A pull system inventory model with carbon tax policies and imperfect quality items

Applied Mathematical Modelling 50 (2017) 450–462

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A pull system inventory model with carbon tax policies and imperfect quality items Tien-Yu Lin a,∗, Bhaba R. Sarker b a b

Department of Marketing and Supply Chain Management, Overseas Chinese University, Taichung 407, Taiwan Department of Mechanical & Industrial Engineering, Louisiana State University, Baton Rouge, LA 70803, United States

a r t i c l e

i n f o

Article history: Received 8 March 2016 Revised 21 May 2017 Accepted 5 June 2017 Available online 13 June 2017 Keywords: Inventory Carbon emissions Quantity discounts Order overlapping Imperfect quality

a b s t r a c t This paper develops a new inventory model with carbon tax policy and imperfect quality items in which the buyer exerts power over its supplier. It employs an order overlapping scheme to avoid shortages, overcomes some flaws in the literature, and develops two efficient solution algorithms. It also investigates different carbon tax systems on the performance of the model. Numerical results are discussed to bring some managerial insights. © 2017 Elsevier Inc. All rights reserved.

1. Introduction One of these unreasonable assumptions of the economic order quantity (EOQ) model [1] is that all units produced are of good quality, e.g., [2]. EOQ models with imperfect quality have received considerable attention in the literature (e.g., [3–10]). An interesting EOQ variant and more reasonable model than the EOQ have been proposed by Salameh and Jaber [11]. In their paper, a lot contains a certain percentage of defective items and a 100% inspection is conducted. At the end of the screening process, the defective items are sold at a discounted price as a single batch. Of interest to this paper is the work of Lin [3] who extended the EOQ model with imperfect quality to include quantity discounts and lot-splitting shipments for the retailer. Chang [4] corrected Lin’s holding cost term and both the works assumed ‘no shortages’ during screening. Some researchers argued that this assumption may not hold when some conditions are present (e.g., [12–14]). This implies that there are flaws in [3,4]. Maddah et al. [13] provided a good idea to overcome shortages by allowing the overlapping shipments. They assumed that good quality and defective items have equal holding costs. In practice, defective items are removed and stored in warehouse with a lower unit-holding cost [15]. The unit-holding costs for the good and defective items are therefore different in this paper. Modern manufacturing is actually a pull system in which deliveries must be made on an as-needed basis only. The ‘order trigger’ is in the retailer’s hand and, thus, the retailer is powerful and has a bulk demand for commodities [16]. One environmental issue that has been given considerable attention is greenhouse-gas (GHG) emissions modeling from transportation and production, energy usage from production and storage activities as well as production waste disposal [17]. Some of the works that modified the EOQ to account for carbon emissions in different inventory situations are Wahab ∗

Corresponding author. E-mail addresses: [email protected] (T.-Y. Lin), [email protected] (B.R. Sarker).

http://dx.doi.org/10.1016/j.apm.2017.06.001 0307-904X/© 2017 Elsevier Inc. All rights reserved.

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et al. [18], Bonney and Jaber [19], Chen et al. [20], Konur [21], Gurtu et al. [22], and Bazan et al. [23]. A framework for reducing GHG (mainly carbon) emissions and/or energy usage are found in Gurtu et al.’s [24] and Bazan et al.’s [25,26]. This paper therefore studies the pull system inventory model in which the retailer is the powerful player who dictates on its supplier the quantity discounts it desires. Moreover, this paper employs an order overlapping scheme to avoid shortages, which could efficiently rectify the shortages flaw shown in Chang’s [4] work. The contributions of this paper are twofold. First, it corrects a flaw shown in the work of Chang [4]. Second, to the best of our knowledge, it is first that incorporates quantity discounts and a flat/progressive carbon tax rate into the pull system inventory model. The remainder of this paper is organized as follows. Section 2 lists the notations and assumptions used in this paper and Section 3 develops a mathematical model with methodology and algorithm to obtain the overall optimal solution. Section 4 explores the impact of full progressive carbon tax on the optimal solution. A numerical example is provided in Section 5 to illustrate the proposed model. Finally, managerial insights are discussed in Section 6. 2. Notations and assumptions The following notations and assumptions are used hereinafter to develop the proposed model: (a) Notation: Q order size D demand rate β tax rate issued by regulatory agencies g fixed amount of carbon emission for the business activities α amount of carbon emission due to purchasing items λ1 amount of carbon emission due to inventory holding for good items λ2 amount of carbon emission due to inventory holding for imperfect items x screen rate, x > D p defective percentage in Q f(p) probability density function of p m selling price per unit v salvage value of per defective item, v < cj d screening cost per unit I percentage of unit price K ordering cost per order N number of shipments per cycle (integer value) R receiving cost per shipment T planning horizon q size of shipment for each delivery which is given by q = Q/N t inventory depletion time for each shipment Qj jth lowest quantity where Qj − 1 < Qj cj unit-purchasing cost of jth level Ig holding cost rate for a unit of good item per period, expressed as a fraction of dollar value and Ig > Id Id holding cost rate for a unit of defect item per period, expressed as a fraction of dollar value hg holding cost for a unit of good item per period, hg > hd hd holding cost for a unit of defect item per period (b) Assumptions: (1) (2) (3) (4) (5) (6) (7) (8) (9)

The demand rate is known and constant. Shortages are not allowed. There are defective items in each lot. The screening rate is greater than the demand rate, x > D. A lot is screened in full (100%). Defective items are withdrawn from inventory and sold as a single batch at a discounted price. The entire lot is produced and delivered in batches. Order overlapping scheme is assumed. Holding costs of good and defective items are different. An all-unit quantity discount scheme is adopted where cj , j = 0,1,…, r, is the discounted price when the order quantity is in the interval [Qj ,Qj + 1 ).

