Pricing and lot-sizing polices for perishable goods when the demand depends on selling price, displayed stocks, and expiration date

Author’s Accepted Manuscript Pricing and lot-sizing polices for perishable goods when the demand depends on selling price, displayed stocks, and expiration date Lin Feng, Ya-Lan Chan, Leopoldo Eduardo Cárdenas-Barrón www.elsevier.com/locate/ijpe

PII: DOI: Reference:

S0925-5273(16)30396-6 http://dx.doi.org/10.1016/j.ijpe.2016.12.017 PROECO6611

To appear in: Intern. Journal of Production Economics Received date: 12 July 2016 Revised date: 7 December 2016 Accepted date: 9 December 2016 Cite this article as: Lin Feng, Ya-Lan Chan and Leopoldo Eduardo CárdenasBarrón, Pricing and lot-sizing polices for perishable goods when the demand depends on selling price, displayed stocks, and expiration date, Intern. Journal of Production Economics, http://dx.doi.org/10.1016/j.ijpe.2016.12.017 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Pricing and lot-sizing polices for perishable goods when the demand depends on selling price, displayed stocks, and expiration date

Lin Feng1, Ya-Lan Chan2, Leopoldo Eduardo Cárdenas-Barrón3,* 1

School of Economics and Management, Southwest Jiaotong University, Chengdu 610031, China. E-mail address: [email protected] 2 Department of International Business, Asia University, Taichung, Taiwan 41354, R.O.C. 3 School of Engineering and Sciences, Tecnológico de Monterrey, E. Garza Sada 2501 Sur, C.P. 64849, Monterrey, Nuevo León, México. *

Corresponding author, E-mail address: [email protected] (L. E. Cárdenas-Barrón)

Abstract Price is a major factor on the demand based on marketing and economic theory. In addition, the demand for perishable products also depends on its freshness. Moreover, it is a well-known fact that increasing stock display (e.g., fresh fruits, vegetables, baked goods) may encourage consumers to purchase more. This paper first proposes an inventory model that stipulates the demand explicitly in a multivariate function of price, freshness, and displayed stocks. It may be profitable to have a closeout sale at a markdown price, and always keep on-hand displayed stocks fresh and plentiful if the demand is freshness-and-stock dependent. Hence, the traditional assumption of zero ending inventories is relaxed to a non-zero ending inventory. As a result, the objective is to determine three decision variables (i.e., unit price, cycle time, and ending-inventory level) in order to maximize the total profit. Then it is demonstrated that the total profit is strictly pseudo-concave in those three decision variables, which reduces the search for solutions to a unique local maximum. Finally, numerical examples to illustrate the theoretical results and to highlight managerial insights are presented.

Keywords: Inventory control; Pricing; Lot-sizing; Perishable products; Expiration date. Acknowledgment The authors deeply appreciate Editor Peter Kelle and three anonymous referees for their encouragement and insightful suggestions. The principal author’s research was supported by the National Nature Science Foundation of China No. 71601162, and the Fundamental Research Funds for the Central Universities 26816WBR01. The second author’s research was funded by the National Science Council of the Republic of China under Grant NSC-105-2420-H-468-004-MY2. The third author was supported by the Tecnológico de Monterrey Research Group in Industrial Engineering and Numerical Methods 0822B01006.

1. Introduction Recently, it is noted that consumers are becoming more health conscious than before as their standard of living continues to improve. Hence, the demand for fresh goods (e.g., vegetables, fruits, baked goods, bread, milk, meat, and seafood) has dramatically increased in recent years. In today’s competitive markets, determining price and order quantity jointly for retailers to better manage perishable products is recognized as an important way to increase profitability and maintain competitiveness in a supply chain. In the literature, Levin et al. (1972) and Silver and Peterson (1985) observed and noted the phenomenon that a large pile of fresh products displayed in a supermarket often induces more sales due to its visibility, freshness, or variety. To incorporate this phenomenon into inventory models, a variety of stock-dependent demand models have been proposed. Baker and Urban (1988) established an economic order quantity (EOQ) inventory model by 2

specifying the stock-dependent demand as a power function with diminishing return of displayed stock level. Taking a simple approach, Mandal and Phaujdar (1989) proposed an EOQ inventory model representing the stock-dependent demand without diminishing return as a linear function of displayed stock level. For generalization, Datta and Pal (1990) further explored the stock-dependent demand as a power function of the on-hand inventory above a certain inventory level, below which it is assumed to be a constant. Since the higher the inventory the higher the demand, it is clear that keeping higher levels of displayed stocks may result in superior sales and profits. As a result, Urban (1992) extended the classical EOQ inventory model from zero ending inventories to non-zero ending inventory when the demand is dependent on the amount of displayed stocks. As noted, the dynamic demand is usually influenced by pricing and timing. Urban and Baker (1997) further generalized the demand rate as a multivariate function of pricing, timing, and displayed stocks level. Teng and Chang (2005) expanded the EOQ inventory model with price- and stock-dependent demand to an economic production quantity (EPQ) inventory model for deteriorating items. Thereafter, the research in this area has been examined in several directions. For example, in the study on stock-dependent demand, the time-dependent partial backlogging was analyzed, see Chang et al. (2006), and Wu et al. (2016). For stock-dependent demand under trade credit financing, see Soni and Shah (2008). To obtain optimal replenishment policies for non-instantaneous deteriorating items when the demand is both price and stock sensitive, see Chang et al. (2010), Dye (2013), Wu et al. (2014b), and Chang et al. (2015). As for supply-chain coordination and trade credit, see Teng et al. (2012), and Teng et al. (2013). In short, notice that research in this area of price- and stock-dependent demand has been

