An economic order quantity model with pa

Computers & Industrial Engineering

journalhomepage:www.elsevier.com/locate/caie

An economic order quantity model with partial backordering and incremental discount

a b c Ata Allah Taleizadeh , , Irena Stojkovska , David W. Pentico ⇑

b School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, Iran Department of Mathematics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, Skopje, Macedonia c Palumbo-Donahue School of Business, Duquesne University, Pittsburgh, PA 15282, USA

article info

abstract

Article history: Determining an order quantity when quantity discounts are available is a major interest of material Received 14 August 2014

managers. A supplier offering quantity discounts is a common strategy to entice the buyers to purchase Received in revised form 2 December 2014

more. In this paper, EOQ models with incremental discounts and either full or partial backordering are Accepted 5 January 2015

developed for the first time. Numerical examples illustrate the proposed models and solution methods. Available online 14 January 2015

Ó 2015 Elsevier Ltd. All rights reserved. Keywords:

EOQ Incremental discounts Full backordering Partial backordering

1. Introduction and literature review all-units discount, purchasing a larger quantity results in a lower unit purchasing price for the entire lot, while incremental dis-

Since Harris (1913) first published the basic EOQ model, many counts only apply the lower unit price to units purchased above variations and extensions have been developed. In this paper we

a specific quantity. So the all-units discount results in the same combine two of those extensions: partial backordering and

unit price for every item in the given lot, while the incremental dis- incremental quantity discounts.

count can result in multiple unit prices for an item within the same Montgomery, Bazaraa, and Keswani (1973) were the first to

lot ( Tersine, 1994 ). In the following we focus on the research using develop a model and solution procedure for the basic EOQ with

only an incremental discount or both incremental and all-units dis- partial backordering (EOQ–PBO) at a constant rate. Others taking

counts together. Since Benton and Park (1996) prepared an exten- somewhat different approaches have appeared since then,

sive survey of the quantity discount literature until 1993, we will including Pentico and Drake (2009) , which will be one of the two

describe newer research, along with a short history of incremental bases for our work here. In addition, many authors have developed

discounts and older research which is more related to this paper. models for the basis EOQ-PBO combined with other situational

The EOQ model with incremental discounts was first discussed characteristics, such as Wee (1993) and Abad (2000) , both of which

by Hadley and Whitin (1963) . Tersine and Toelle (1985) presented included a finite production rate and product deterioration,

an algorithm and a numerical example for the incremental dis- Sharma and Sadiwala (1997) , which included a finite production

count and examined the methods for determining an optimal order rate with yield losses and transportation and inspection costs,

quantity under several types of discount schedules. Güder, Zydiak, San José, Sicilia, and García-Laguna (2005) , which included models

and Chaudhry (1994) proposed a heuristic algorithm to determine with a non-constant backordering rate, and Taleizadeh, Wee, and

the order quantities for a multi-product problem with resource Sadjadi (2010) , which included production and repair of a number

limitations, given incremental discounts. Weng (1995) developed of items on a single machine. Descriptions of all of these models

different models to determine both all-units and incremental dis- and others may be found in Pentico and Drake (2011) .

count policies and investigated the effects of those policies with Enticing buyers to purchase more by offering either all-units or

increasing demand. Chung, Hum, and Kirca (1996) proposed two incremental quantity discounts is a common strategy. With the

coordinated replenishment dynamic lot-sizing problems with both incremental and all-units discounts strategies. Lin and Kroll (1997) extended a newsboy problem with both all-units and incremental

⇑ Corresponding author. discounts to maximize the expected profit subject to a constraint E-mail addresses: taleizadeh@ut.ac.ir (A.A. Taleizadeh), irenatra@pmf.ukim.mk , irena.stojkovska@gmail.com (I. Stojkovska), pentico@duq.edu (D.W. Pentico).

that the probability of achieving a target profit level is no less than http://dx.doi.org/10.1016/j.cie.2015.01.005

0360-8352/Ó 2015 Elsevier Ltd. All rights reserved.

22 A.A. Taleizadeh et al. / Computers & Industrial Engineering 82 (2015) 21–32

a predefined risk level. Hu and Munson (2002) investigated a

Parameters

dynamic demand lot-sizing problem when product price schedules

A Fixed cost to place and receive an order offer incremental discounts. Hu, Munson, and Silver (2004) contin-

b The fraction of shortages that will be backordered ued their previous work and modified the Silver-Meal heuristic

The purchasing unit cost at the jth break point algorithm for dynamic lot sizing under incremental discounts.

D Demand quantity of product per period Rubin and Benton (2003) considered the purchasing decisions

g The goodwill loss for a unit of lost sales facing a buying firm which receives incrementally discounted price

Holding cost rate per unit time

schedules for a group of items in the presence of budgets and space

Number of price breaks

limitations. Rieksts, Ventura, Herer, and Sun (2007) proposed a

Lower bound for the order quantity for price j

serial inventory system with a constant demand rate and incre-

Selling price of an item

mental quantity discounts. They showed that an optimal solution

Backorder cost per unit per period is nested and follows a zero-inventory ordering policy. Haksever

p 0 The lost sale cost per unit at the jth break point of unit

and Moussourakis (2008) proposed a model and solution method

j þg>0 constraint inventory systems from suppliers who offer incremental

purchasing cost, p 0

to determine the ordering quantities for multi-product multi-

Decision variables

quantity discounts. Mendoza and Ventura (2008) incorporated

B The back ordered quantity

quantity discounts, both incremental and all-units, on the pur-

F The fraction of demand that will be filled from stock chased units into an EOQ model with transportation costs.

The order quantity

Taleizadeh, Niaki, and Hosseini (2009) developed a constrained

The length of an inventory cycle multi-product bi-objective single-period problem with incremen-

tal discounts and fully lost-sale shortages. Ebrahim, Razm, and

Dependent variables

Haleh (2009) proposed a mathematical model for supplier

ATC Annual total cost

selection and order lot sizing under a multiple-price discount

ATP Annual total profit

environment in which different types of discounts including

CTC

Cyclic total cost

all-unit, incremental, and total business volume are considered.

CTP

Cyclic total profit

Taleizadeh, Niaki, Aryanezhad, and Fallah-Tafti (2010) developed

a multi-products multi-constraints inventory control problem with stochastic period length in which incremental discounts and par-

2.1. EOQ models with no discount

tial backordering situations are assumed. Munson and Hu (2010) proposed procedures to determine the optimal order quantities

In this section we briefly discus EOQ models with fully or par- and total purchasing and inventory costs when products have

tially backordered shortages when discounts are not available. either all-units or incremental quantity discount price schedules.