3. Mathematical model We develop a mathematical model with the above assumptions in this section. The objective is to minimize the total cost, which the buyer’s perfect item inventory from the previous shipment is just enough to meet the demand during the

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T.-Y. Lin, B.R. Sarker / Applied Mathematical Modelling 50 (2017) 450–462

Fig. 1. Behavior of the inventory level for a setup and lot-splitting shipments.

screening process. The behavior of inventory is shown in Fig. 1, similar to [13]. The triangle area BCE in the top part of Fig. 1 is the same as KJM (e.g., the area of triangle with dotted line), which could sufficiently meet the demand during the screening process; that is, when the screening process is completed for each delivery, the imperfect items are removed from the stock and the inventory level is then reduced by (1 − p)Q/N. This implies that inventories from two consecutive deliveries ‘overlap’ and thus the total quantity will be Q for each delivery, which may improve the flaw in Chang’s [4] work. The inventory system for items with imperfect quality in an overlapping cycle is illustrated in Fig. 1 in which a single order and lot-splitting shipments are considered (say N shipments) during the planning horizon, T. For each shipment the delivery interval and the amounts are the same. The 100% lot screening process is finished at time te for each delivery of q units where te = q/x. It can be found that the inventory level will reduce bypq at time te due to defective items stock

T.-Y. Lin, B.R. Sarker / Applied Mathematical Modelling 50 (2017) 450–462

453

withdrawal. Furthermore, to meet the real world situation, the defective percentage (i.e., p) in Q taken from Salameh and Jaber [11] is considered as

p ≤ 1 − D/x

(1)

where p is a random variable and in nature D < x. Let TR(Q, N) be the total revenue per cycle from selling good and defective items and is given by

T R(Q, N ) = N [mq(1 − p) + vqp] = mQ (1 − p) + vQ p.

(2)

Let TCj (Q, N) be the total cost per cycle if we assume the carbon tax is flat and the unit-purchasing cost cj , is valid for all Q. Considering the shaded areas and blank triangle areas are similar to CFG but not including  BCE in Fig. 1, the holding cost per cycle can be expressed as follows:

HC = c j Ig

N   qte i=1

(Nq )

2

=

2



2

+



c j Ig







 qte tq(1 − p) + c j Id + t (i − 1 ) pq 2 2

1 ( 1 − p )2 + xN DN

N

i=1



+ c j Id

1

+

xN

(N − 1 ) (1 − p) p



.

DN

Therefore, the total cost for the retailer is given as





T C j (Q, N ) ≡ T C Q c j , N = c j Nq + K + dNq + NR +

(Nq )2



2



c j Ig

1 (1 − p)2 + xN DN



+ c j Id

1 xN

+



(N − 1 ) (1 − p) p DN

   1  1 (Nq )2 (1 − p)2 (N − 1 ) (1 − p) p + βα Nq + β g + + + βλ2 + βλ1 2 xN DN xN DN 

= c j + βα Q + dQ + β g + K + NR      1

 1 Q2

(1 − p)2 (N − 1 ) (1 − p) p + c j Ig + βλ1 + + c j Id + βλ2 + 2

xN

DN

xN

DN

j = 1, 2, ..., r.

(3)

The net profit per unit time, TPUj (Q, N), is determined by the revenue per unit time, TR(Q, N)/T, less the cost per unit time, TCj (Q, N)/T. Thus, the average profit is given by









T PU j (Q, N ) ≡ T PU Q c j , N = T R(Q, N ) − T C j (Q, N ) /T , j = 1, 2, ..., r.

(4)

The replenishment cycle length is T = {(1 − p)Q /D}. Using the renewal reward theorem [27], the expected total profit per unit of time (ETPU) is given by



lim

t→∞



 E [T R(Q, N )] − E T C j (Q, N ) T PUt = E T PU j (Q, N ) ≡ E T PU Q c j , N = t E [T ]



D c j + d + βα D(K + β g + NR) DvE0 − − E1 Q E1 E1

 D + x E 2 

 D + x(N − 1 )(E0 − E3 )  Q

− c j Ig + βλ1 + c j Id + βλ2 2E1 xN xN j = 1, 2, ..., r = Dm +

E[(1 − p)2 ],

(5)

E[p2 ].

where E0 = E[p], E1 = 1 − E[p], E2 = E3 = Note that (E0 − E3 ) > 0 provides E0 < 0.5 [4], which often holds in general. Furthermore, if we neglect the environmental costs and suppose that the imperfect and perfect items have the same holding cost (i.e., I = Ig = Id ), Eq. (5) can be reduced as follows:







Ic j Q 2D 1 D ( vE0 − d ) D(K + NR ) ET PU j (Q, N ) = Dm + − Dc j + + + E2 + ( N − 1 ) ( E0 − E3 ) E1 E1 Q 2N x

.