3

extensive. The traditional EOQ models such as Harris (1913) and Wilson (1934) assume that products can be stored indefinitely to meet future demand. However, many perishable products such as fruits, vegetables, medicines, and volatile liquids degrade or deteriorate continuously due to evaporation, spoilage, and obsolescence, among other reasons. Consequently, Ghare and Schrader (1963) proposed an EOQ inventory model by assuming an exponentially decaying inventory. Nahmias (1982) provided a review on the perishable inventory theory. Later, Raafat (1991) presented a survey of literature on a continuously deteriorating inventory model. Subsequently, Goyal and Giri (2001) provided a survey on the recent trends in modeling of deteriorating inventory. Recently, Bakker et al. (2012) studied a review of inventory systems with deterioration since 2001. Most researchers modeled the effect of perishability in the past only from the retailer’s perspective as the on-hand stocks shrink due to damage, spoilage, dryness, vaporization, etc. Researchers seldom developed the effect of perishability into their models from the consumer’s perspective. In fact, freshness is one of the most critical criteria affecting consumers’ purchasing decisions. Fujiwara and Perera (1993) took the effect of product freshness on consumer demand into consideration. Subsequently, Sarker et al. (1997) studied the inventory policies for perishable products when the demand is negatively impacted by the age of the on-hand stocks. Bai and Kendall (2008) then proposed an EOQ inventory model for perishable goods by extending the demand to be freshness and shelf-space dependent. Many perishable goods (e.g., raspberries, strawberries, fruit and vegetable salads, donuts, milk, etc.) have only a few days of shelf life, and their demand rates gradually decline to zero

4

as the expiration dates approach. By taking the expiration date into consideration, Wu et al. (2016a) established the retailer’s optimal replenishment cycle time and ending stock level when the demand rate is a multivariate function of the product freshness and displayed stock level. Chen et al. (2016) further expanded the inventory model by Wu et al. (2016a) to obtain the retailer’s optimal lot-size, shelf-space, and ending-stock by considering the shelf-space size as another decision variable. Other recent research publications related to maximum lifetime span are studied in Sarker (2012), Wang et al. (2014), and Wu et al. (2014a). To better manage food near its expiry date at the retail stage, Aiello et al. (2014) studied the optimal time to withdraw the food from the shelf. Muriana (2015) further extended the hypothesis of deterministic and constant shelf-life in Aiello et al. (2014) to consider products affected by uncertainties of shelf-life. In addition, deterioration costs and loss of profits are included in Muriana (2015). Concurrently, Aiello et al. (2015) discussed alternatives on food recovery for human non-profit organizations. It is well known that the consumer’s demand for perishable products is better modeled through stochastic rather than deterministic models. In this context, Muriana (2016) developed an EOQ model for perishable goods under stochastic demand conditions which allows consideration of changes in consumer behavior that can be dependent on exogenous factors such as weather, purchasing power, etc. Although those previous studies have contributed much to our understanding of the effect of the selling price and the stock level on the demand, most of them have not taken into consideration the combined effect of the selling price, the freshness index (which is determined by the expiration date), and the displayed stocks on the demand. In our view, failing to consider this combined effect may lead to a biased solution. Therefore, explicit

5

examination of the joint effect is necessary. This paper develops an EOQ inventory model for perishable products to capture the following relevant and important facts: (1) the demand for a perishable product is dependent on its price, freshness, and stock level; (2) the age of the product not only reduces on-hand stocks, but also decreases the demand rate; and (3) the perishable product is not suitable for consumption or cannot be sold after its expiration date or maximum lifetime. Notice that the selling price vitally affects not only the demand rate but also the total profit. As a result, the selling price is given as an endogenous decision variable in this paper in contrast to an exogenous constant in Wu et al. (2016b) and Chen et al. (2016). To the best of our knowledge, no studies in the existing literature have considered the demand for a perishable product as a multivariate function of its unit price, freshness condition, and stock level all together. In practice, many bakery stores have closeout sales on baked goods each night before expiration dates. Thus for generality, the traditional assumption of zero ending inventories is relaxed to non-zero ending inventory in order to enhance the potential profit. The objective of this study is to determine jointly the optimal unit price, cycle time and ending-inventory level that maximize the total profit. The remainder of this paper is structured as follows. Section 2 first defines the notations and then specifies assumptions. Section 3 formulates the mathematical model by incorporating the demand as a multivariate function of unit price, displayed stocks, product freshness, and expiration date. Then the objective is to determine selling price, replenishment cycle time and ending-inventory level so that the total profit is maximized. Section 4 demonstrates that the total profit is strictly pseudo-concave in all three decision variables, which simplifies the search to find a local optimum. Furthermore, the necessary and sufficient conditions for

6

finding the optimal solution are established. Thereafter, Section 5 provides some numerical examples to highlight the theoretical results and managerial insights. Finally, Section 6 concludes the paper with a brief discussion on future research directions.

2. Notations and assumptions For convenience, the following notations and assumptions are adopted throughout the entire paper. 2.1. Notations For simplicity, the symbols are defined into three groups: parameters, decision variables, and functions. Parameters: c

Purchasing cost per unit.

h

Holding or carrying cost per unit per unit of time.

m

Maximum lifetime (i.e., the time to its expiration date) in units of time.

o

Ordering cost per order.

s

Salvage price per unit.

w

Size of shelf space in units.

t1

Time at which the inventory level reaches w.

Decision variables: E

Ending-stock (or inventory) level in units, with E  0 .

P

Price per unit.

T

Replenishment cycle time in units of time. 7

Functions: a(t)

Age-dependent freshness at time t, which is a decreasing function within [0, 1].

D(t,P)

Demand rate is affected by price, freshness, and inventory level directly. It is noted from Equations (2) and (3) below that both freshness index and inventory level are functions of time. For simplicity, the demand rate is denoted by D(t,P) or D.

I(t)

Inventory level at time t.

TP( E, P, T ) Total profit, which is a multivariate function of E, P, and T.