For the first case, the EOQ models with fully backordered shortages Bai and Xu (2011) considered a multi-supplier economic lot-sizing

(see Fig. 1 ), Pentico and Drake (2009) derived the optimal values of problem in which the retailer replenishes his inventory from

F and T as:

several suppliers who may offer either incremental or all-units quantity discounts. Chen and Ho (2011) developed an analysis

ð1Þ method for the single-period (newsboy) inventory problem with

p r þ iC ffiffiffiffiffiffiffiffi r ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

fuzzy demands and incremental discount. Taleizadeh, Barzinpour,

2A p þ iC

ð2Þ fuzzy demand, incremental discounts, and lost-sale shortages.

¼ iCD

and Wee (2011) discussed a constrained newsboy problem with

For the second case, the EOQ model with partial backordering, Taleizadeh, Niaki, and Nikousokhan (2011) developed a multi-

Pentico and Drake (2009) showed that the values of F and T that constraint joint-replenishment EOQ model with uncertain unit

minimize annual total cost are

cost and incremental discounts when shortages are not permitted. Bera, Bhunia, and Maiti (2013) developed a two-storage inventory

Þ p 0 þb p T

model for deteriorating items with variable demand and partial

ð iC þ b p ÞT

backordering. Lee, Kang, Lai, and Hong (2013) developed an

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

2A iC þ b p ½ ð

integrated model for lot sizing and supplier selection and quantity

ð4Þ discounts including both all units and incremental discounts.

iCD

b iC p

Archetti, Bertazzi, and Speranza (2014) studied the economic only if b is at least as large as a critical value b 0 given by Eq. (5) lot-sizing problem with a modified all-unit discount transportation

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

cost function and with incremental discount costs.

2AiCD

D p 0 ð5Þ

According to the above mentioned research, it is clear that no

researchers have developed an EOQ model with partial backorder- ing and incremental discounts. Taleizadeh and Pentico (2014) developed an EOQ model with partial backordering and all-units

discounts. In this paper we develop EOQ models with fully and par- tially backordered shortages when the supplier offers incremental discounts to the buyer.

DFT

2. Model development

( 1F − ) T

t In this section we model the defined problem under two differ-

FT

D ( 1F − ) T

ent conditions: full backordering and partial backordering. But first

we briefly discuss the EOQ model with full or partial backordering when discounts are not assumed. We use the following notation.

Fig. 1. EOQ model with fully backordered shortages.

A.A. Taleizadeh et al. / Computers & Industrial Engineering 82 (2015) 21–32

2.2. EOQ model with incremental discount without shortages where C 0 j is the purchasing cost per unit given by Formula (10) . Substituting Formula (10) into (13) and dividing by T we get the Consider an EOQ model in which the supplier offers the

annual total cost for ordering the quantity from the interval volume-based unit purchasing costs shown in Eq. (6) (Q ¼ DT).

½q j ;q jþ1 Þ:

8 > C 1 > q 1 ¼06Q<q > 2

AþX j iX j F 2 iC j DF 2 T p

2 þC ð14Þ j ¼

< > C 2 q 2 6 Q<q 3 ATC j ðT; FÞ ¼ T

> > > .. .. Thus, the cost function that has to be minimized has the form : C n q n 6 Q

> > ATC 1 ðT; FÞ ; 0 < DT < q where C 2 1 >C 2 > n and q

< > > ATC 2 ðT; FÞ ; q 2 6 cost per order is: DT < q 3

1 ¼0<q 2 <

n . The purchasing

ð15Þ M j ¼X j þC j DT; j ¼ 1; 2; . . . ; n;

ATCðT; FÞ ¼ >

: ATC n

ðT; FÞ ; q

n 6 DT

where

The minimization is performed over the region T > 0; 0 6 F 6 1 (see

X j ¼ q k ðC

k Þ; j ¼ 2; 3; . . . ; n and X 1 ¼ 0;

Fig. 1 ).

k¼2

From the definitions of C j ;q j and X (14) and (15) , is

j , we have that:

Proposition 1. The function ATCðT; FÞ, defined by continuous.

X j P 0; j ¼ 1; 2; . . . ; n:

Then the purchasing cost per unit is ( Tersine, 1994 )

Proof. See Appendix B .h

As a consequence of Proposition 1 , the minimization problem DT ¼ DT þ

can be transformed into

and the optimal cycle length for ordering from the quantity from the jth interval ½q j ;q jþ1 Þ is

T>0;06F61 min ATCðT; FÞ ¼ min 16j6n ðT;FÞ2X min j ATC j ðT; FÞ ð16Þ s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T

2ðA þ X j Þ :

iC j D ð11Þ

where

X 1 ¼ fðT; FÞj 0 < T 6 q 2 =D; 0 6 F 6 1 g;

The optimal order quantity is Q j ¼ DT j , with minimal annual total

X j ¼ fðT; FÞj q j =D 6 T 6 q jþ1 =D; 0 6 F 6 1 cost of

X n ¼ fðT; FÞj q n =D 6 T; 0 6 F 6 1 g:

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi iX j ATC j

2DiC j ðA þ X j ð16aÞ

C j D:

Note that the sign < is changed into 6 in the upper bounds,

which is allowed by the continuity of ATCðT; FÞ. In what follows Q j <q j , then the optimal acceptable order quantity is Q j ¼q j . If

The optimal order quantity Q j is acceptable if q j 6 Q j <q jþ1 . If

we will use the notation T j and F j for T and F, respectively, when Q j P q jþ1 , then the optimal acceptable order quantity is

we are minimizing the annual total cost for ordering the quantity Q j ¼q jþ1 . For the latter two cases, the corresponding annual total

from the interval ½q j ;q jþ1 Þ defined by Eq. (14) . To solve the jth subproblem in (16) , i.e. the problem

cost, calculated using Eq. (A1) in Appendix A , is the new optimal annual total cost ATC j . Finally, ATC j for j ¼ 1; 2; . . . ; n are compared

ð17Þ to find the minimal value among ATC j ;j ¼ 1; 2; . . . ; n, which will be

ðT;FÞ2X min ATC j j ;F j ðT j Þ;

the optimal annual cost for the EOQ model with incremental dis- we first find the first partial derivatives of ATC j ðT j ;F j Þ with respect count, and the corresponding Q j will be the optimal order quantity

to T j and F j .

dure is justified, because we can prove that if Q Þ

for the EOQ model with incremental discount. This solution proce-

@ATC j

AþX j iC j DF 2 j p

2 ð18Þ is an order quantity which costs less to order than Q j does (see

j P q jþ1 , then there

@T j ðT

j ;F j

Appendix A ).