(6)

It was not possible to show that Eq. (5) is jointly concave in Q and N. The Hessian matrix was too complex. Therefore, we develop algorithms in the next section to find the optimal and feasible solutions. 4. Solution methodology According to Lin [3] and Papachristos and Koustantars [12], maximizing ETPUj (Q, N) is equivalent to minimizing the expected ‘relevant’ cost. Thus, omitting the terms independent of Q, and N, β and cj , Eq. (5) can be expressed as

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D c j + βα D(K + β g + NR) EC j (Q, N ) = + Q E1 E1

 D + xE2 

 D + x(N − 1 )(E0 − E3 )  Q

+ c j Ig + βλ1 + c j Id + βλ2 2E1 xN xN j = 1, 2, ..., r

(7)

which is basically controlling the optimality of the profit or cost function. For the derivation of the optimal solution in the expected relevant cost ECj (Q, N), the following property is needed. Property 1. For fixed N, ECj (Q, N) is convex in Q. Proof. For fixed N, ECj (Q, N)in Eq. (7) becomes a function of single variable Q. Hence, the sign of ∂ 2 ECj (Q, N)/∂ Q2 characterizes its convexity. Thus, we have

  ∂ 2 EC j [Q, N] 2 D(K + β g + NR) = > 0, j = 1, 2, . . . r E1 ∂Q2 Q3

which indicates that, for a fixed value of N, ECj (Q, N) is convex in Q for all j =1,2,…,r. Property 1 shows that there is a unique solution that minimizes Eq. (7) when the number of shipments is given. However, we cannot directly obtain Q∗ from Eq. (6) because the unit-purchasing cost depends on ordering lots and thus there are different cost curves for each cost level. This means the values for Qj (the order quantities corresponding to each purchasing cost) in this procedure are not over-all optimal solutions but feasible solutions by letting ∂ EC j (Q, N )/∂ Q = 0. Therefore, if we set ∂ EC j (Q, N )/∂ Q = 0 and let it be Q˜ j corresponding to each purchasing cost, we have

 Q˜ j (N ) =



2D(K + β g + NR)

c j Ig + βλ1

 D+xE2 xN



+ c j Id + βλ2

 D+x(N−1)(E0 −E3 ) xN

j = 1, 2, ..., r.

(8)

Substituting Eq. (8) into Eq. (5) and rearranging of the result yield to



1 EC j (N ) = E1

c jD +





 D + xE2 

 D + x(N − 1 )(E0 − E3 )  2D(K + β g + NR) c j Ig + βλ1 + c j Id + βλ2 xN

xN

j = 1, 2, ..., r

(9)

Ignoring the terms independent of N and taking square of ECj (N), we know that minimizing ECj (N) is equivalent to minimizing ϕ j (N):

 D + xE2 

 D + x(N − 1 )(E0 − E3 )  φ j (N ) = 2D(K + β g + NR) c j Ig + βλ1 + c j Id + βλ2 xN

xN

j = 1, 2, ..., r

(10)

Treating N as a continuous variable and taking the first and the second derivative of ϕ j (N) with respect to N, we have



 d φ j (N ) = 2D c j Id + βλ2 (E0 − E3 )R dN 

1 − 2 (K + β g) N



c j Ig + βλ1 [D + xE2 ] x



+ c j Id + βλ2

 D x

− E0 + E3

 

j = 1, 2, ..., r and

d φ j (N ) 4D ( K + β g ) = dN N3 2

j = 1, 2, ..., r

(11)





c j Ig + βλ1 [D + xE2 ] x



+ c j Id + βλ2

 D x

− E0 + E3

  (12)

Based on Eq. (1), it is clear that (D/x − E[ p] ) > 0 yields E [p] < 0.5 which often holds in practice. This leads to (dφ j 2 (N)/dN) > 0 and ϕ j (N) is convex in N. This implies ECj (N) is also convex in N. Therefore, the first order condition ∂ φ j (N )/∂ N = 0 leads to

T.-Y. Lin, B.R. Sarker / Applied Mathematical Modelling 50 (2017) 450–462

 N¯ j =

455





  (K + β g) c j Ig + βλ1 D+xxE2 + c j Id + βλ2 Dx − E0 + E3

 R c j Id + βλ2 (E0 − E3 )

j = 1, 2, ..., r

(13)

Since N is an integer value of the discrete variable, Eq. (13) does not guarantee that an integer could be obtained. After some manipulations for Eq. (10) we have the following forms:

Mink1 z + k2 /z,

(14)

where k1 and k2 are positive and z is a positive integer. García-Laguna et al. [28] showed that the discrete solution to the minimization of Eq. (14) is



z=



−0.5 +



k2 0.25 + , k1

(15)

where π means the smallest integral value greater than or equal to π . ˜ j is given by Following the result of Eq. (15), the close-form to determine the candidate optimal value for N





˜j = N

−0.5 +





   (K + β g) c j Ig + βλ1 D+xxE2 + c j Id + βλ2 Dx − E0 + E3

 0.25 + , j = 1, 2, ..., r R c j Id + βλ2 (E0 − E3 )

(16)

˜ j ) corresponding to unit-purchasing cost (i.e., cj ) in which Q˜ j (N ˜ j) Therefore, we can plug Eq. (16) into Eq. (8) to find Q˜ j (N ˜ ˜ ˜ ˜ is valid when Q j−1 ≤ Q j (N j ) < Q j holds. We note that if Q j (N j ) is invalid corresponding to cj , then two cases occur: ˜ j ) > Q j , where Qj is the maximum order quantity corresponding to unit-purchasing cost cj . Case I: Q˜ j (N ˜ j ). In this case, the retailer may employ the lower unit-purchasing cost (say ck for ck < cj ) to obtain the order lot-size Q˜ j (N ˜ ˜ From Eq. (9), we knowECk (Nk ) < EC j (N j ). This implies the possible candidates under cj could not be the overall optimal and thus no additional computation procedures are needed. ˜ j ) < Q j−1 , where Qj − 1 is the minimal order quantity corresponding to the unit-purchasing cost cj . Case II: Q˜ j (N In this case one may fix Q and N when cj is given, and then two scenarios occur: Scenario 1: The perspective of breakpoint Qj − 1 corresponding to cj . In this scenario, the candidate optimal lot size may occur at the breakpoint Qj − 1 with its corresponding candidate optimal number of shipments. Thus, for given Qj − 1 corresponding to cj , the close form to determine the candidate integer shipments ˆ j is given by forN