The next sub section proposes some necessary assumptions in order to build up the mathematical model. 2.2. Assumptions According to traditional marketing and economic theory, price is a vital factor on the demand of a product. It is evident that the higher the price, the lower the demand. Following Robinson and Lakhani (1975), Thompson and Teng (1984), Teng and Chang (2005), and Chang et al. (2006), we assume that the demand rate D is proportional to an exponential function of the price P. Thus, D(t,P) is proportional to e  P ,

(1)

where  is the maximum number of potential consumers with  ,   0 . Product freshness is another critical factor on consumers’ purchasing decision. The consumer’s freshness index of a perishable product may be affected by time, temperature, humidity, refrigeration, etc. The famous Arrhenius’s formulae can be applied to establish the 8

shelf life for products at specified temperature conditions. Giuseppe et al. (2010) provided an example of the application of the Arrhenius model to perishable products. In fact, most perishable products (e.g., vegetables, milk, meat, and seafood) have their own expiration dates. It is evident that health-conscious consumers prefer a perishable product with a longer remaining shelf life to a shorter one. As a result, it is assumed that the consumer’s freshness index is age-dependent. That is, it begins with 1 at time 0, and then gradually degrades closer to 0 as the expiration date of the product approaches. Therefore, following Wu et al. (2016a), we assume that the age-dependent freshness at time t is linearly decreasing from 1 at the beginning to 0 at the maximum lifetime m as follows: a(t ) 

mt , 0  t  m. m

(2) (Insert Figure 1 here)

Levin et al. (1972) observed that “large piles of consumer goods displayed in a supermarket will lead customers to buy more.” A variety of inventory models have thus been proposed to quantify this phenomenon and to explore optimal inventory policies. Baker and Urban (1988) proposed specifying the demand pattern as a power function of displayed stock level. Hence, D(t,P) is proportional to [ I (t )] ,

(3)

where 0    1 because the effect of the stock level on the demand rate is positive but gives diminishing return. Therefore, the demand for fresh products is also a function of displayed on-hand stocks. As a result, the demand is a multivariate function of the selling price, product freshness, displayed stocks, and expiration date. At the beginning (i.e., at time 0), the retailer receives Q units, displays w units on the shelf,

9

and stores the remainder of the product (i.e., Q  w units) in the backroom. Whenever sales are made, the retailer moves stock in the backroom to fill up shelf space until no more stock is in the backroom at time t1 . Please see Figure 1 for illustration. Consequently, during the time interval [0, t1 ], the shelf space is full (i.e. a constant w), and the demand only depends on both the price and the freshness index. Following Kalish (1983), Thompson and Teng (1984), and Urban and Baker (1997), we assume that the demand is a deterministic, multiplicative form of price, freshness, and stock level. Combining these Equations (1) – (3), we obtain the demand rate as follows: D(t , P)  w

m  t  P e ; 0  t  t1  T , m

(4)

where  ,  , and  are fixed parameters with   0 , 0    1 , and   0 . In addition, it is clear from (4) that the replenishment cycle time T must be less than or equal to the expiration date of the product m (i.e., T  m ). Otherwise, D(t , P)  0 if t  m . In contrast to zero ending inventories in the classical EOQ inventory model, it may be more profitable to keep displayed stock fresh and plentiful (i.e., non-zero ending inventory) if the demand rate depends on the product freshness and displayed stock level. Thus, for generality, the classical assumption of zero ending inventories is extended to a non-negative ending inventory (i.e., the ending inventory level E  0 ). Also, if the replenishment occurs on a daily basis, then non-zero ending inventory policy is more relevant than the impact of freshness on the demand. Note that the backroom is empty at time t1 . After time t1 to T, the shelf space is partly stocked and the demand depends on the selling price, freshness index, and displayed units. At the replenishment cycle time T (i.e., the ending inventory level reaches E units), the retailer

10

sells all E units at a salvage price s per unit, obtains a new order quantity Q units, and then operates a new replenishment cycle. Consequently, the demand rate at time t during the time interval period [ t1 , T] is governed as follows: D(t , P)   [ I (t )]

m  t  P e ; t1  t  T , m

(5)

where   0 and 1    0 . To fully utilize the shelf space, it is assumed that the order quantity Q is larger than or equal to the shelf-space size w (i.e., Q  w ). Otherwise, the retailer just reduces the size of the shelf space to Q units to save the shelf-space cost. Since Q  w , it is obvious that t1  0 . In addition, it is also assumed that shortages and quantity discounts are not allowed. Finally, replenishment rate is instantaneous and complete. Using the above notations and assumptions, now the retailer’s total profit is modeled as a multivariate function of the ending inventory E, the selling price P, and the cycle time T for perishable goods with expiration date.

3. Mathematical model From the above assumptions, the inventory level I (t ) at time t during the time interval [0, t1 ] is governed by the following differential equation:

dI (t ) m  t  P  w e ; 0  t  t1 , dt m

(6)

with the boundary conditions: I (0)  Q and I (t1 )  w . By solving the differential equation in (6), then the inventory level is  t2   , 0  t  t1 . I (t )  Q  w  e P  t  2 m  

(7)

11

Substituting t1 into (7), using the boundary condition I (t1 )  w , the fact that t1  T  m , and re-arranging terms, thus the time at which the inventory level is w (or the backroom is empty) is represented as

t1  m  m2 

2m(Q  w)  0. w e P

(8)

Solving (8), hence the order quantity is given as follows:  t2  Q  I (0)  w  w  e P  t1  1   w. 2m  

(9)

Similarly, the inventory level I (t ) at time t during the time interval [t1 , T ] is described by the following differential equation: dI (t ) m  t P (10)   [ I (t )] e ; t1  t  T , dt m with the boundary conditions: I (t1 )  w and I (T )  E . Solving the differential equation in

(10), yields 1

 (1   )e P 2 1  I (t )   t  2m(T  t )  T 2  E1   , t1  t  T . 2m  





(11)

For detailed derivation, see Appendix A. Substituting t1 into (11), using the facts I (t1 )  w and t1  T  m , and simplifying terms, thus the time at which the inventory level reaches to w units (or the backroom is running out of inventory) is t1  m  (m  T )2 

2m( w1   E1  )  0.  (1   )e P

(12)

Substituting (12) into (9), and simplifying terms, thus the order quantity is Q  w

w e P  2m

2m( w1   E1  )  2 2 m  ( m  T )   w.   (1   )e P  

(13)

Hence, the holding cost during the time interval [0, t1 ] is given by:      t2   t3 t2  H1  h  I (t )dt  h  w e P   t   Q dt  h w e P  1  1   Qt1  .  2m   6m 2  0 0     t1

t1

12

(14)

Likewise, the holding cost during the time interval [t1 , T ] is as follows: 1

 (1   )e P 2 1  H 2  h  I (t )dt  h   t  2m(T  t )  T 2  E1   dt . 2m  t1 t1  T



T



(15)

The integration of (15) seems too complicated and tedious to derive an explicit analytical solution. For simplicity, a simple approximation is used to calculate it as follows. The average inventory level during the time interval [t1 , T ] approximately equals (w+E)/2. Therefore, the holding cost during the time interval [t1 , T ] approximately equals h (16) ( w  E )(T  t1 ) . 2 Notice that the error between H 2 in (15) and (16) is less than 1% of the total profit based on H2 

our numerical results. If the demand depends on the stock level (i.e., a higher stock level results in a higher demand rate as well as purchasing cost), then the retailer’s objective must maximize the total profit. Otherwise, if the objective is to minimize the total cost, then both the stock level and the demand rate decrease, thus resulting in sales and profits decreasing too. The total profit = the revenue received + the salvage value – the purchasing cost – the ordering cost – the holding cost.