@ATC j

j ÞT j : ð19Þ In the following sub-sections we model the defined problem

@F j ðT j ;F j Þ ¼ iX j F j þ iC j DF j T j p

under two different conditions: full backordering and partial Setting the first derivatives (18) and (19) equal to 0, and solving backordering, which are developed in Sections

the corresponding system with respect to T j and F j , remembering respectively.

2.3 and 2.4

that T j > 0, we get

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ðA þ X j Þ

j 2 Þ ð20Þ We will consider an EOQ model in which all shortages will be

2.3. EOQ model with full backordering and incremental discounts

¼T j ðF j Þ¼

D½iC j F 2 j þ p

ð21Þ unit purchasing cost discounts. Then, according to Fig. 1 , the cyclic

j j þ D iC j j p

j Þ j ¼ 0:

iX F F T

backordered and the supplier offers incremental volume-based

To find the solution of the system (20) and (21) , we substitute total cost for ordering the quantity from the interval ½q j ;q jþ1 Þ is

(20) in (21) , and obtain an equation with respect to F j :

ð22Þ z}|{

Holding Cost

Backordering Cost

iX j F j

þ D iC

ðF Þ ¼ 0;

2 T 2 which can be solved numerically with a solver like MatLab, ðT; FÞ ¼

Fixed Cost Purchasing Cost

zfflfflfflfflfflffl}|fflfflfflfflfflffl{ 0 2 2 zfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflffl{

zffl}|ffl{ 0 iC j DF T

CTC j

A C DT

Mathematica, or Excel Solver. Let us denote the solution of (22) Mathematica, or Excel Solver. Let us denote the solution of (22)

F j . If we denote the

left

side of

j Þ T j ðF j Þ, then from wð1Þ ¼ iX j þ

DiC j T j

p T j ð0Þ < 0, and wðF j Þ being continuous,

we have that there exists a solution F j of (22) in the interval [0, 1]. So, we can formulate the following proposition.

Proposition 2. There exists a solution F j of Eq. (22) , for which 06F j 6 1. Then, from Eq. (20) we have:

T j ¼T j ðF j Þ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðA þ X

D½iC j F j þ p

ð23Þ

If ðT j ;F j Þ2 X j , then it is the optimal solution of the Subproblem (17) . The following proposition stands. Proving the global optimal-

ity of ðT j ;F j Þ can be also done as in Stojkovska (2013) .

Proposition 3. Assume that ðT j ;F j Þ2 X j , where F j is the solution of (22) ,T j is defined by (23) , and X j is the feasible region of

Subproblem (17) . Then ðT j ;F j Þ is the global optimal solution of Subproblem (17) .

Proof. See Appendix C .h Note that wðF j Þ is a monotone nondecreasing function since

@w ðF j Þ=@F j ¼@ 2 / ðF j Þ=@F 2 j > 0 (see (C6) in Appendix C ). Thus the solution F j of Eq. (22) is the unique solution in the interval [0, 1]. If ðT j ;F j Þ2 X j , then ðT j ;F j Þ is the unique global minimizer of the

function ATC j ðT j ;F j Þ on the set X j .

From Proposition 2 we have that 0 6 F j 6 1, but for T j it might not be always true that q j =D 6 T j 6 q jþ1 =D. If T j <q j =D, then the

global solution ðT j ;F j Þ of Subproblem (17) lies on the lower bound- ary of T j , i.e. T j ¼q j =D, and F j ¼ p q j = ððiC j þ p Þq j þ iX j Þ. If

T j >q jþ1 =D, then the global solution ðT j ;F j Þ of Subproblem (17) lies on

the upper boundary

F j ¼ p q jþ1 = ððiC j þ p Þq jþ1 þ iX j Þ. This is true because of the convex- ity of ATC j ðT j ;F j Þ with respect to T j (see (C1) in Appendix C ), and minimizing ATC j ðq j =D; F j Þ and ATC j ðq jþ1 =D; F j Þ, respectively, in order to obtain the last two values for F j .

From the above discussion we can conclude that the global optimal solution ðT ;F Þ that minimizes the annual total cost given in Eq. (15) is the pair ðT j ;F j Þ for which the corresponding

ATC j ðT j ;F j Þ is minimal over all j = 1,2, . . ., n. That is, ðT ;F Þ ¼ arg min 16j6n ATC j ðT j ;F j Þ n o :

ð24Þ

We have the following solution procedure for EOQ model with incremental discount and full backordering. Solution procedure for the EOQ model with incremental discounts and full backordering

1. For j = 1, 2, . . . ,n:

1.1. Solve (22) using some numerical procedure, to obtain F j . Calculate T j from (23) .

1.2. If q j =D 6 T j 6 q jþ1 =D (with q 1 ¼ 0 and q nþ1 ¼ 1), then

ðT j ;F j Þ is an acceptable solution (or Q j ¼ DT j is an accept- able order quantity).

1.3. If T j <q j =D and j

2 f2; . . . ; ng, then calculate the new

ðT j ;F j Þ using T j ¼q j =D and F j ¼ p q j = ððiC j þ p Þq j þ iX j Þ

1.4. If T j >q jþ1 =D and j ðT j ;F j Þ using T j ¼q jþ1 =D and F j ¼ p q jþ1 = ððiC j þ p Þq jþ1 þ iX j Þ

1.5. Calculate ATC j ðT j ;F j Þ.

2. Find the optimal solution as the pair ðT j ;F j Þ for which the corresponding ATC j ðT j ;F j Þ is minimal over all j = 1, 2, . . . , n.

3. Calculate Q ¼ DT and B

ÞT .

2.4. The EOQ with incremental discounts and partial backordering Unlike the full backordering model in which we minimized the

annual total cost to obtain the optimal solutions, in the partial backordering model, in order to facilitate reaching the optimal solution using the approach in Pentico and Drake (2009) , we will first model the profit function and then by maximizing it we will get the optimal solutions. According to Fig. 2 , in which it is clear

chasing cost becomes

þC j

ð25Þ where X j is given by Eq. (8) . Then the cyclic total profit for ordering

the quantity from the interval ½q j ;q jþ1 Þ is

z}|{ Fixed Cost z}|{ þC 0 zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ j Purchasing Cost þ iC 0 j DF 2 T 2 zfflfflfflfflfflffl}|fflfflfflfflfflffl{ 2

Holding Cost

|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} 2

Backordering Cost

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Lost Sale Cost

ð26Þ Substituting (25) in (26) and dividing by T gives the average

annual profit for ordering the quantity from the interval ½q j ;q jþ1 Þ:

ATP j

AþX j T þC j

iX j F 2

þ iC j DF 2 2 T þ p b 2 2 T

; ð27Þ

After some algebraic transformations and letting p 0 j

j þ g, we

have:

ATP j

AþX j

iX j F 2

iC j DF 2 T

) ð28Þ

The function to be maximized over the region T > 0; 0 6 F 6 1, has the form

FT

1F ) − T

βD

( ) D 1F β T − ( )() 1 D 1F β − β −

DFT

Fig. 2. EOQ model with partially backordered shortages.