Nˆ j =



−0.5 +



0.25 + Q j−1

2 

c j Ig + βλ1

 D+xE2 x



+ c j Id + βλ2

 D x

− E0 + E3





/2DR ,

(17)

j = 1, 2, ..., r Scenario 2: The perspective of fixed N. For a given reasonable N, in this scenario, the candidate optimal solution may occur at (Q j (N ), N ) in which the condi-

tion of Q j−1 ≤ Q j (N ) < Q j is satisfied. Therefore, plugging Eq. (8) into Q j−1 ≤ Q j (N ), we have

ω N 2 + ψ j N − γ j > 0,

(18)

where ω = 2DR, ψ j = 2D(K + β g) − {(Qj − 1



γ j = Q j−1

2

c j Ig + βλ1

)2 (c

 D + xE2  x

j Id + βλ2 )[E0 − E3 ]},



+ c j Id + βλ2

 D x

and

− E0 + E3



Solving the above inequality for N, we have

Nj >

−ψ j +



ψ j 2 + 4ω γ j ≡ N¯ j , j = 1, 2, ..., r. 2ω

(19)

Taking N j = N¯ j  in which N¯ j  is the smallest integer larger than N¯ j , we therefore obtain Q j (N ) from Eq. (8). Note j

that if Q j (N ) > Q j , no feasible solution occurs and thus, no additional computation procedures are needed. j We have, until now, obtained the candidate optimal lot size and the number of shipments corresponding to the unitpurchasing cost but could not obtain the overall optimal solution. Therefore, to achieve benefit of the overall optimal lot-size and the number of shipments, Algorithm 1 is developed (see Appendix A). Note that when the buyer losses its dominance, the supplier may not offer quantity discounts. This implies that the holding costs for perfect and imperfect items are modified as hg = cIg and hd = cId . Furthermore, if we neglect the environmental

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T.-Y. Lin, B.R. Sarker / Applied Mathematical Modelling 50 (2017) 450–462 Table 1 Quantity-discounts regression-tax scheme. g 1 2 3 . . . b

Q g−1 ≤ Q < Q g Q

Q

1≤ < 1 Q 1 ≤ Q < Q 2 Q 2 ≤ Q < Q 3 . . . Q b−1 ≤ Q < ∞

cg

βg

c1 c2 c3 . . . cb

β1 β2 β3 . . .

βb

costs, Eq. (8) can then be reduced to

 Q˜ j (N ) =

2D(K + NR ) . {(hd + hg )D/Nx + [hg E2 − hd (E0 − E3 )]/N + hd [E0 − E3 ]}

(20)

We found Eq. (20) is similar to Eq. (17) shown in Chang’s [4] work except the first term in the denominator. This is because this model employs the order overlapping policy to avoid shortages from occurring, which could rectify the flaws in shortages that appears in Chang’s [4] work. Furthermore, this result intuitively leads to the fact that the order lot-size in the model is less than that in Chang’s [4] model. We note that the carbon tax rate in this section was assumed to be a flat tax. However, in practice, the purpose of levying environmental tax is to reduce carbon emissions and thus a ‘progressive’ tax policy is employed by agencies to achieve sustainability. The greater the quantity of carbon emissions, the higher the carbon tax rate is. Therefore, in the next section, we will explore the impact of progressive tax policy on the optimal solution

5. The impact of full progressive carbon tax policy Many countries (e.g., Finland, Denmark) recently implemented environmental regulations to reduce carbon emissions by imposing taxes. From an economic perspective, the carbon tax is a type of Pigovian tax which is a tax levied on any market activity that generates negative externalities [29]. This implies the carbon tax could be a progressive tax system, which we explore in our model. We changed the flat tax (β ) discussed in the previous section into a full progressive tax (β k , k = 1,2,…,l). This means that when the order quantity between [Qj − 1 , Qj ), the progressive tax rate is β j, . Thus, if an agency employs the progressive tax rates with l intervals, we have order quantity [Q0 , Q1 ) corresponding to unit purchase cost β 1 , [Q1 , Q2 ) corresponding to unit purchase cost β 2 ,…, and [Ql − 1 , Ql ) corresponding to unit purchase cost β l , where β 1 < β 2 < … < βl. The same mathematical model developed in the previous section is employed again but with a different tax rate corresponding to different order quantities. Observing all-unit quantity discounts scheme in Assumption (9) and full progressive tax rates scheme described in this section, we found they have similar structures but with different cost trends. We note that both all-unit quantity discounts scheme and full progressive tax rates scheme employ the lot size as their index for determining the unit-purchasing cost and carbon tax rate. To obtain the optimal solution, we need to combine these two schedules into a new scheme (Table 1), which could include possible lot size intervals; that is, Table 1 named quantitydiscounts progressive-tax schedule as a new scheme accomplished by identifying all possible lot size intervals simultaneously corresponding to the unit-purchasing cost and carbon tax rate. Algorithm 2 is developed to model the restructured scheme (see Appendix B). We further employ Algorithm 1 developed in Appendix A in which the indicators j and k are combined as the new indicator g using Algorithm 2. This means the order lot sizes are separated into several new independent intervals, and thus, breakpoints are redefined. A new scheme is developed as in Table 1 and thus Algorithm 1 in Appendix A is also modified according to this new scheme. The progressive carbon tax policy has a significant impact on the optimal solution compared with the flat carbon tax policy, as we will see later on.