(17)

Therefore, the problem here is to determine the optimal ending inventory level E, unit price P, and replenishment cycle time T simultaneously in order to maximize the total profit. Therefore, the EOQ inventory model for perishable products with price-freshness-and-stock dependent demand is:

Max TP( E , P, T ) 

  P  t13 t12   1  P ( Q  E )  sE  cQ  o  h    Qt1   w e  T   6m 2   

h   ( w  E )(T  t1 ) , 2  13

(18)

subject to Q  w

w e  P  2m

2m( w1  E1 )  2 2 m  ( m  T )   w,   (1   )e P  

t1  m  (m  T ) 2 

2m( w1  E1 )  0,  (1   )e P

0  E  w , and 0  t1  T  m . Notice that the proposed model in (16) is a generalized form of many previous studies such as Baker and Urban (1988), Urban (1992), Urban and Baker (1997), and Teng and Chang (2005). Then this paper analyzes the retailer’s optimal solution and derives the optimal ending stock, selling price, and cycle time in the next section.

4. Theoretical results and optimal solution According to Cambini and Martein (2009, p. 245) the real-value function

q( x) 

y ( x) z ( x)

(19)

is (strictly) pseudo-concave, if y(x) is non-negative, differentiable and (strictly) concave, and z(x) is positive, differentiable and convex. For simplicity, to solve the problem in (18), the following is defined J 

w  e P m

  P h  w E  w E 2  w e ( P  c  ht1 )   m  T   .  m  t1  2  2(m  t1 ) 2   m

(20)

It is assumed without loss of generality (WLOG) that J  0 because P  c  ht1  0 in general and J has only one relatively small positive term (i.e., h(w  E) /[ 2(m  t1 )] ). Given fixed E and P, using the result in (19) it can be proved that the retailer’s total profit TP( E, P, T ) in (18) is strictly pseudo-concave in T if J  0 . Consequently, there exists a

unique optimal solution T * such that TP( E, P, T ) is maximized.

14

Theorem 1.

Given the ending-inventory level E and the selling price P,

if J  0 then TP( E, P, T ) in (18) is a strictly pseudo-concave function in T, and there exists a unique maximum solution T * . Proof. See Appendix B. Note that the proof of Theorem 1 is similar to that of Wu et al. (2016a). Given the ending-inventory level E and the unit price P, taking the first-order derivative of TP( E, P, T ) in (18) with respect to T, setting the result to zero, and simplifying terms, it is obtained from Theorem 1 that the necessary and sufficient condition for the optimal replenishment cycle time T * is as follows: w  e  P  h m T (m  T )( P  c  ht1 )   ( w  E )  w  E  T  2  m  t1   m

  h  t3 t2   P(Q  E )  sE  cQ  o  h w  e P  1  1   Qt1   ( w  E )(T  t1 ).  6m 2    2

(21)

For detailed derivation, see Appendix C. By using (12) and (13), the following results are easily obtained. Corollary 1.

Given the ending-inventory level E and the unit price P, both t1 and Q are

increasing and concave downward in T. Proof. It immediately follows (B3), (B4), (B6), and (B7). Next, it is proved that TP( E, P, T ) in (18) is a strictly concave function in both E and P for any given T. Similarly, for simplicity, let

K

 w  1  meP ( w1   E 1  ) hmeP ( w  E )  meP ( w1   E 1  )  w      1  h  1  , 2   (1   )(m  t1 ) E 2 (m  t1 ) E    (1   )(m  t1 ) 2  E

L  ( P  c  ht1 )

w  E1



hmeP  (m  t1 ) E 

 w meP ( w  E )  (w  E)  (  1 )   ,  E 2  2 (m  t1 ) E 2 E  

and 15

(22)

(23)

3  w ( w1   E1  )  t12  2   P  t1  M  [2  2 ( P  c  ht1 )]Q  w   h  w e    6m 2  1     



hm2eP ( w1   E1  )   ( w  E )  meP ( w1   E1  )   Q  w     (1   )(m  t ) 2  1  .  (1   )(m  t1 )  2 1  

(24)

In general, the unit profit P  c  ht1  0 . So, L in (23) has only one positive term (i.e., the last term) which is related to the holding cost after time t1 . Hence, we may assume WLOG that L  0 . Likewise, [2   ( P  c  ht1 )]  0 in general. By using an analogous argument for L, the only positive term in (24) is the last term related to the holding cost after t1 . Thus we may assume WLOG that M  0 . As a result, we have the following result. Theorem 2.

Given the replenishment cycle time T,

if L  0 , M  0 , and LM  K 2  0 , then TP( E, P, T ) in (18) is a strictly concave function in both E and P, and hence there exists a unique maximum solution E * and P * . Proof.

See Appendix D.

Given the replenishment cycle time T, taking the first-order partial derivative of TP( E, P, T ) in (18) with respect to E and P, setting the result to zero, and simplifying terms,

it is obtained from Theorem 2 that the necessary and sufficient condition for the optimal ending-inventory level is

( P  c  ht1 )

w h meP ( w  E )   s  P  T  t  1    0, 2 E  (m  t1 ) E  

(25)

and the necessary and sufficient condition for the optimal price per unit is 3  t12  w  ( w1   E 1  )   P  t1  Q  E   ( P  c  t1 ) Q  w    h  w e    6m 2  1     



hmeP ( w1   E1  ) ( w  E )  0. 2 (1   )(m  t1 )

16

(26)

Please see Appendix E for detailed derivation. Likewise, the following results can be easily obtained from (12) and (13). Corollary 2.