24 A.A. Taleizadeh et al. / Computers & Industrial Engineering 82 (2015) 21–32

A.A. Taleizadeh et al. / Computers & Industrial Engineering 82 (2015) 21–32

8 ATP > >

which can be solved numerically with a solver like MatLab, > >

< ATP 2 ðT; FÞ ; q 2 6 3 Mathematica, or Excel Solver. Let us denote the solution of (37) ATPðT; FÞ ¼

by F . If we denote the left side of Eq. (37)

by nðF j Þ¼

iX j F

> .. F j

: > > iX j F j 2 ATP

n ðT; FÞ ; q n 6 2F j

j Þ 2 þ D iC j F j p b j Þ j ðF j p 0 j

then we

j þ g > 0, Appendix D ).

The function ATPðT; FÞ defined by (28) and (29) , is continuous (see

q ffiffiffiffiffiffiffiffiffiffiffiffi 2ðAþX Þ Thus, the maximization problem can be written as

and nð1Þ ¼ iX j

iX

2 þ DiC j T j

p j 0 j þ DiC j

DiC j

iX j

T>0;06F61 max ATPðT; FÞ ¼ max 16j6n max ATP j

ðT; FÞ

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð38Þ with

ðT;FÞ2 ~ X j

iX j

þ 2ðA þ X j ÞDiC j

b >1

iX j

0 D ¼b j

X ~ 1 2 ; T > 0; 0 6 F 6 1 g;

is satisfied. Then, because of the continuity of the function nðF j Þ, we can formulate the following proposition.

X ~ j ¼ fðT; FÞj q j 6 jþ1 ; T > 0; 0 6 F 6 1 g;

Proposition 4. If Condition (38) is satisfied, then there exists a and X ~ n

n 6 solution F j of Eq. (37) ¼ fðT; FÞj q , for which 0 6 F j 6 1. For F j given by the solution of Eq. (37) ð30aÞ , we define T j by

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð39Þ the order quantity, which is allowed by the continuity of ATPðT; FÞ

Note that the sign < is changed into 6 in the upper bounds for

2ðA þ X j

j ¼T j ðF j Þ¼

D½iC j F j þ p b j Þ

(see Appendix D ). Since maximizing ATP j ðT; FÞ is equivalent to minimizing the

Thus, if Condition (38) is satisfied, ðT j ;F j Þ is the solution of the system (35) and (36) for which 0 6 F j 6 1. Note that, when b function 0 j < 0, it is clear that b > b 0 (since b P 0) and Condition

is satisfied. AþX

iX j F 2

iC j DF 2 T

Also note that if b ¼ b j 0 , where b ðT; FÞ ¼ 0 þ j is defined by (38) , then T þ ð

2 ð1Þ ¼ 0, and F j ¼ 1 is the solution of Eq. j ¼ 1, from

(37) . For F

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ

p j 0 ð31Þ

we have that T j ¼

2ðA þ X j Þ=ðDiC j Þ which is the optimal

2 cycle length for the cost of C j in the EOQ model with incremental Problem (30) is transformed into

discount without shortages (see Section 2.2 , Eq. (11) ). (

We can also prove that if Condition (38) is satisfied and T>0;06F61 max ATPðT; FÞ ¼ max 16j6n

ðT;FÞ2 ~ X j j ðT; FÞ

ðT j ;F j Þ2~ X j , then it is the global minimizer of the function

u j ðT j ;F j Þ over the domain ~ X j . The following proposition stands.

As in Section 2.3 , we will use T j and F j for T and F respectively Proving the global optimality of ðT j ;F j Þ can be also done as in

when we are minimizing the function u

j ðT; FÞ defined by Eq.

Stojkovska (2013) .

In order to minimize the function u j ðT j ;F j Þ, we first take the first partial derivatives:

Proposition 5. Assume that Condition (38) is satisfied and ðT j ;F j Þ2~ X j , where F j is the solution of (37) ,T j is defined by (39) ,

and ~ X j is the feasible region defined by (30 a). Then ðT j ;F j Þ is the @T j ðT T 2 þ

j ;F j AþX

iC j DF j 2 p b j 2

2 j ð33Þ

global minimizer of the function u ðT

j j ;F j Þ over the domain ~ X j .

2iX j F j F j

@F ðT j ;F j

Þ¼ Proof. See Appendix E j .h 2 2F j

h þ iC i

j F j T j p b j ÞT j p 0 j

D:

As in the full backordering case, note that nðF j Þ is a monotone non-decreasing function since @nðF j Þ=@F j

¼@ 2 Þ=@F j > 0 (see Setting the first derivatives (33) and (34) equal to 0, and solving

2 g ðF j

(E6) in Appendix E ). Thus, if Condition (38) is satisfied, the solution the corresponding system with respect to T j and F j , remembering

F j of Eq. (37) is the unique solution in the interval [0, 1], and if that T j > 0, we have:

ðT j ;F j Þ2~ X j , then ðT j ;F j Þ is the unique global minimizer of the s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðA þ X

function u j ðT j ;F j Þ on the set ~ X j . If Condition (38) is not satisfied, T j ¼T j ðF j Þ¼

D½iC 0

j F 2 j ð35Þ þ p b j 2 Þ then 0 6 b < b j , which is equivalent to nð1Þ < 0, and from nðF j Þ

being a monotonic function, we have that there is no solution of iX j F j

iX j F 2 j

F 2 þ D iC j F j

Eq. (37) in the interval [0, 1]; consequently, partial backordering

jj

2F j

cannot be optimal. So, in this case (0 6 b < b j 0 ), the optimal decision

p j 0 ð36Þ

is either meeting all demand (EOQ model with incremental discount and no shortages, Section 2.2 ) with the optimal value of

Substituting (35) into (36) , we obtain an equation with respect to

the cycle length T

2ðA þ X Þ=ðDiC Þ the fill rate F j ¼ 1, or losing all sales with T j ¼ þ1 and F j ¼ 0.

