⎩TP2i (T ); T < M i

for i ∈ {1, 2, 3}.

(7)

On summation and simplification, the following result is achieved: ⎡ bT cT 2 ⎞ sI e ⎧ aM i2 bM i3 cM i4 ⎫ ⎛ TP1i (T ) = ⎢( s – c p ) ⎜ a + + + + ⎨ ⎬ ⎟+ 2 3 ⎠ T ⎩ 2 6 12 ⎭ ⎝ ⎣ ⎧1 ⎛ b 2T 3 c 2T 5 bcT 4 2acT 3 ⎞ aT bT 2 + + abT 2 + + + – h ⎨ ⎜ a 2T + ⎟+ 4 9 3 3 ⎠ 2 3 ⎩ 2k ⎝ 3 2 2 3 2 3 4 2 5 2acT cT a T abT bT bcT cT ⎫ + – – – – – – ⎬ 4 3k 4k 3k 9k ⎭ k k aM i2 ⎧ aT bT 2 cT 3 + + – aM i + – c p Ic ⎨ 4 2T 3 ⎩ 2 –

bTM i bM i3 cT 2 M i cM i4 ⎫ C0 ⎤ – + + ⎬ – ⎥, 2 6T 3 12T ⎭ T ⎦

(8)

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P. Mahata and G.C. Mahata ⎡ bT cT 2 ⎞ ⎛ TP2i (T ) = ⎢( s – c p ) ⎜ a + + ⎟ 2 3 ⎠ ⎝ ⎣ bTM i cT 2 M i aT bT 2 cT 3 ⎫ ⎧ – – – + sI e ⎨aM i + + ⎬ 2 3 2 3 4 ⎭ ⎩ ⎧1 ⎛ b 2T 3 c 2T 5 bcT 4 2acT 3 ⎞ aT bT 2 – h ⎨ ⎜ a 2T + + + abT 2 + + + ⎟+ 4 9 3 3 ⎠ 2 3 ⎩ 2k ⎝ +

(9)

cT 3 a 2T abT 2 2acT 3 b 2T 3 bcT 4 c 2T 5 ⎫ C0 ⎤ – – – – – – ⎬ – ⎥. k k 4 3k 4k 3k 9k ⎭ T ⎦

Therefore, it is easy to verify that TP1i(Mi) = TP2i(Mi), TPi(T) is continuous and well-defined on T > 0. In the next section, we characterise the retailer’s optimal solution and determine its optimal cycle time T* for both the cases of T ≥ M and T ≤ M.

5

Theoretical results and optimal solutions

In order to find the optimal solution T* for the case of T ≥ M, we derive that the first-order necessary condition for TP1i(T) in equation (8) to be maximised is ⎡ 2cT ⎞ sI e ⎧ aM i2 bM i3 cM i4 ⎫ ⎛ dTP1i (T )dT = ⎢( s – c p ) ⎜ b2 + + + ⎨ ⎬ ⎟– 3 ⎠ T2 ⎩ 2 6 12 ⎭ ⎝ ⎣ ⎧1 ⎛ 3b 2T 2 5c 2T 4 4bcT 3 ⎞ a + + 2abT + + 2acT 2 ⎟ + – h ⎨ ⎜ a2 + 4 9 3 ⎠ 2 ⎩ 2k ⎝ 2 2 2 2 3 2 4 2bT 3cT a 2abT 3b T 4bcT 5c T ⎫ + + – – – – – ⎬ 3 4 k k 4k 3k 9k ⎭ ⎧ a 2bT 3cT 2 aM i2 bM i + – – c p Ic ⎨ + – 2T 2 2 3 4 ⎩2 –

(10)

bM i3 2cTM i cM i4 ⎫ C0 ⎤ – – ⎬+ ⎥ 6T 2 3 12T 2 ⎭ T 2 ⎦

1 [ X1T 6 + X 2T 5 + X 3T 4 + X 4T 3 + X 5T 2 + X 6 ] T2 = 0, =

which implies that X 1T 6 + X 2T 5 + X 3T 4 + X 4T 3 + X 5T 2 + X 6 = 0,

where X1 =

5c 2 h , 18k

X2 =

2bch , 3k

⎧⎛ 3b 2 ⎞ h 2 3c ⎫ X 3 = ⎨⎜ 2ac + – ( h + c p Ic ) ⎬ , ⎟ 4 ⎠ 2k 4⎭ ⎩⎝

2c hab 2b 2c p cM i I c ⎫ ⎧ X 4 = ⎨( s – c p ) + – ( h + cp Ic ) + ⎬, 3 k 3 3 ⎩ ⎭