Given the replenishment cycle time T,

(i) t1 is increasing in E, while decreasing in P, and (ii) Q is increasing and concave in E, but is decreasing and convex in P. Proof. From (D1), (D2), (D7), (D8), (D10), and (D11), it immediately follows. Based on previous results, the following solution algorithm is proposed in order to find the optimal de3cision variables:

An algorithm for finding optimal solution Step 1. Input the values of parameters. Set i = 1, and choose a trial value of T, say Ti  m / 2 . Step 2. Substitute Ti into Equations (25) and (26) to get Ei and Pi . Step 3. Substitute Ei and Pi into Equation (21) to obtain Ti 1 . Step 4. If  0.001  Ti 1  Ti  0.001 , then set the optimal solution E * = Ei , P * = Pi , and

T * = Ti , and then stop. Otherwise, set i = i + 1, and go to Step 2.

5. Numerical examples This section illustrates the theoretical results and managerial insights obtained by using two numerical examples run with MATHEMATICA 10.2. The details of sensitivity analysis on the optimal solution with respect to each parameter is given. Example 1. Assume that the maximum number of potential consumers   6000 , the shelf-space efficiency of demand   0.6 , the price efficiency of demand   0.1 , c = $20/unit, h = $5/unit/year, m = 0.04 years, o = $10/order, s = $10/unit, and w = 25 units. By using software MATHEMATICA 10.2, the local optimal solution to maximize TP( E, P, T )

17

in (18) is obtained as follows:

E *  6.5156 units, P*  $28.1920 , Q *  39.6586 units, T *  0.0232 years, t1  0.0065 years, and TP* = $8,366.46. We calculate

 2TP( E, P, T ) / T 2  (2.17904) 107  0 at the optimal

E *  6.5156

units, P*  $28.1920 , and T *  0.0232 years. Thus, given E *  6.5156 units and

P*  $28.1920 , TP( E, P, T ) in (18) is a strictly quasi-concave function in T, which implies it is also a strictly pseudo-concave function in T. Furthermore, Figure 2 reveals that TP( E, P, T ) in (18) is a strictly pseudo-concave function in T as shown in Theorem 1.

Similarly, given T *  0.0232 years, we obtain K  1.2486 , L  1.6874 , M  4.8244 , and LM  K 2  6.5817 at the optimal solution. Hence, TP( E, P, T ) in (18) is a strictly quasi-concave function in both E and P. In addition, Figure 3 demonstrates TP( E, P, T ) in (18) is a strictly quasi-concave function in both E and P as proven in Theorem 2. (Insert Figures 2 and 3 here) Example 2. Using the same data as those in Example 1, the sensitivity analysis of the optimal solution with respect to each parameter is obtained. The computational results of the sensitivity analysis are shown in Table 1. (Insert Table 1 here) In the first case of the sensitivity analysis, the value of  is permitted to vary while keeping the other parameters constant. As shown in Table 1, the ending inventory, the selling price, the order quantity, and the total profit all gradually increase as the value of  increases (which implies higher demand and slightly shorter replenishment cycle time). Similarly, in the second case, the ending inventory, the selling price, the order quantity, and the total profit increase if the stock efficiency of demand  increases (which causes a higher demand and hence a shorter cycle time). Conversely, in the third case, the ending 18

inventory, the selling price, the order quantity, and the total profit decrease if the price efficiency of demand  increases (which makes a lower demand while a longer cycle time). The numerical results in the fourth case reveal that a higher purchasing cost c causes the ending inventory, the order quantity, and the total profit to lower but the selling price and the cycle time increase. Likewise, the fifth case reveals that an increase in the inventory holding cost h has similar effects as an increase in the purchasing cost c but on a much smaller scale. In the next case, the ending inventory, the selling price, the order quantity, the cycle time, and the total profit all increase as the maximum lifetime m increases. In the seventh case, the results indicate that the ending inventory, the selling price, the order quantity, and the cycle time increase but the total profit decreases as the ordering cost o increases. In the sequential case, the ending inventory, the selling price, the order quantity, and the total profit all increase meanwhile the cycle time decreases when the salvage price s increases. In the final case, an increase in shelf space w causes an increase in the ending inventory, the order quantity, the cycle time, and the total profit while causing a decrease in the selling price. In short, the numerical results in Table 1 have demonstrated that the total profit is very sensitive to the stock efficiency of demand  , the price efficiency of demand  , the purchasing cost c, the maximum lifetime m, and the salvage price s. Hence, the retailer must pay more attention to those parameters than the others. An increase in the maximum lifetime elevates all decision variables and the total profit. Consequently, the retailer may invest in preservation technology such as refrigeration to prolong the product lifespan and increase the total profit. The sensitivity analysis on the salvage price indicates that if an earlier closeout sale increases the markdown (or salvage) price, then the retailer should try it to increase the total profit as well as the order quantity.

6. Conclusions

19

This paper has developed an EOQ inventory model to capture the following relevant and important facts: (1) An increase in shelf space for an item induces consumers to buy more, (2) Consumers’ purchasing decisions are vitally influenced by product freshness, (3) Pricing strategy is an important competitive tool to increase sales and profits, and (4) The demand is diminished when a product is approaching its expiration date. For generality, the traditional assumption of zero ending inventories is relaxed to non-zero ending inventory in order to boost the total profit. This paper has demonstrated that the total profit is strictly pseudo-concave in all three decision variables (i.e., unit price, replenishment time, and ending-stock level), which reduces the search for the global solution to a local optimum. Finally, some numerical examples to illustrate the proposed model and its managerial approaches are provided. An applied mathematical model is always a simplification of the complicated real-life problem. To strengthen the applicability, this paper can be extended in several forms. For instance, advertising strategy, quantity discount, and time-varying demand may be added into the model. Furthermore, there is usually a significant cost for the additional effort of restocking the shelf space. This extra cost should be taken into consideration in future study. Finally, the single player local optimal solution could be expanded to an integrated cooperative solution for multiple players in a supply chain.