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

and the optimal value of

iX j F j iX j F 2 j From Proposition 4 and the above discussion about the values

2 þ D iC j F j p b j Þ j ðF j Þ

2F j

for T j and F j , we always have T j Þ > 0 and 0 6 F j 6 1 when Condi- tion (38) is met, but it might not be always true that

p j 0 ð37Þ

q j 6 DT j ðF j

j ÞÞ 6 q jþ1 , in which case the pair ðT j ;F j Þ would

26 A.A. Taleizadeh et al. / Computers & Industrial Engineering 82 (2015) 21–32

be infeasible and cannot be the minimizer of the function u j ðT j ;F j Þ

then the minimizer lies on the upper boundary T j ¼q j1 =D, and on the set ~ X j . We have the following proposition.

j Þ. Note that, in the second case, we can exclude the point ðT j ;F j Þ from the set of

Proposition 6. Assume that Condition (38) is satisfied, but candidates for the optimal solution, since the corresponding ðT j ;F j ÞR~ X j , where F j is the solution of (37) ,T j is defined by (39) ,

order quantity is not the overall optimal order quantity (see and ~ X j is the feasible region defined by (30a) . Then the minimizer

Appendix A ).

ðT j ;F j Þ of the function u j ðT j ;F j Þ over the domain ~ X j is defined by:

When Condition (38) is met (b P b 0 j ) and ðT j ;F j ÞR~ X j , but neither (40a) or (40b) nor (41a) or (41b) is satisfied, this means

(i) If DT j ðF j j ÞÞ < q j and one of the following conditions that partial backordering cannot be optimal, so the optimal are met

decision is meeting all demand from the EOQ model with incremental discount and no shortages (see Section 2.2

) or losing

2D p j q j

0 AþX j

0 AþX j

q P 0 and 1þb >

all sales. In this case we should search for the optimal decision

ð40aÞ

iX j

þ iC 0 case.

as in the b < b j and b P

We can conclude that the global optimal solution ðT ;F Þ that or

maximizes the annual total profit, Function (29) , is as one of the points ðT j ;F j Þ for which the corresponding profit is maximal over

2D AþX j q

0 AþX j

1þb

q < 0 and

ð40bÞ

all j = 1, 2, . . . , n.

The following solution procedure for the EOQ model with incre- mental discounts and partial backordering summarizes the details

then F j is the solution of of the preceding theoretical results and their implications for the !

ðA þ X optimal solution. j Þ j q

0 iX

q F j þ iC j F j p b j Þ DT j ðF j

Solution procedure for the EOQ model with incremental discounts

and partial backordering

iX j

j þ iC

1. For j ¼ 1; 2; . . . ; n:

D 2 1.1. Calculate b j according to Formula (38) 2 . j ðF j Þ¼0

ð40cÞ

1.2. If b P b j 0 P

0 or b j 0 < 0, solve Eq. (37) to obtain F j and

2q j

calculate T j according to Formula (39) .

where T q j ðF j Þ¼

, and T j ¼

1.2.1. If q 6 DT

j ðF j

j ÞÞ 6 q (with q 1 ¼ 0 and

(ii) If DT j ðF j j ÞÞ > q jþ1 and one of the following condi-

q nþ1 ¼ 1), then ðT j ;F j Þ is an acceptable solution tions are met

(or Q j ¼ DT j ðF j

j ÞÞ is acceptable). Calculate

2D p j 0 AþX j the profit ATP j ðT j ;F j Þ, using Formula (28) 0 . Compare AþX j 1þb q p

j Þ with the profit from not stock- q jþ1

P 0 and

jþ1

the profit ATP ð41aÞ j j ;F

ðT

D, and take the higher profit. If the profit or

iX j þ iC j q jþ1

from not stocking is higher, set T j ¼ þ1 and F j ¼ 0.

0 1.2.2. If DT j ðF j

j ÞÞ < q j

and j 2 f2; . . . ; ng, then

j 0 AþX < 0 and 1þb >

AþX j

2D q

jþ1

ð41bÞ

(1.2.2.i) If one of the Conditions (40a) or (40b) is satisfied,

find F j as the solution of Eq. (40c) , and set then F j is the solution of

jþ1

b p q jþ1

j ÞÞÞ. Calculate the profit !

T j ¼q j = ðDðF j

ðA þ X ATP Þ j ðT j ;F j Þ using Formula (28) .

iX j

F j þ iC j F j p b j Þ DT j j

ðF Þ

(1.2.2.ii) Set F j ¼ 1, and calculate T j ¼ 2ðA þ X j Þ=ðDiC j Þ . If

T j <q j =D, set T j ¼q j =D, and if T j >q =D, set

iX j

jþ1

j þ iC j F 2 j þ p b j 2 Þ

T j ¼q jþ1 =D. Calculate the profit PD j Þ, where

jþ1

ðT Þ is given by Formula

j ðF j Þ¼0

2 ð41cÞ

j 0 D, and set

2q

jþ1

T j ¼ þ1 and F j ¼ 0.

where T

(1.2.2.iv) Compare the profits from (1.2.2i), (1.2.2.ii), (1.2.2.iii)

to determine the optimal (highest) profit if Proof. See Appendix F .h

DT j ðF j

j ÞÞ < q j , and set T j ¼T j and F j ¼F j

for the optimal solution.

When b < b j 0 and b j 0 P

0, as we saw earlier, the optimal decision

j ÞÞ > q jþ1 (1.2.3.i) If one of the Conditions (41a) or (41b) is satisfied,

1.2.3. If DT

j ðF j

is meeting all demand from the EOQ model with incremental dis-

count and no shortages, i.e., the minimizer of u j ðT j ;F j Þ lies on the

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi find F j as the solution of Eq. (41c) , and set

boundary F j ¼ 1, so T j ¼

j ÞÞÞ. Calculate the profit convexity of

2ðA þ X j Þ=ðDiC j Þ and F j ¼ 1. From the

T j ¼q jþ1 = ðDðF j

ATP j ðT j ;F j Þ, using Formula (28) . AþX j iX j iC j DT j

h ðT j

j Þ ¼ ATC . If ðT j ;1 Þ¼ j ¼ 1, and calculate T j ¼ 2ðA þ X j Þ=ðDiC j

þ 2þ 2 þC

(1.2.3.ii) Set F

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

T j <q j =D, set T j ¼q j =D, and if T j >q jþ1 =D, set (see (C1) in Appendix C for F j ¼ 1), if DT j ðF j

T j ¼q jþ1 =D. Calculate the profit PD j Þ, where DT j

j ÞÞ ¼

h ðT j Þ is given by Formula (42) . boundary T

j <q j , then the minimizer lies on the lower

j ¼q j =D, and the corresponding optimal profit is

D, and set

j Þ. If DT j ðF j

j ÞÞ ¼ DT j

j >q

jþ1 ,

j ¼ þ1 and F j ¼ 0.

27 (1.2.3.iv) Compare the profits from (1.2.3i), (1.2.3.ii), (1.2.3.iii)

A.A. Taleizadeh et al. / Computers & Industrial Engineering 82 (2015) 21–32

Table 1

to determine the optimal (highest) profit if Results for EOQ model with full backordering and incremental discounts ( Example 1 ). DT j ðF j

F j ¼F j for the optimal solution.