(11)


573

b ha 2 a c p bM i I c ⎫ ⎧ X 5 = ⎨( s – c p ) + – ( h + c p Ic ) + ⎬, k 2 2 2 ⎩ ⎭ 2 3 4 ⎛ aM i bM i cM i ⎞ + + X 6 = C0 + ⎜ ⎟ ( c p I c – sI e ) . 3 12 ⎠ ⎝ 2

For the second-order condition, it is clear from equation (10) that d 2TP1i (T ) ⎡ 2c 2 sI ⎛ aM i2 bM i3 cM i4 ⎞ = ⎢( s – c p ) + 3 e ⎜ + + ⎟ 2 dT 3 T ⎝ 2 6 12 ⎠ ⎣ ⎧ 1 ⎛ 3b 2T 20c 2T 3 ⎞ 2b 3cT + + 2ab + 4bcT 2 + 4acT ⎟ + + – h⎨ ⎜ 9 2 ⎠ c ⎩ 2k ⎝ 2 2 2 2 3 2ab 4acT 3b T 4bcT 20c T ⎫ – – – – – ⎬ k k 2k k 9k ⎭

(12)

⎛ 2b 3cT aM i2 bM i3 2cM i cM i4 ⎞ 2C0 ⎤ + 3 + + – c p Ic ⎜ + – ⎟– ⎥ 6T 3 ⎠ T 3 ⎦ 2 T 3T 3 3 ⎝ 3 1 = 3 [ 4 X 1T 6 + X 2T 5 + X 3T 4 + X 4T 3 – 2 X 6 ]. T

Consequently, the following theoretical results for TP1i(T) can be derived. Theorem 1: For TP1i(T), the following theoretical results can be obtained: 1

Equation (11) has not more than one positive solution.

2

If the only positive solution T1i to equation (11) exists for T1i ∈ [Mi, ∞), then T* = T1i is the optimal solution to TP1i(T) in equation (8) if 4X1(T*)6 + X2(T*)5 + X3(T*)4 + X4(T*)3 – 2X6 < 0; otherwise the optimal solution to TP1i(T) is at T * = {T | 3c T 3 + b2 T 2 + aT – kM i = 0, T > 0} if X 1M i6 + X 2 M i5 + X 3 M i4 + X 4 M i3 + X 5 M i2 + X 6 > 0 holds.

Proof: See Appendix A. Similarly, for the alternative case of T ≤ M, the first-order necessary condition for TP2i(T) in equation (9) to be maximised is dTP2i (T ) ⎡ ⎛ 2cTM i bM i a 2bT 3cT 2 ⎞ ⎛ b 2cT ⎞ = ⎢( s – c p ) ⎜ + + – – – ⎟ – sI e ⎜ ⎟ dT 3 ⎠ 2 2 3 4 ⎠ ⎝2 ⎝ 3 ⎣ ⎧1 ⎛ 3b 2T 2 5c 2T 4 4bcT 3 ⎞ + + 2abT + + 2acT 2 ⎟ – h ⎨ ⎜ a2 + 4 9 3 ⎠ ⎩ 2k ⎝

=

+

a 2bT 3cT 2 a 2 2abT + + – – 2 3 4 k k

–

2acT 2 3b 2T 2 4bcT 3 5c 2T 4 ⎫ C0 ⎤ – – – ⎬+ ⎥ k 4k 3k 9k ⎭ T2 ⎦

1 [ X1T 6 + X 2T 5 + X 7T 4 + X 8T 3 + X 9T 2 + C0 ] = 0, T2

(13)

574


which implies that X 1T 6 + X 2T 5 + X 7T 4 + X 8T 3 + X 9T 2 + C0 = 0,

(14)

⎧⎛ 3b 2 ⎞ h 3c ⎫ X 7 = ⎨⎜ 2ac + – ( h + c p Ie ) ⎬ , ⎟ 4 ⎠ 2k 4⎭ ⎩⎝

(15)

where

{

X8 = ( s – cp )

}

2c hab 2b 2 sI e cM i – ( h + sI e ) + , + 3 3 3 k

b ha 2 a sI bM ⎫ ⎧ – ( h + sI e ) + e i ⎬ . X 9 = ⎨( s – c p ) + 2 2 2 ⎭ k ⎩

(16) (17)