Appendix A. Solution to differential equation (10) Re-arranging (10), yields t   [ I (t )]  dI (t )  e P   1  dt. m 

(A1)

Performing integration on both sides of (A1), yields

20

 1 t2    C. [ I (t )]1   e P   t  1  2m  

(A2)

Substituting I (T )  E into (A2), and re-arranging terms, thus C

 1 T2  . E1   e P  T  1  2 m  

(A3)

Then substituting (A3) into (A2), and re-arrange terms, the inventory level is 1

 (1   )e P 2 1  I (t )   t  2m(T  t )  T 2  E1   . 2m  





(A4)

Appendix B. Proof of Theorem 1 Given E and P, taking the first- and second-order derivatives of (12) and (13) with respect to T, the following results are obtained:

 dt1 2m( w1  E1 )   (m  T ) (m  T ) 2  dT  (1   )e P  

0.5

d 2t1 2m( w1  E1 )  2 2  ( m  T ) ( m  T )   dT 2  (1   )e P  

 0, 1.5

dQ w e  P  (m  T )  0, dT m

 2m( w1  E1 )   (m  T ) 2   (1   )e P  

(B1) 0.5

.

(B2)

(B3)

and

d 2Q w e P    0. dT 2 m

(B4)

It is clear from (12) that

(m  t1 ) 2  (m  T ) 2 

2m( w1  E1 ) .  (1   )e P

(B5)

Substituting (B5) into (B1) and (B2), yields

dt1 m  T   0, dT m  t1

(B6)

21

and

d 2t1 (m  T ) 2 1    0, 2 3 dT (m  t1 ) m  t1

(B7)

respectively. From (18), set

    h  t3 t2  y(T )  P(Q  E )  sE  cQ  o  h w e P  1  1   Qt1   ( w  E )(T  t1 ).    6m 2    2

(B8)

and z (T )  T  0 .

(B9)

Consequently,

q(T ) 

y (T )  TP( E , P, T ). z (T )

(B10)

Given E and P, and taking the first-order derivative of y(T), thus   dt  dy (T ) dQ dQ    P  t12  y ' (T )   ( P  c)  hw e   t1   Q 1  t1  dT dT dT    2m   dT  

h dt  ( w  E )(1  1 ). 2 dT

(B11)

From (7), the following is obtained:  t2  I (t1 )  w e P  1  t1   Q  w,  2m 

(B12)

Substituting (B12) into (B11), and re-arranging terms, y' (T )  ( P  c  ht1 )

dQ h dt h  ( w  E ) 1  ( w  E ). dT 2 dT 2

(B13)

Taking the derivative of (B13) with respect to T,

y" (T ) 

d 2 y(T ) d 2Q dt1 dQ h d 2t1  ( P  c  ht )  h  ( w  E ) . 1 dT 2 dT 2 dT dT 2 dT 2

(B14)

Substituting (B3) - (B4), (B6), and (B7) into (B14), and simplifying terms, the following 22

result is obtained: y" (T )  

w  e  P m

( P  c  ht1 ) 

  P h  w E  w E 2  w e   m  T     J . (B15)   m  t1  2  2(m  t1 ) 2   m

As a result, if J  0 then y" (T )  0 and hence y(T) is non-negative, differentiable and strictly concave. Thus, if J  0 then TP( E, P, T ) as in (18) is a strictly pseudo-concave function in T, and there exists a unique optimal solution. Appendix C. The optimal replenishment cycle time T * It is obvious from (B8) - (B10) that TP( E, P,T )  y(T ) / T . Hence, given E and P, taking the first-order derivative of TP( E, P, T ) with respect to T, and setting the result to zero, thus the necessary and sufficient condition to find T * is given as follows: d y' (T ) y(T ) TP(T , E )   2  0  y' (T )T  y(T )  0 . dT T T

(C1)

From (B3), (B6), (B8), (B13), and (C1), if J  0 then the necessary and sufficient condition for T * is w  e  P  h m T (m  T )( p  c  ht1 )   ( w  E )  w  E  T  2  m  t1   m

  P  t13 t12   h  P(Q  E )  sE  cQ  o  h w e     Qt1   ( w  E )(T  t1 ).  6m 2    2

(C2)

Appendix D. Proof of Theorem 2 For any given T, taking the first- and second-order partial derivatives of (10) with respect to E and P, and applying (B5), the following results are obtained:

t1 meP   0, E  (m  t1 ) E 

(D1)

t1  m( w1  E1 )eP   0, P  (1   )(m  t1 )

(D2)

23

 2t1 meP  EP  (m  t1 ) E 

 meP ( w1   E 1  )  1  ,  (1   )(m  t1 ) 2  

(D3)

 2t1 (meP ) 2 meP   , E 2  2 E 2  (m  t1 ) 3 E  1 (m  t1 )

(D4)

and

 2t1 2 meP ( w1   E 1  )  meP ( w1   E 1  )    1 .   (1   )(m  t1 )   (1   )(m  t1 ) 2 P 2 

(D5)

From (8) and (12) one gets: 2m( w1   E 1  ) 2m(Q  w) m  (m  T )   .  (1   )e P w  e P 2

2

(D6)

For any given T, taking the first- and second-order partial derivatives of (13) with respect to E and P, and applying (D6), one obtains: 

Q  w      0, E  E 

(D7)

Q w  ( w1   E 1  )   (Q  w)   0, P 1 

(D8)

 2Q  0, PE

(D9)

 2 Q  w   1   0, E 2 E

(D10)

and  2Q 2 w  ( w1   E 1  ) 2   (Q  w)   0. 1  P 2

(D11)

From (18), let

  h  t3 t2  z ( E , P)  P(Q  E )  sE  cQ  o  h w  e P  1  1   Qt1   ( w  E )(T  t1 ).  6m 2    2 Consequently, for any given T, thus the total profit is

24

(D12)

TP( E , P, T ) 

1 z ( E, P). T

(D13)

Taking the first- and second-order partial derivatives of z ( E, P) with respect to E and P, applying (B12), (D1) - (D5), (D7) - (D11), and simplifying terms, one gets:

t  z ( E, P) Q h  ( P  c  ht1 )  s  P  T  t1  ( w  E ) 1 , E E 2 E 

(D14)

 t3 t2  h t z ( E, P) Q  Q  E  ( P  c  ht1 )  hw  e P  1  1   ( w  E ) 1 , P P P  6m 2  2

(D15)