1 0.562731 > q 2 /D = 0.375

0.555294 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðA þ X j Þ=ðDiC j Þ .

1.3. If 06b<b j ,

2 1.10621 > q 3 /D = 0.75

Correction (2)

1.3.1. If q

j =D 6 T j 6 q

jþ1 =D (with q 1 ¼ 0 and q nþ1 ¼ 1),

then ðT j ;F j Þ is acceptable (or Q j ¼ DT j is accept- able). Calculate the profit PD - hðT j

jþ1 Þ, and if the correction is possible, i.e., if Conditions þ

j =2

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

interval ½q

j ;q

(40a) or (40b) or Conditions (41a) or (41b) is satisfied. Then, ‘‘profit Formula (42) .

2ðA þ X ÞDiC

Þ is given by

(j)’’ is calculated for those corrected values for T j and F j . If correc-

1.3.2. If T j <q j =D and j

2 f2; . . . ; ng, then set T j ¼q j =D and

tions of the PBO model are done, then the row indicated with ‘‘NBO

by Formula (42) . Compare the profit with the profit j ¼ DT j from NBO model is not in the

j Þ, where hðT j Þ is given

model (j)’’ is filled, and if Q

p 0 D, and take the higher profit,

interval ½q j ;q jþ1 Þ, then the ‘‘NBO correction (j)’’ is done, and ‘‘profit

(j)’’ is calculated for those corrected values for T j and F j . For each j, If the profit from not stocking is higher, set the profit from not stocking is calculated and is displayed in the T j ¼ þ1 and F j ¼ 0. row ‘‘not stocking (j)’’. The highest profit is taken as the over-all

1.3.3. If T j >q jþ1 =D and j

j ¼q jþ1 =D

profit.

given by Formula (42) . Compare the profit with the , when b ¼ 0:95, the annual profit is

j Þ, where hðT j Þ is

According to Table 2

maximized for j = 3, under the partial backordering policy, with

T ¼T 3 ¼ 1:85987 and F ¼F 3 ¼ 0:636287, with the optimal profit profit. If the profit from not stocking is higher, set

p j 0 D, and take the higher

ATPðT ;F Þ ¼ ATP 3 ðT 3 ;F 3 Þ ¼ 686:411. The optimal order quantity is T j ¼ þ1 and F j ¼ 0.

Q ¼ DT ðF

ÞÞ ¼ 365:21 and the maximum backordered

quantity is B

ÞT ¼ 128:528.

2. Identify the maximum profit; the point ðT j ;F j Þ at which it is attained is the global optimal solution ðT ;F Þ.

3. If the optimal policy is partial backordering, calculate For b ¼ 0:80, the annual profit is maximized for j = 3, under the Q ¼ DT ðF

partial backordering policy, with T ¼T 3 ¼ 1:74378 and F ¼F 3 ¼ policy is meeting all demand with incremental discount, calcu-

ÞÞ and B

ÞT . If the optimal

0:805105, with the optimal profit ATPðT ;F Þ ¼ ATP 3 ðT 3 ;F 3 Þ¼ late Q ¼ DT . If the optimal policy is losing all sales, then

630:197. The optimal order quantity is Q ¼ DT ðF ÞÞ ¼ Q ¼ 0.

backordered quantity is

B ÞT ¼ 54:3765.

3. Numerical examples For b ¼ 0:50, the annual profit is maximized for j = 3, under the policy of meeting all demand, with T ¼T 3 ¼ 1:45774; F ¼F 3 ¼ 1, We give numerical examples for both the full and partial back-

with the optimal profit ATPðT ;F Þ ¼ 616:393. The optimal order ordering models with incremental discounts proposed in the above

quantity is Q ¼ DT ¼ 291:548.

sections. The solution procedures are coded in Wolfram Mathem- All examples showed that if the jth optimal quantity is not in atica, using built-in functions to solve nonlinear equations.

the jth interval ½q j ;q jþ1 Þ, then it cannot be the overall optimal quan- tity, even if it is corrected to the relevant interval endpoint. This

Example 1 (EOQ model with incremental discounts and full backor- was proved for the EOQ model with incremental discount and no dering). We will use the values of all common parameters from the

backordering (see Appendix A ). It is left to be proven that this numerical example for Taleizadeh and Pentico’s (2014) all-units

might be also true for the proposed EOQ models with incremental discount model: P = $9/unit, D = 200 units/period, i = 0.3/period,

discount – full and partial backordering respectively. From the

p = $2/unit/period,

C ¼ ðC 1 ;C 2 ;C 3 Þ = $(6, 5, 4)/unit,

g = $2/unit,

examples we can see that keeping all parameters fixed and by

varying the backordering rate, the total profit decreases when We set the fixed order cost A to $30/order. Values for T j ;F j and

q ¼ ðq 1 ;q 2 ;q 3 Þ = (0, 75, 150) units, p 0 ¼ð p 0 1 ; p 0 2 ; p 0

3 Þ = $(5, 6, 7)/unit.

the backordering rate is decreasing.

ATC j ðT j ;F j Þ for each j ¼ 1; 2; 3, are displayed in Table 1 . Rows that are noted as ‘‘correction (j)’’, display the values of T j and F j after correcting T j

4. Sensitivity analysis

for not being in the interval q =D 6 T

q jþ1 =D. Then,

ATC j ðT j ;F j Þ is calculated for those corrected values for T j and F j . There are at least two possible objectives for sensitivity According to Table 1 , the annual total cost is minimized for j = 3,

analysis:

so the overall optimal solution is T ¼T 3 ¼ 1:83941; F ¼

1. Assess the relative impact of mis-estimation of different model 1089:06. The optimal order quantity is Q ¼ DT ¼ 367:881, with

F 3 ¼ 0:591107, with the optimal cost ATCðT ;F Þ ¼ ATC 3 ðT 3 ;F 3 Þ¼

parameters on the model’s performance. the maximum backordered quantity B

ÞT ¼ 150:424.

2. Assess the relative importance of the different model parame- ters in determining the values of the decision variables and the performance function.

Example 2 (EOQ model with incremental discounts and partial back- ordering). We use the same values for the parameters as in Exam-

4.1. Study plan

ple 1 , and we will

vary the backordering parameter

b ¼ 0:95; 0:80; 0:50. The results are displayed in Table 2 . Rows that Both objectives can be addressed by changing a single parame- are noted as ‘‘PBO correction (j)’’, display the values of T j and F j

ter’s value by given percentages, repeating the analysis for each after correcting Q j ¼ DT j ðF j

j ÞÞ for not being into the

parameter of interest, using the same percentage changes.