6. Comparison with Maddah et al.’s [13] model We compare our model to that of Maddah et al.’s [13] model by combining the number of shipments into a single delivery, with no quantity discounts, no receiving cost and the same holding cost for both good and imperfect quality items. The optimal order quantity in Eq. (8) is then reduced to

 QSingle =

2D ( K + β g ) (h + βλ)(E2 + 2D/x )

(21)

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457

Substituting Eq. (21) into Eq. (5), the expected total profit is obtained as





ET PU (Q, 1 )sin gle

D c j + d + βα 1 DvE0 = Dm + − − E1 E1 E1



 2D

2D ( K + β g )

x



+ E2 (h + βλ )

(22)

If the carbon tax rate issued by the regulatory agency is free, the optimal order lot size in Eq. (21) is then reduced to Maddah et al.’s [13] result:



QMaddah =

2DK h(E2 + 2D/x )

(23)

and the expected total profit in Maddah et al.’s [13] model is given by



ET PUMaddah



D cj + d 1 DvE0 = Dm + − − E1 E1 E1



2DKh

 2D x

+ E2

 (24)

Thus, a simple sensitivity of Q∗ single with respect to Q∗ Maddah yields ∗ Qsin gle ∗ QMad dah



=

Kh + β hg , Kh + β K λ

(25)

where the following set of relationships hold: ∗ ∗ (i ) Qsin gle = QMaddah ⇔

K g = h λ

(26)

K g < h λ

(27)

K g > h λ

(28)

∗ ∗ (ii ) Qsin gle > QMaddah ⇔

∗ ∗ (iii ) Qsin gle < QMaddah ⇔

Hence, the optimal ordering quantity for the proposed model given in a single delivery (i.e., Q∗ single ) may be larger or smaller than that in Maddah et al.’s [13] work (i.e., Q∗ Masddah ) as it might be equal to it as well depending on the value of the cost components. The difference in the expected total profits in Eqs. (22) and (24) is evaluated as

E T PUMaddah − E T PU (Q, 1 )sin gle 1 = E1

 

− 2DKh

 2D x

  2D  + E2 + 2D ( K + β g ) + E2 (h + βλ ) + βα > 0  

x

(29)

This implies that the expected total profit in Maddah et al.’s work is larger than that in the proposed model with a single delivery policy. Further, flat tax rate β (see Eqs. (25)–(29)) does not have an impact on the optimal order quantity but it does affect the total profit if the single delivery policy is adopted which confirms that taxing carbon emissions gives incentive to identify emission sources, estimate the emission parameters and curb the emissions to achieve lower operating cost [30]. Therefore, employing a flat carbon tax schedule with the macroeconomic policy of a county is a useful tool for regulatory agencies. However, this paper assumed the buyer exerts power over its supplier possibly resulting in several shipments for each order. Observing Eq. (13), we know the carbon emission tax rate does have an impact on the number of shipments and thus affects the optimal order quantity. In the following section we will give two examples and then make analysis to illustrate the proposed model. 7. Numerical examples and sensitivity analysis We adopt two examples to verify the model and algorithms developed. A sensitivity analysis is conducted to investigate the efforts of four important parameters. Managerial insights are also explored. Example 1. Retailers in auto-industry. Retailers in many industries are powerful enough to control the vendors for their goods and services as the vendors depend on the mercy of the retailer’s decisions for the purchase and transactions. As indicated in the introduction, a powerful retailer in the automobile industry is considered in this example. All necessary variable costs and fixed costs are estimated from existing data and listed in Table 2. Relevant data of the system inventory including the demand information are assumed from standard practices. Because of the proprietorship of the company’s sensitive data the information on emission is not disclosed and/or is kept at very minimal level to satisfy the public and the government emission regulations. The various emission data compiled considering different situations and the environment. In addition, the supplier provides a price discount schedule as the following intervals: [1, 3500) corresponding to c1 = 25.019, [3500, 7000) corresponding to c2 = 25.017, [70 0 0, 10, 50 0)corresponding to c3 = 25.015, [10, 500, 14, 000) corresponding to c4 = 25.013, and [14, 0 0 0, ∞) corresponding to c5 = 25.011.

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T.-Y. Lin, B.R. Sarker / Applied Mathematical Modelling 50 (2017) 450–462 Table 2 All necessary variable costs and fixed costs. Description and parameters

Value

Unit

Demand rate (D) Ordering cost (K) Receiving cost (R) Screening rate (x) Screening cost (d) Holding cost for a unit of good item (a fraction of dollar value) (Ig ) Holding cost for a unit of defect item (a fraction of dollar value) (Id ) Selling price of good quality items (m) Salvage value of defect items (v) Tax rate issued by regulatory agencies (β ) Fixed amount of carbon emission for the business activities (g) The amount of carbon emission due to purchasing items (α ) The amount of carbon emission due to holding good items (λ1 ) The amount of carbon emission due to holding imperfect items (λ2 )

3500 100 200 175,200 0.50 0.40 0.20 50 20 0.02 200 75 50 25

units/year $/cycle $/shipment units/year $/unit $/unit/year $/unit/year $/unit $/unit $/g g/cycle g/unit g/unit g/unit