 2t1  2 z ( E , P)  2Q Q  Q 1  t1 h  ( P  c  ht1 )   1  h    (w  E ) PE PE E PE  E 2  P 2 

 w  1  meP ( w1   E 1  ) w      1  h 2   (1   )(m  t1 ) E E 

hmeP ( w  E )  meP ( w1   E 1  )  1    K, 2 (m  t1 ) E    (1   )(m  t1 ) 2 

(D16)

t1  Q  h  2t1  2 z ( E , P)  2Q  ( P  c  ht )  h  1  ( w  E )   1 E  E  2 E 2 E 2 E 2  ( P  c  ht1 )

w  E 1 

hmeP   (m  t1 ) E 

 w meP ( w  E )  (w  E )     1    L, (D17) 2  2 E E 2  ( m  t ) E 1  

and 3 t12  t1 h  2 t1  2 z ( E , P) Q  2Q 2  P  t1    2  ( P  c  ht )   h  w e    h ( Q  w )  ( w  E ) 1  6m 2  P P 2 P 2 P 2 P 2  

3  t12  w  ( w1   E 1  )  2  P  t1   [2  2 ( P  c  ht1 )]Q  w    h  w e    6m 2  1      

hmeP ( w1   E 1  )   ( w  E )  meP ( w1   E 1  )     1   M . Q  w   2  (1   )(m  t1 )  2   (1   )(m  t1 ) 

(D18)

If L  0 , M  0 , and LM  K 2  0 , then the Hessian matrix associated with z ( E, P) is negative definite. Consequently, for any given T, if L  0 , M  0 , and LM  K 2  0 , then 25

TP( E, P, T ) in (18) is a strictly concave function in E and P. Hence, there exists a unique

optimal solution. Appendix E. Optimal ending-inventory level and shelf-space size For any given T, substituting (D1) and (D7) into (D14), and setting the result to zero, the necessary and sufficient condition for E * is given as follows:

w h meP ( w  E )  ( P  c  ht1 )   s  P  T  t1    0. 2 E  (m  t1 ) E  

(E1)

Similarly, substituting (D2) and (D8) into (D15), and setting the result to zero, thus the necessary and sufficient condition for P * is 3  t12  w  ( w1   E 1  )   P  t1  Q  E   ( P  c  ht1 ) Q  w    h  w e    6m 2  1     



hmeP ( w1   E1  ) ( w  E )  0. 2 (1   )(m  t1 )

(E2)

References Aiello G., Enea M., Muriana C. Alternatives to the traditional waste management: food recovery for human non-profit organizations. International Journal of Operations and Quantitative Management 2015; 21(3), 215-239. Aiello G., Enea M., Muriana C. Economic benefits from food recovery at the retail stage: An application to Italian food chains. Waste Management 2014; 34(7), 1306- 1316. Aiello G., La Scalia G., Micale R., Muriana C. Experimental analysis of shelf life based inventory management policies for RIFD enabled supply chains. Proceedings of the Flexible Automation and Intelligent Manufacturing 20th international conference 2010; 1150-1155. July 12-14, California, USA. ISBN: 9681617827143.

26

Bai R., Kendall G. A model for fresh produce shelf-space allocation and inventory management with freshness-condition-dependent demand. Informs Journal on Computing 2008; 20(1), 78-85. Baker R.C., Urban TL. A deterministic inventory system with an inventory level dependent demand rate. Journal of the Operational Research Society 1988; 39(9), 823-831. Bakker M., Riezebos J., Teunter R. Review of inventory systems with deterioration since 2001. European Journal of Operational Research 2012; 221(2), 275-284. Cambini A., Martein L. Generalized convexity and optimization: Theory and application 2009. Springer-Verlag Berlin Heidelberg, USA. Chang C.T., Cheng M.C., Ouyang L.Y. Optimal pricing and ordering policies for non-instantaneously deteriorating items under order-size-dependent delay in payments. Applied Mathematical Modelling 2015; 39(2), 747-763. Chang C.T., Goyal S.K., Teng J.T. On “An EOQ model for perishable items under stock-dependent selling rate and time-dependent partial backlogging” by Dye and Ouyang. European Journal of Operational Research 2006; 174(2), 923-929. Chang C.T., Teng J.T., Goyal S.K. Optimal replenishment policies for non-instantaneous deteriorating items with stock-dependent demand. International Journal of Production Economics 2010; 123(1), 62-68. Chang H.J., Teng J.T., Ouyang L.Y., Dye C.Y. Retailer’s optimal pricing and lot-sizing policies for deteriorating items with partial backlogging. European Journal of Operational Research 2006; 168(1), 51-64. Chen, S.C., Min, J., Teng, J. T., and Li, F., Inventory and shelf-space management for fresh

27

produce with freshness-and-stock dependent demand and expiration date. Journal of the Operational Research Society 2016; 67(6), 884-896. Datta T.K., Pal A.K. A note on an inventory model with inventory level dependent demand rate. Journal of the Operational Research Society 1990; 41(10), 971-975. Dye C.Y. The effect of preservation technology investment on a non-instantaneous deteriorating inventory model. Omega 2013; 41(5), 872-880. Fujiwara O, Perera U.L.J.S.R. EOQ models for continuously deteriorating products using linear and exponential penalty costs. European Journal of Operational Research 1993; 70(1), 104-114. Ghare P.M., Schrader G.P. A model for an exponentially decaying inventory. Journal of Industrial Engineering 1963; 14(5), 238-243. Goyal S.K., Giri B.C. Recent trends in modeling of deteriorating inventory. European Journal of Operational Research 2001; 134(1), 1-16. Harris, F. W. How many parts to make at once. Factory, The Magazine of Management 1913; 10(2), 135-136. Kalish S. Monopolist Pricing with Dynamic Demand and Production Cost. Marketing Science 1983; 2(2), 135-159. Levin R.I., McLaughlin C.P., Lamone R.P., Kottas F.J. Productions / Operations Management: Contemporary Policy for Managing Operating Systems 1972, McGraw-Hill, New York, p. 373. Mandal B.N., Phaujdar S. An inventory model for deteriorating items and stock-dependent consumption rate. Journal of the Operational Research Society 1989; 40(5), 483-488.