28 A.A. Taleizadeh et al. / Computers & Industrial Engineering 82 (2015) 21–32

Table 2 Results for EOQ model with partial backordering and incremental discounts ( Example 2 ).

Profit (j) b = 0.95

108.67 > q 2 PBO correction (1)

75 466.148 NBO model (1)

1 81.6496 > q 2 NBO correction (1)

1 75 452.5 Not stocking (1)

217.952 > q 3 PBO correction (2)

570.496 NBO model (2)

1 167.332 > q 3 NBO correction (2)

536.25 Not stocking (2)

686.411 Not stocking (3)

NBO model (1)

1 81.6496 > q 2 NBO correction (1)

1 75 452.5 Not stocking (1)

180.49 > q 3 PBO correction (2)

536.394 NBO model (2)

1 167.332 > q 3 NBO correction (2)

536.25 Not stocking (2)

630.197 Not stocking (3)

NBO model (1)

1 81.6496 > q 2 NBO correction (1)

1 75 452.5 Not stocking (1)

2 0.774202 > b

NBO model (2)

1 167.332 > q 3 NBO correction (2)

536.25 Not stocking (2)

3 0.708905 > b

616.393 Not stocking (3)

NBO model (3)

The parameters in our model can be divided into two groups:

4.2. Study results

(1) Parameters that have known values. (2) Parameters that are estimated. The second group can again be divided into at least

4.2.1. Effects of parameter changes on ATP two groups: those for which the estimates are probably fairly accu-

The details of the results of the changes in the estimated param- rate and those that are less certain. For this model the breakdown

eters are shown in Table 3 . The percentage changes in ATP are is:

shown graphically in Fig. 3 . From these results we can draw the following conclusions about how the estimated parameter changes

Known: selling price ðPÞ, purchase cost ({C j }), number of differ-

affected the ATP:

ent unit costs ðnÞ, cost breakpoints ({q j }) Estimated:

1. As would be expected, the further the changed parameter’s More confident: ordering cost ðAÞ, demand ðDÞ, holding cost

value is from the value in the base case, the greater the decrease rate ðiÞ.

in the value of the ATP. There is one exception to this conclu- Less confident: backordering rate (b), goodwill loss for

sion, b, for which the percentage changes in ATP are identical

stockout ðgÞ, backordering cost ( p ) There is one other relevant parameter group, the lost sale cost

is that changes in b by these percentages bring b below its crit-

ical value for which partial backordering is optimal. As can be not need to consider it separately.

per unit ðf p 0 j gÞ, but that is derived from P, {C j }, and g, so we do

seen in those rows of Table 3 , the optimal values of T and F We use the problem solved in Example 2 with b = 0.80 as the

for those cases are 1.45774 and 1.0, giving Q = 291.548, and base case and then resolve it with changes of ±25%, ±20%, ±15%,

B = 0. That is, the optimal solution for those cases is to use the ±10%, and ±5% in each of the estimated parameters, keeping all

basic EOQ with no stockouts for these parameter sets. As shown the other parameters constant. The performance measure is

in Example 2 , the minimum value of b for which partial backor- percent reduction in the average profit per period (ATP) for the

dering is optimal when j = 3 is 0.708905, a reduction of 11.39 variation relative to the optimal ATP from using the original

percent from the base case value of 0.80. Note also that an parameter values.

increase of 25% in the value of b increases its value to 1.0, which Base case parameters: P = $9/unit, D = 200 units/period, A = $30/

means that all shortages will be backordered. This solution,

order, i = 0.3/period, p = $2/unit/period, b ¼ 0:80, C ¼ ðC 1 ;C 2 ;C 3 Þ=

which is shown in the last row of the b section of Table 3 , results

$(6,5,4)/unit, q ¼ ðq 1 ;q 2 ;q 3 Þ = (0, 75, 150) units, g = $2/unit, p 0 ¼

in a decrease in ATP of over 3.5 percent.

2. For all parameters except b and g, decreases in the parameter Base

ð p 0 1 ; p 2 0 ; p 0 3 Þ = $(5, 6, 7)/unit.

value resulted in greater reductions from the base case value Q ⁄ = 335.161, B ⁄ = 54.3765, ATP ⁄ = 630.197/period.

case optimal

than did the same-sized increases. The reason for this difference

29 Table 3

A.A. Taleizadeh et al. / Computers & Industrial Engineering 82 (2015) 21–32

Sensitivity analysis for Example 2 problem with b = 0.80. Parameter

Change (%)

Values of variables

Changes in variables

for b was just discussed. The reason for g is unclear, but we note Since changes in the value of b in 5 percent decrements, which that the reductions in ATP for the same-sized negative and posi-

means changes of 4 percentage points, quickly resulted in solutions tive changes are very close and less than 0.04 percent.

that did not use partial backordering, we looked at the effects of

3. Changes in A result in the least reduction in ATP, followed by g, p , D; i, and b, in that order. Note, however, that, with the excep-

demands that will not be backordered. b = 0.80 for the base case, tion of b and negative changes in i, the reductions in ATP are less than one percent from the base case, even for 25 percent

changes in the parameter value. the changes in b, so we looked at the effect of 10 percent changes

30 A.A. Taleizadeh et al. / Computers & Industrial Engineering 82 (2015) 21–32

Percent Reduction in ATP When Parameters Are Changed One at a Time Reduction in ATP when (1-β) changes

e duction (%

P r 0.5

ATP Reduction (%) 0.5

0 5 10 15 20 25 (1-β) change (%) Parameter Change (%)

-25 -20 -15 -10

Fig. 3. Percent reduction in ATP when parameters are changed one at a time.

Table 5

(2 percentage points each). As shown in Table 4 and Fig. 4 , only a Direction of changes in decision variable values as a parameter increases.

Increase in

Change in

parameter

backordering. Since b = 0.70 is less than the minimum value of b

F Q B for which partial backordering is optimal, this large an increase

A Increase

Decrease Increase Increase

Increase Decrease Decrease the EOQ with no stockouts (F = 1.0).

D Decrease

Decrease

Decrease Decrease Increase

g Decrease

Increase Decrease Decrease

4.2.2. Effects of parameter changes on decision variable values p

Decrease

Increase Decrease Decrease

Decrease Nonmonotone Increase The percentage changes in the values of the four decision vari-

b Nonmonotone

ables that resulted from changing the parameter values are also shown in Tables 3 and 4 . As was the case with the percentage changes in ATP, there are similarities and differences among the

by g, p

variables. values of the four decision variables are very similar, with A; g, and p in some order having the least impact and i; D, and b (or

1. For all the parameters except b, the changes in T; F; Q , and B the sizes and directions of the effects of a parameter change

to +25%. However, these changes were not necessarily in the graphically, the relative changes in the four decision variables

same direction for all four variables. For A; D; g, and p , the

as D changes are shown in Fig. 5 .

changes for T; Q, and B were in the same direction with F in the opposite direction. For i the changes in T; F, and Q were in

4.2.3. Implications

the same direction, with B in the opposite direction. This is Our analysis of the effects of changes in the six unknown summarized in Table 5 , which shows the direction of the

parameters values on ATP and the four decision variables – changes for the variables as each parameter increases in value.