Table 3 Quantity-discounts regression-tax scheme. g

Q g−1 ≤ Q < Q g

cg

1 2 3 4 5 6 7 8

1 ≤ Q < 3500 3500 ≤ Q < 4500 4500 ≤ Q < 7000 70 0 0 ≤ Q < 8500 8500 ≤ Q < 10, 500 10500 ≤ Q < 11, 500 11500 ≤ Q < 14, 000 Q ≥ 14, 0 0 0

c1 c2 c3 c4 c5 c6 c7 c8

βg = = = = = = = =

25.019 25.017 25.017 25.015 25.015 25.013 25.013 25.011

β1 β2 β3 β4 β5 β6 β7 β8

= = = = = = = =

0.0020 0.0020 0.0022 0.0022 0.0024 0.0024 0.0026 0.0026

For an illustrative purpose, assume the percentage defective p can take any value in the range [L, U] with L = 0 and U = 0.04; that is, p is uniformly distributed as



f ( p) =

25, 0 ≤ p ≤ 0.04 0, otherwise

Considering the type of cost structure and using the algorithm developed in Section 4, one has the optimal ordering quantity, number of shipments and minimum annual total cost (maximum annual total profit) as Appendix C. Based on the above discussion, we know that the optimal number of shipments is 14 times per cycle and the optimal order lot size is 14467.7 units. The unit procurement cost is $25.011. The expected total cost is $903705.20 (i.e., the expected total profit is $835688.70) per year. Compared with Chang’s [4] work, we know that the carbon emission tax for a multiple delivery policy could significantly influence the optimal solution because Chang’s [4] work showed Q∗ = 10608.5 units and N∗ = 9 deliveries corresponding to the unit-purchasing cost c5 (c5 = $25.011). We also have EC5 [10608.5,9] = $897992.1 (note: the actual value is $896961.70 because Chang’s work omitted the irrelevant terms). This means that when regulatory agencies impose a carbon tax the retailer’s operating costs will increase. Note that if a single delivery policy is adopted, the optimal order quantity obtained from Eq. (21) is then 1275.20 units. The expected total cost is EC5 [1275.2,1] = $906, 187 (i.e., ETPU5 [1275.2,1] = $833207). Thus, multiple deliveries, in general, for the powerful retailer could result in less cost (or larger profit). Example 2. Progressive carbon tax If the agency adopts a progressive carbon tax system with the following structure where β 1 < β 2 < β 3 < β 4 and each interval has a small increase [i.e., β k = β 1 ∗ (1 + 10%∗ (k − 1))], say, 10%: [1, 4500) corresponding to β 1 = 0.0 020, [450 0, 850 0) corresponding to β 2 = 0.0 022, [850 0, 11, 50 0) corresponding to β 3 = 0.0024, and [11, 500, ∞) corresponding to β 4 = 0.0026. Employing Algorithm 2, we have the quantity-discounts regression-tax scheme as follows (Table 3): Using Algorithm 1 developed in Appendix A, one has the optimal solution as follows:

Q ∗ = 4389.6. units, N ∗ = 4 deliveries, ET PU4 [4389.6, 4] = $904246 under β2 = 0.002 and c3 = 25.017. Compared with Example 1, if a progressive tax rate scheme is adopted by the regulatory agency, we know the optimal order lot size, the optimal number of shipments and the expected total profit will decrease. This illustrates that a carbon tax imparts a significant cost burden upon the retailer.

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459

Table 4 The values of Q∗ , N∗ and ETPU∗ corresponding to 16 combinations of D, x, Ig, R. D

x

Ig

R

N∗

Q∗

ci

βi

ETPU∗

35,0 0 0

175,200

0.32

160 200 160 200 160 200 160 200 160 200 160 200 160 200 160 200

7 3 4 4 7 3 4 7 7 3 4 7 3 3 7 3

7412.3 3712.3 3980.5 4389.6 7730.3 3847.7 4143.6 7751.7 8047.5 4028.3 4326.9 8101.1 3858.2 4238.4 7659.8 3830.6

25.015 25.017 25.017 25.017 25.015 25.017 25.017 25.015 25.015 25.017 25.017 25.015 25.017 25.017 25.015 25.017

0.0022 0.0020 0.0020 0.0020 0.0022 0.0020 0.0020 0.0022 0.0022 0.0020 0.0020 0.0022 0.0020 0.0020 0.0022 0.0020

837549.9 836311.2 836499.6 835148.0 838028.9 836870.5 837017.1 835505.7 1,048,086.1 1046697.1 1046940.6 1045119.9 1048768.7 1047458.8 1047407.6 1045904.1