For the second-order condition, it is clear from equation (13) that d 2TP2i (T ) ⎡ 2c ⎛ 2cM i 2b 3cT ⎞ = ⎢( s – c p ) – sI e ⎜ – – ⎟ 2 dT 3 3 2 ⎠ ⎝ 3 ⎣ ⎧ 1 ⎛ 3b 2T 20c 2T 3 ⎞ 2b 3cT + + 2ab + 4bcT 2 + 4acT ⎟ + + – h⎨ ⎜ 9 2 ⎠ 3 ⎩ 2k ⎝ 2 2ab 4acT 3b 2T 4bcT 2 20c 2T 3 ⎫ 2C0 ⎤ – – – – – ⎬– ⎥ 2k 9k ⎭ T 3 ⎦ k k P =

(18)

1 [5 X1T 6 + X 2T 5 + 2 X 7T 4 + X 8T 3 – 2C0 ]. T3

Consequently, the following theoretical results for TP2i(T) can be derived. Theorem 2: For TP2i(T), the following results can be derived: 1

Equation (14) has not more than one positive solution.

2

If the positive solution T2i to equation (14) exists for T2i ∈ [0, Mi], then T* = T2i is the optimal solution to TP2i(T) in equation (9) if 5X1(T*)6 + X2(T*)5 + 2X7(T*)4 + X8(T*)3 – 2C0 < 0; otherwise the optimal solution to TP2i(T) is T* = Mi if M i6 X 1 + M i5 X 2 + M i4 X 7 + M i3 X 8 + M i2 X 9 + C0 > 0.

Proof: See Appendix B.

6

Numerical example

The following numerical examples are given to illustrate the above solution procedure. Example 1: We consider the values of the following parameters in appropriate units: C0 = $250 per order, h = $20/unit/year, M1 = 120 / 365 year, M2 = 150/365 year, M3 = 180/365 year, σ1 = 10%, σ2 = 5%, σ3 = 2%, a = 100 units, b = 50 units, c = 10 units, d(t) = a + bt + ct2, cm = $120/units, s = $180/units, Ic = 0.16/$/year, Ie = 0.13/$/year, P = 500 units.


575

Then the optimal solutions are {TP11 = $7,522.33, T* = 0.845044 year, t1 = 0.208737 year}, {TP12 = $6,980.84, T* = 0.716632 year, t1 = 0.171458 year}, {TP13 = $6,760.68, T* = 0.655712 year, t1 = 0.15452 year}, {TP21 = $7,426.42, T* = 0.667352 year, t1 = 0.15772 year}, {TP22 = $6,947.22, T* = 0.62593 year, t1 = 0.14641 year}, {TP23 = $6,751.5, T* = 0.610456 year, t1 = 0.142241 year}. Among the above optimal solutions, the better optimal solution is TP11 = $7,522.33, T* = 0.845044 year, t1 = 0.208737 year. For this type of demand pattern, the average profit function is highly non-linear. So it is impossible to find closed type formula for T. But Figure 3 and Figure 4 show the concavity of the function. Hence, the better optimal solution is a global maximum. Figure 3

Case 1: average profit versus cycle length for quadratic increasing demand

Figure 4

Case 2: average profit versus cycle length for quadratic increasing demand

The sensitivity analysis is performed by varying different parameters and is given in Table 1. It is important to discuss the influence of key model parameters on the optimal solutions. The effect of changing the parameters are shown in Table 1. Based on Table 1, we have the following comments. 1

If ordering cost increases, the replenishment cycle time increases and the total relevant profit decreases. The economic interpretation is as follows: the retailer needs to order more quantity to reduce the number of orders if the ordering cost is more costly.

576


2

If the unit holding cost per unit item increases, both the replenishment cycle time and the total profit of the system decrease. So it is reasonable that when the holding cost increases the retailer will shorten the cycle time and order quantity in an effort to maintain his profit gained by keeping the threshold credit period.

3

Increasing value of maximum retailer price increases the total purchasing cost of the whole system which decreases the total profit.

4

The larger the value of the selling price per item, the smaller value of the optimal cycle time and the higher the value of the annual total profit. This result implies that the retailer will order less quantity to take the benefits of the trade credit more frequently.

5

As production rate increases, the annual total profit decreases, so it not advisable to increase the production rate without the prior knowledge about the demands. Sensitivity analysis of the numerical example