28

Muriana C. Effectiveness of the food recovery at the retailing stage under shelf life uncertainty: an application to Italian food chains. Waste Management 2015; 41, 159-168. Muriana C. An EOQ model for perishable products with fixed shelf life under stochastic demand conditions. European Journal of Operational Research 2016; 255(2), 388–396. Nahmias S. Perishable inventory theory: a review. Operations Research 1982; 30(4), 680-708. Raafat F. Survey of literature on continuously deteriorating inventory model. Journal of the Operational Research Society 1991; 42(1), 27-37. Robinson B., Lakhani C. Dynamic price models for new-product planning. Management Science 1975; 21(6), 1113-1122. Sarkar B., 2012. An EOQ model with delay in payments and stock dependent demand in the presence of imperfect production. Applied Mathematical and Computation 2012; 218(17), 8295-8308. Sarker B.R., Mukherjee S., Balan C.V. An order-level lot size inventory model with inventory-level dependent demand and deterioration. International Journal of Production Economics 1997; 48(3), 227-236. Silver E.A., Peterson R. Decision Systems for Inventory Management and Production Planning, 2nd edition, 1985, Wiley, New York. Soni H.N., Shah N.H. Optimal ordering policy for stock-dependent demand under progressive payment scheme. European Journal of Operational Research 2008; 184(1), 91-100. Teng J.T., Chang C.T. Economic production quantity models for deteriorating items with

29

price- and stock-dependent demand. Computers and Operations Research 2005; 32(2), 297-308. Teng J.T., Min J., Pan Q. Economic order quantity model with trade credit financing for non-decreasing demand. Omega 2012; 40(3), 328-335. Teng J.T., Yang H.L., Chern M.S. An inventory model for increasing demand in a supply chain with an upstream trade credit linked to order quantity. Applied Mathematical Modelling; 2013; 37(14), 7624-7632. Thompson G.L., Teng J.T. Optimal pricing and advertising policies for new product oligopoly models. Marketing Science 1984; 3(2), 148-168. Urban T.L. An inventory model with an inventory-level-dependent demand rate and relaxed terminal conditions. Journal of the Operational Research Society 1992; 43(2), 721-724. Urban T.L., Baker R.C. Optimal ordering and pricing policies in a single-period environment with multivariate demand and markdowns. European Journal of Operational Research 1997; 103(3), 573-583. Wang W.C., Teng J.T., Lou K.R. Seller’s optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime. European Journal of Operational Research 2014; 232(2), 315-321. Wilson R.H. A Scientific Routine for Stock Control. Harvard Business Review 1934; 13, 116–128. Wu J., Chang C.T., Cheng M.C., Teng J.T., Al-Khateeb F.B. Inventory management for fresh produce when the time-varying demand depends on product freshness, stock level and

30

expiration date. International Journal of Systems Science: Operations & Logistics 2016a; 3(3), 138-147. Wu J., Ouyang L.Y., Cárdenas-Barrón L.E., Goyal S.K. Optimal credit period and lot size for deteriorating items with expiration dates under two-level trade credit financing. European Journal of Operational Research 2014a; 237(3), 898-908. Wu J., Skouri K., Teng J.T., Hu Y. Two inventory systems with trapezoidal-type demand rate and time-dependent deterioration and backlogging. Expert Systems with Applications 2016b; 46, 367-379. Wu J., Skouri K., Teng J.T., Ouyang L.Y. A note on “Optimal replenishment policies for non-instantaneous deteriorating items with price and stock sensitive demand under permissible delay in payment”. International Journal of Production Economics 2014b; 155, 324-329.

Figure 1. Graphical representation of the system

Quantity

31

Q Inventory level Displayed stock level w

E Time 0

t1

m

T

Figure 2. TP( E, P, T ) with respect to T for given E = 6.5156 and P = 28.19

32

Figure 3. TP( E, P, T ) with respect to E and P given T = 0.0232

Table 1. Sensitivity analysis with respect to each parameter Parameters   5000   6000   7000   0.50   0.60   0.70   0.09   0.10   0.11 c = $15.00 c =$20.00 c =$25.00 h = $4.00 h = $5.00 h = $6.00 m = 0.03 m = 0.04 m = 0.05

E*

P*

Q*

T*

TP*

6.3092 6.5156 6.6616 4.5507 6.5156 8.2537 7.4696 6.5156 5.6038 12.1321 6.5156 3.1874 6.5338 6.5156 6.4974 6.1928 6.5156 6.6986

27.9153 28.1920 28.3891 27.5550 28.1920 28.6709 29.5365 28.1920 27.0023 24.3929 28.1920 31.2179 28.1904 28.1920 28.1936 27.7287 28.1920 28.4711

35.1292 39.6586 43.7655 29.4590 39.6586 51.2392 45.2377 39.6586 34.1995 53.4550 39.6586 24.7289 39.7191 39.6586 39.5984 32.7181 39.6586 45.6094

0.0253 0.0232 0.0215 0.0255 0.0232 0.0207 0.0205 0.0232 0.0261 0.0151 0.0232 0.0314 0.0232 0.0232 0.0232 0.0200 0.0232 0.0260

6033.86 8366.46 10790.40 5125.27 8366.46 13414.40 13336.80 8366.46 5051.78 20919.60 8366.46 2353.32 8386.79 8366.46 8346.17 6577.28 8366.46 9601.24

33

o = $5.00 o = $10.00 o = $15.00 s = $5.00 s = $10.00 s = $15.00 w = 20 w = 25 w = 30

6.5011 6.5156 6.5290 3.9449 6.5156 11.9123 5.2875 6.5156 7.7196

28.1697 28.1920 28.2128 27.4941 28.1920 29.1432 28.3199 28.1920 28.0801

39.2535 39.6586 40.0446 37.8834 39.6586 41.8648 33.7735 39.6586 45.2203

0.0227 0.0232 0.0236 0.0250 0.0232 0.0193 0.0224 0.0232 0.0239

8584.16 8366.46 8152.81 7312.31 8366.46 10449.70 7681.95 8366.46 8926.97

Highlights

Demand rate for perishable products depends on price, freshness and displayed stocks. Product freshness starts with 1 and ends at 0 as its expiration date is approaching. The profit is strictly pseudo-concave in price, cycle time and ending stock level. Optimal selling price, cycle time, and non-zero ending inventory level are derived.

34