0.40 262,800

0.32 0.40

43,750

175,200

0.32 0.40

262,800

0.32 0.40

To realize the model parameter effects on optimal policy in Example 2, the parameters including D, K, R, x, d, Ig , Id , m, v, g, β , λ1 , λ2 , α , and f(p) theoretically should be explored. However, exploring the impact of these parameters on optimal policy would be a laborious computational undertaking. We, therefore, only investigate the D, x, Ig , R parameters under a progressive carbon tax system using Table 3. All of the four parameters are set at two levels (low and high) and shown as follows: D = (35,0 0 0, 43,750); x = (175,20 0, 262,80 0); Ig = (0.32, 0.40); and R = (160, 200). The other parameters remain unchanged except for the above four parameters. Table 4 lists the optimal solution under 16 combinations of D, x, Ig , and R. We obtained the findings shown below from Table 4: (1) ETPU∗ increases with D; while, by combining Q∗ and N∗ , three cases occur when D increases: Case I: N∗ remains unchanged and Q∗ is increasing corresponding to ci and β i unchanged. Case II: N∗ and Q∗ are both increasing with D corresponding to ci decreasing and β i increasing. Case III: N∗ and Q∗ are both decreasing with D corresponding to ci increasing and β i decreasing. As to the Case I, N∗ , ci , and β i remains unchanged and Q∗ is increasing with D. It is intuitively that the higher demand is, the larger ordering quantity is, which matches the traditional EOQ model results; that is, the retailer orders additional quantities to meet the customer needs as the demand increases. The retailer could sell more quantities and thus ETPU∗ is increasing with D. In Case II N∗ , Q∗ and ETPU∗ are increasing with D corresponding to ci decreasing and β i increasing. Obviously, the trade-off effect occurs between ci and β i . In this case, the quantity discounts effect is larger than that of taxing carbon emissions, which significantly occurs at the low level demand rate; that is, the retailer orders more quantities and employs a greater number of shipments to obtain the maximum expected total profit, which agrees with the result in Lin’s [5] work. It is intuitive that ETPU∗ increases with D because the retailer could sell more quantities. In Case III, N∗ and Q∗ are both decreasing with D corresponding to ci increasing and β i decreasing. Obviously, the taxing carbon emission effect is larger than that of quantity discounts in this case. This leads the retailer to orders fewer quantities and reduces the number of shipments to obtain the maximum expected total profit. We note that this case usually occurs at the high level demand rate because the retailer bears higher operational costs in producing under carbon emission tax conditions. In summary, the powerful retailer employs the optimal combinations of Q and N corresponding to ci and β i to achieve his/her maximum benefit. (2) Similar to (1) in the findings, we have ETPU∗ increasing with x; while, by combining Q∗ and N∗ , three cases occur in which x increases (see findings in (1) and replace D with x). In the traditional EOQ model a higher screening rate results in the quick removal of defective items, which increases quantity orders and reduces the inventory holding cost. However, the trade-off between quantity discounts and taxing progressive carbon emission leads to the above result no longer holding. In general, the lot size increment for each delivery increases the length of the time interval between successive deliveries and thus decreases the receiving cost but increases the holding cost. Thus, two possible combinations of Q∗ and N∗ (i.e., Cases II and III in this finding) corresponding to ci and β i are employed to find the maximum ETPU∗ . The reason is similar to (1) in the findings. (3) It is also found that ETPU∗ decreases as Ig increases. Three combinations of Q∗ and N∗ occur with Ig : Case I: N∗ and Q∗ are both decreasing with Ig corresponding to ci increasing and β i decreasing.

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Case II: N∗ and Q∗ are both increasing with Ig corresponding to ci and β i unchanged. Case III: N∗ remains unchanged and Q∗ is decreasing with Ig corresponding to ci and β i unchanged. Observing Case III in this finding, we found that N∗ , ci and β i remain unchanged and Q∗ decreases with Ig . This illustrates that the higher the holding cost for good items is, the fewer the number of quantity orders and the less expected total profit are, which matches the result in the traditional EOQ model. This illustrates that when the holding cost increases, the retailer orders fewer quantities for each batch to reduce the inventory holding cost. As to Case I in this finding, N∗ and Q∗ both decrease with Ig corresponding to ci increasing and β i decreasing. Obviously, the trade-off between progressive carbon emission taxation and quantity discounts works and the effort of the former is larger than that of the latter. Thus, N∗ and Q∗ both decrease with Ig . It is reasonable that the retailer will order less quantity to reduce the holding cost. This implies that the retailer enjoys the benefit from the lower carbon tax rate, which agrees with our expectation. As to Case II in this finding, N∗ and Q∗ both increase with Ig corresponding to ci and β i remaining unchanged. We note that in the traditional EOQ model, Q∗ decreases with Ig . To our knowledge, the fewer the number of quantity orders, the less the holding cost is. However, in this case, the higher Ig is, the larger N∗ and Q∗ are. This illustrates that when the retailer orders more quantity under the same value of ci and β i , he/she should add the number of shipments to reduce the holding cost. The expected total profit is maximized through increasing N∗ and Q∗ given ci and β i remain unchanged. In summary, the optimal combinations of Q and N corresponding to ci and β I facilitate the retailer obtaining the maximum expected total profit, which is the same as (1) in the findings. (4) ETPU∗ decreases with R; while, by combining Q∗ and N∗ , three cases similar to (1) in the findings occur with R (see findings in (1) and replacing D with R). It is intuitive that the higher the receiving cost is, the larger the ordering quantity is, which agrees the result for Case I in this finding. This result matches the outcome in the traditional EOQ model. However, if the interaction effect exists for ci and β i , Cases II and III occur. In Case II in this finding, the quantity discounts effect is larger than that of taxing the carbon emissions. This result leads the retailer to order a larger quantity, and thus, a greater number of shipments are required to deliver the goods. Alternatively, for Case (III) in this finding, the quantity discounts effect is less than that of taxing the carbon emissions. The trends for Q∗ , N∗ , ci , and β i are opposite those in Case (II). We further note that ETPU∗ decreases with R for all of the above cases in this finding. This is because the higher the receiving cost is, the higher the total cost is and the less the expected total profit. 8. Conclusion This paper proposed a pull system inventory model with carbon tax policy in which a powerful retailer requests that the supplier provides a procurement cost discount rate to maintain a cooperative relationship. Two efficient algorithms were developed to find the optimal individual solutions for flat and progressive carbon taxation. The theoretical results show that, for a powerful retailer, either a flat or progressive carbon tax could present an impact on the number of shipments, order quantity and the expected total profit. However, if the retailer losses its dominant position, a single delivery could occur and thus the optimal order quantity is not affected by the flat carbon tax system. This result shows that taxing carbon emissions offers an inducement to curb emissions approaching lower operating costs, estimating the emission parameters and recognizing emission resources. From the numerical examples illustrated, we see that taking the carbon tax into the traditional EOQ model could significantly affect the manager’s decisions. The numerical examples show that (1) the number of shipments and the lot size depend on the quantity discounts and the carbon tax system; (2) the operating cost increases if the regulatory agency imposes carbon emission taxation; (3) unlike a flat carbon tax system, a progressive one provides flexibility in deciding the number of shipments and the sizes; (4) The lowest unit purchasing cost and the lowest carbon tax rate may not guarantee that the costs for the whole system will be minimized (or profit maximized) because a tradeoff relationship exists among progressive carbon emission taxation and quantity discounts; (5) a multiple delivery policy is better than a single delivery policy. The proposed model can be further enriched by considering more practical issues to match the different real world situations, such as deteriorating items, incremental quantity discounts, probability demand and different carbon emission taxation policies. Acknowledgments The authors are sincerely thankful to the Editors and the anonymous reviewers for their critical comments and constructive suggestions for the improvement of the paper. This research was partially supported by the Ministry of Science and Technology of ROC under Grant #MOST 103-2410-H-240-005.