Table 1

Optimal duration of replenishment

Optimal cycle time

Optimal average profit

125

0.1668434

0.7002026

7,683.238

Parameter C0

h

Cm

s

P

187.50

0.1902255

0.7822109

7,599.081

250

0.2087369

0.8450444

7,522.334

312.50

0.2244370

0.8969836

7,450.616

375

0.2382617

0.9417464

7,382.658

10

0.4297871

1.486767

8,136.666

15

0.2889888

1.098849

7,773.826

20

0.2087369

0.8450444

7,522.334

25

0.1625143

0.6846770

7,334.482

30

0.1342445

0.5804673

7,184.154

60

1.255286

3.256242

16,325.28

90

0.7925701

2.275550

11,713.83

120

0.2087369

0.8450444

75,22.334

150

0.1086158

0.4814190

4,328.387

180

0.07139209

0.3287559

1,311.381

90

0.07142577

0.3288992

1,954.495

135

0.09955894

0.4452826

2,245.407

180

0.2087369

0.8450444

7,522.334

225

0.8317499

2.348360

14,110.53

270

1.7525472

3.953252

21,235.26

250

0.4165208

1.423521

7,712.326

375

0.3228509

0.9541404

7,615.952

500

0.2087369

0.8450444

7,522.334

625

0.1548216

0.7935481

7,473.031

750

0.1232128

0.7635191

7,442.404


7

577

Conclusions

A widespread approach to inventory modelling is to associate costs and profits with measures of system performance and determine the control policy which maximises the average profit per unit time. In this paper, we develop an EPQ model of time varying quadratic non-decreasing demand pattern for a retailer where supplier’s trade offer gives the retailer a credit period and price-discount on the purchase of merchandise. From the view point of the supplier, price discount on the purchasing cost of the items by retailer is given at different delay period to motivate the retailer to buy more. In this context, σ / Mi net Mi approach is applied in the model. Generally, suppliers allow maximum delay period, after which they would not take a risk of getting back money from retailers or any other loss of profit. As a result, the purchasing cost cp is infinite when M is greater than M3, i.e., the supplier will not agree to sale items to retailers after the delay period M3. In practice, Ic is greater than Ie from any reliable source such as government sectors or renowned private sectors. From retailer side, he always chooses the optimum delay period/credit period, keeping in mind the profit maximisation. In addition, we establish some effective and easy to use theorems and figures to help the retailer to take optimal strategy for his marketing policy. As far as the knowledge of authors goes, such type of extension has not yet been discussed in the literature. Further extension of the model may be generalised by considering shortages, deterioration, stochastic demand, time-value for money, finite time horizon, etc.

Acknowledgements The authors greatly appreciate the anonymous referees for their very valuable and helpful suggestions on an earlier version of this paper. This research work is fully supported by the University Grants Commission (UGC), INDIA, for providing a Minor Research Project (MRP, UGC) under the Research Grant No. PHW-249/11-12.

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Appendix A Proof of Theorem 1 First part of the theorem is obvious from the above calculations of

dZ1i (T ) dT

and

d 2 Z1i (T ) . dT 2

For the second part, if Z1i(T) does not have any stationary points in [Mi, 1), then Z1i(T) is either monotonic increasing or monotonic decreasing function of dZ1i (T ) T ∈ [Mi, 1). Here dT → ∞ as T → ∞ because X1 > 0. So, Z1i(T) will be monotonic increasing if t1 =

1 P

dZ1i (T ) dT T =Mi

> 0 as Z1i(T) does not have stationary points in [Mi, 1). Here,

( aT + b2 T 2 + 3c T 3 ) , t1 increase with increasing value of T. According to our model,

t ≤ Mi, so for the feasibility of the model, t1* = M i and T* will be obtained from c (T * )3 + b2 (T * ) 2 + a (T * ) = PM i . By Descarte’s rule, 3c (T * )3 + b2 (T * ) 2 + a (T * ) – PM i = 0 3 may have atmost one positive root and two negative roots. Therefore, Z1i(T) has a maximum value at T * = {T | 3c T 3 + b2 T 2 + aT – PM i = 0, T > 0} if M12 [ X 1M i6 + X 2 M i5 i

+ X 3 M i4 + X 4 M i3 + X 5 M i2 + X 6 ] > 0. Here T* is unique if it exists, by Descarte’s rule. Hence, the proof.

Appendix B Proof of Theorem 2 First part of the theorem is obvious from the calculations of

dZ 2 i (T ) dT

and

d 2 Z 2 i (T ) . dT 2

For the

second part, if Z2i(T) does not have any stationary points in [0, Mi], then Z2i(T) is either dZ 2 i (T ) monotonic increasing or monotonic decreasing function of T ∈ [0, Mi]. Here dT →∞ as T → 0 because C0 > 0. So, Z2i(T) will be monotonic increasing if

dZ 2 i (T ) dT T =Mi

> 0 as

Z2i(T) does not have stationary points in [0, Mi]. Hence, Z2i(T) has a maximum value at T = Mi if M i6 X 1 + M i5 X 2 + M i4 X 7 + M i3 X 8 + M i2 X 9 + C0 > 0. Hence, the proof of the theorem.