T; F; Q, and B – leads to two basic conclusions: The inconsistent results for b are, as discussed above, due to the fact that large decreases in the value of b led to the basic EOQ

1. As is shown for the basic EOQ model in many introductory texts without backordering being optimal and an increase in b of

on inventory control model, even relatively large changes in or

25 percent to 1.0 led to full backordering being optimal. As mis-estimation of the value of a model parameter have rela- can be seen in Table 4 , these inconsistencies with respect to b

tively small effects on the value of the model’s performance disappear when looking at the effects of changes in the value

measure. Our conclusion here is basically the same. The only model parameter that generated changes in ATP of more than

2. The columns of Tables 3 and 4 that give the percentage changes approximately one percent for a parameter change of ±25% in the decision variables also make it possible to see which vari-

was b. Thus, if the user’s interest is primarily finding a solution ables have the greatest impact on the values of ATP and the

that will give a value of ATP close to the optimal without wor- decision variables. Looking only the results for ±25%, although

rying about whether the values of the decision variables are the same conclusions would be reached if the other sizes are

approximately correct, keeping the parameter estimates within considered, changes in A have the least effect on ATP, followed

about 25% of the true values should be sufficient.

Table 4

Change (%) Value Values of variables

Changes in variables

B (%) ATP (%) 0.90 1.85457

B ATP

T (%)

F (%)

Q (%)

A.A. Taleizadeh et al. / Computers & Industrial Engineering 82 (2015) 21–32

Changes (%) in output variables, when parameter D changes

References

Abad, P. L. (2000). Optimal lot size for a perishable good under conditions of finite 30

production and partial backordering and lost sale. Computers & Industrial 20

F Engineering, 38, 457–465 10 .

Archetti, C., Bertazzi, L., & Speranza, M. G. (2014). Polynomial cases of the economic Changes (%) -10 -25

0 5 10 15 20 25 lot sizing problem with cost discounts. European Journal of Operational Research, -20

Bai, Q. G., & Xu, J. T. (2011). Optimal solutions for the economic lot-sizing problem with multiple suppliers and cost structures. Journal of Applied Mathematics and

Computing, 37, 331–345 . Benton, W. C., & Park, S. (1996). A classification of literature on determining the lot size Fig. 5. Percent changes in T; F; Q , and B when D changes by a given percent.

D change (%)

under quantity discount. European Journal of Operational Research, 92, 219–238 . Bera, U. K., Bhunia, A. K., & Maiti, M. (2013). Optimal partial backordering two- storage inventory model for deteriorating items with variable demand.

2. If, on the other hand, the user is equally as interested in having International Journal of Operational Research, 16, 96–112 . Chen, S. P., & Ho, Y. H. (2011). Analysis of the newsboy problem with fuzzy demands the values of T; F; Q , and B be approximately correct, then less

and incremental discounts. International Journal of Production Economics, 129,

attention can be paid to estimating the values of A; g, and p

and more attention needs to be paid to estimating the values Chu, P., & Chung, K.-J. (2004). The sensitivity of the inventory model with partial backorders. European Journal of Operational Research, 152, 289–295 . of i; D, and b.

Chung, C. S., Hum, S. H., & Kirca, O. (1996). The coordinated replenishment dynamic lot-sizing problem with quantity discounts. European Journal of Operational

One final comment on sensitivity analysis is relevant. Due to the

Research, 94, 122–133 . Ebrahim, R. M., Razm, J., & Haleh, H. (2009). Scatter search algorithm for supplier

relative complexity of the equations for T and Fand, as a result, for selection and order lot sizing under multiple price discount environment. the ATP, we used, as is most frequently done in assessing the sen-

Advances in Engineering Software, 40, 766–776 . sitivity of a model to changes in its inputs, a numerical approach in

Güder, F., Zydiak, J., & Chaudhry, S. (1994). Capacitated multiple item ordering with this study. As was pointed out by incremental quantity discounts. Journal of the Operational Research Society, 45, Chu and Chung (2004) in their

discussion of sensitivity analysis of a basic EOQ with partial back- Hadley, G., & Whitin, T. M. (1963). Analysis of inventory systems. Englewood Cliffs, NJ, ordering, ‘‘the conclusions made by the analyses of sensitivities

USA: Prentice-Hall International, Inc., USA .

based on the computational results of a set of numerical examples Haksever, C., & Moussourakis, J. (2008). Determining order quantities in multi- product inventory systems subject to multiple constraints and incremental are questionable since different conclusions may be made if differ-

discounts. European Journal of Operational Research, 184, 930–945 . ent sets of numerical examples are analyzed.’’ While we are confi-

Harris, F. W. (1913). How many parts to make at once. Factory, The Magazine of Management, 10, 135–136. Reprinted in (1990), Operations Research, 38, 947–950 dent that our conclusions above are fairly general, any user of this . Hu, J., & Munson, C. L. (2002). Dynamic demand lot-sizing rules for incremental or a similarly complex model needs to conduct his or her own

quantity discounts. Journal of the Operational Research Society, 53, 855–863 . study.

Hu, J., Munson, C. L., & Silver, E. A. (2004). A modified silver-meal heuristic for dynamic lot sizing under incremental quantity discounts. Journal of the Operational Research Society, 55, 671–673 .

Lee, A. H. I., Kang, H. Y., Lai, C. M., & Hong, W. Y. (2013). An integrated model for lot

5. Conclusion sizing with supplier selection and quantity discounts. Applied Mathematical Modelling, 37, 4733–4746 .

Lin, C. S., & Kroll, D. E. (1997). The single-item newsboy problem with dual We extended the basic EOQ model with incremental discounts

performance measures and quantity discounts. European Journal of Operational by combining the basic solution procedure for that problem with

Research, 100, 562–565 .

repeated use of Pentico and Drake’s (2009) models for the EOQ Mendoza, A., & Ventura, J. A. (2008). Incorporating quantity discounts to the EOQ with full or partial backordering at a constant rate b to determine model with transportation costs. International Journal of Production Economics,

the best order quantity for each possible cost. Minimum cost (or Montgomery, D. C., Bazaraa, M. S., & Keswani, A. K. (1973). Inventory models with a maximum profit) was then used to choose among the best full