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Algorithm 1 Finding overall optimal lot-size and the number of shipments. ˜ j from Eq. (16) and Q˜ j (N ˜ j ) from Eq. (8), where j = 1, 2, . . . , r. Step 1: Obtain N Step 2: For j = 1 to r ˜ j ) > Q j , NEXT j IF Q˜ j (N ELSE { ˜ j ) < Q j , DO { IF Q j−1 ≤ Q˜ j (N ˜ j ) from Eq. (7) and record it} {Compute EC j (Q˜ j , N {NEXT j} } } ELSE { ˜ j ) < Q j−1 , DO { IF Q˜ j (N {Obtain Nˆ j from Eq. (17) and compute EC j (Q j−1 , Nˆ j ) from Eq. (7) and record it} {Obtain N¯ j from Eq. (19) and take N j = N¯ j ; Compute Q j (N j ) from Eq. (8). If Q j−1 ≤ Q j (N j ) < Q j , { Compute EC j (Q j , N j ) from Eq. (7) and record it } ELSE { EC j (Q j , N j ) = ∞ and record it } } {Min[EC j (Q j−1 , Nˆ j ), EC j (Q j , N j )] and then record it}. {NEXT j} } } Step 3: Compare all of the expected costs recorded in Step 2. The solution corresponding to the lowest expected relevant cost provides the optimal order lot size, the number of shipments, and the unit-purchasing cost, which simultaneously minimizes the expected total cost. Substituting optimal policy into Eq. (5), the optimal expected total profit is obtained.

Algorithm 2 Developing model for restructured scheme. Step 1: Take all values occurring at break points inquantity discounts and carbon tax systems. Step 2: Arrange all values in non-decreasing order obtained from Step 1, where Q1 < Q2 < ... < Qb and b ≤ (k + l). Step 3: For g = 1 to b {Labeling Qg which belongs [Qj − 1 , Qj ) in Table 1, then record cg = cj } {Labeling Qg which belongs [Qk − 1 , Qk ) in Table 3, then record β g = β k } NEXT g Step 4: Structure the quantity-discounts regression-tax schedule as Table 3.

Appendix A Algorithm 1. Appendix B Algorithm 2. Appendix C Illustration for Example 1: Computing optimal order quantity, number of shipments and minimal total cost. ˜ j from Eq. (16) and Q˜ j from Eq. (8), we have Step 1: Obtain N

˜ j = 12, for j = 1, 2, . . . , 5 N and Q˜1 = 11485.6; Q˜2 = 11483; Q˜3 = 11483.5; Q˜4 = 11484; Q˜5 = 11484.4 Step 2: Because Q˜1 > Q1 , Q˜2 > Q2 , and Q˜3 > Q3 we directly skip them. However, the value of Q˜4 is 11969.4 and is lo˜ 4 ) from Eq. (7) and record it as EC4 (11484,11) = $903764.6 cated in the interval of [10,50 0, 14,0 0 0), we compute EC4 (Q˜4 , N (recorded). ˆ 5 ) from Eq. (7). These results We note that Q˜5 < Q(5−1 ) . Therefore, we obtain Nˆ 5 from Eq. (17) and compute EC5 (Q5−1 , N ˆ are N5 = 13 and EC5 (140 0 0,13) = $903708.3. Alternatively, we can N¯ 5 from Eq. (19) and take N5 = N¯ 5 . One has N¯ 5 = 14 Compute Q5 (14 ) from Eq. (8), we have Q5 (14 ) = 14467.7

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Because Q5 (15 ) > Q5−1 , we employ Eq. (7) to get EC5 (14467.7,14) = $903705.2









Take Min EC5 Q5−1 , Nˆ 5 , EC5 Q5 , N5



, we have

EC5 (14467.7, 14) = $903705.2 (Recorded ). Step 3: Compare all of the expected costs recorded in Step 2. We can obtain the optimal order lot size, the number of shipments and the expected total cost corresponding to the unit- purchasing cost c5 = 25.011 as follows:

Q ∗ = 14467, N ∗ = 14, EC5 (14467.7, 14) = $903705.2.

(i.e., ET PU5 (14467.7, 14) = $835688.7 corresponding to c5 = 25.011 ). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

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