# Impacts of Transportation Cost on Distribution-Free Newsboy Problems

Authors: Ming-Hung Shu; Chun-Wu Yeh; Yen-Chen Fu
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

A distribution-free newsboy problem (DFNP) has been launched for a vendor to decide a product’s stock quantity in a single-period inventory system to sustain its least maximum-expected profits when combating fierce and diverse market circumstances. Nowadays, impacts of transportation cost on determination of optimal inventory quantity have become attentive, where its influence on the DFNP has not been fully investigated. By borrowing an economic theory from transportation disciplines, in this paper the DFNP is tackled in consideration of the transportation cost formulated as a function of shipping quantity and modeled as a nonlinear regression form from UPS’s on-site shipping-rate data. An optimal solution of the order quantity is computed on the basis of Newton’s approach to ameliorating its complexity of computation. As a result of comparative studies, lower bounds of the maximal expected profit of our proposed methodologies surpass those of existing work. Finally, we extend the analysis to several practical inventory cases including fixed ordering cost, random yield, and a multiproduct condition.


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## Body

## 1. Introduction

A newsboy (newsvendor) problem has been initiated to determine the stock quantity of a product in a single-period inventory system when the product whose demand is stochastic has a single chance of procurement prior to the beginning of selling period. Aiming to maximize expected profit, decisive quantity trades off between the risk of underordering, which fails to gain more profit, and the loss of overordering, which compels release below the unit purchasing cost.Traditional models for the newsboy problem assume that a single vendor encounters the demand of a product complying with a particular probability distribution function with known parameters, such as a normal, Schmeiser-Deutsch, beta, gamma, or Weibull distribution [1]. With this assumption, several recent studies have to a certain extent succeeded in resolution of certain practical problems. For example, Chen and Ho [2] and Ding [3] analyzed the optimal inventory policy for newsboy problems with fuzzy demand and quantity discounts. Arshavskiy et al. [4] performed experimental studies by implementing the classical newsvendor problem in practice. Ozler et al. [5] studied a multiproduct newsboy problem under value-at-risk constraint; with loss-averse preferences, Wang [6] introduced a problem of multinewsvendors who compete with inventories setting from a risk-neutral supplier. When confronting myriad conditions in markets, however, in many occasions this designated distributional demand failed to best safeguard the vendor’s profit.To cope with the failure, models for the distribution-free newsboy problem (DFNP) have been broadly introduced over the past two decades. Gallego and Moon [7] first outlined a compacted analysis procedure for arranging optimal order quantities to certain inventory models, such as the single product, fixed ordering, random yield, and a multiproduct case. Alfares and Elmorra [8] further employed the procedure for the inventory model which considers shortage penalty cost. Moon and Choi [9] derived an ordering rule for the balking-inventory control model, where probability of per unit sold declines as inventory level falls below balking level. More recently, Cai et al. [10] provided measurements for deployment of multigenerational product development with the project cost accrued from different phases of a product life cycle such as development, service, and associated risks. Lee and Hsu [11] and Güler [12] developed an optimal ordering rule when an effect of advertising expenditure was reckoned on the inventory model. Kamburowski [13] presented new theoretical foundations for analyzing the best-case and worst-case scenarios. Due to prevalence of purchasing online, Mostard et al. [14] studied a resalable-return model for the distant selling retailers receiving internet orders from customers who have right to return their unfit merchandise in a stipulated period.Over the past few years, energy prices have risen significantly and become more volatile; transportation of goods has become the highest operational expense as noted by Barry [15]. Many evidences indicate that in the US inbound freight costs for domestically sourced products and imported products typically range from 2 to 4% and from 6 to 12% of gross sales, respectively, and outbound transportation costs typically average 6% to 8% of net sales. In addition, Swenseth and Godfrey [16] reported that depending on the estimates utilized, upwards of 50% of the total annual logistic cost of a product could be attributed to transportation and that these costs were going up. UPS recently announced a 4.9% increase in its net average shipping rate. Ostensibly, the expenditure of the inbound/outbound material transportation has become a critical component of a total annual logistic cost function for determining purchase quantities. Effects of transportation have gained substantial recognition in vendor-buyer coordination problems.Swenseth and Godfrey [16] unified two freight rate functions into a total annual cost function to understand their brunt on purchasing decisions. For integration of inventory and inbound/outbound transportation decisions, Çetinkaya and Lee [17] enabled an optimal inventory policy and Toptal et al. [18] carried out ideal cargo capacity and minimal costs. Toptal and Çetinkaya [19] further studied a coordination problem between a vendor and a buyer under explicit transportation consideration. More recently, Zhang et al. [20] generalized a standard newsboy model to the freight cost proportional to the number of the containers used. Toptal [21] studied exponentially/uniformly distributed demands and trucking costs. Mutlu and Çetinkaya [22] developed an optimal solution when inventory replenishment and shipment scheduling under common dispatch costs are considered.Although impacts of the transportation cost on determination of the optimal inventory quantity have become attentive, its influence on the DFNP has not been fully investigated. To bridge the gap, this paper develops analytical and efficient procedures to acquire optimal policies for the DFNP in which the transportation cost function is explicitly joined into the vendor’s expected profit structure. We borrowed the idea from the transportation management models [23] that the transportation cost is modeled as a function of delivery quantities; as a result of the computational studies, our proposed optimal-ordering rules increase lower bound of maximized expected profit as much as 4% on average, as opposed to the optimal policies recommended by Gallego and Moon [7]. Moreover, in order to determine and implement the optimal policies in practice, we perform comprehensive sensitivity analyses for the vital parameters, such as the demand mean and variance, unit cost of product, and transportation cost.Lastly, this paper is organized as follows. Section2 describes our model formulation for the DFNP in presence of transportation cost whose optimal order quantity Q* along with lower bound of maximized expected profit E(Q*) is resolved in Section 3. In Section 4, we study sensitivity analyses and comparative studies. A fixed-ordering cost case is analyzed in Section 5, while a random-yield case is considered in Section 6. In Section 7, we further contemplate a multiproduct case with budget constraint. Conclusions and Implications make up Section 8.


## 2. Model Formulation for the DFNP with Transportation Cost

For investigating impacts of the DFNP in consideration of the transportation cost, we briefly depict its model assumptions and notations used in this paper. Demand rate from a specific buyer is denoted byD, whose distribution G is unknown with mean μ and variance σ2. Note that the unknown distribution G is equal to or better off the worst possible distribution ϑ. With a product’s unit cost c, a vendor orders size of Q, which arrive before delivering to the buyer. Intuitively, in one replenishment cycle, min⁡{Q,D} units are sold with unit price p and the unsold items (Q-D)+ are salable with unit salvage value s, where s<p, where (Q-D)+ defined as the positive part of Q-D are equivalent to max{Q-D,0}. This implies Q=min⁡{Q,D}+(Q-D)+.Furthermore, we assume transportation cost is a function of the order quantityQ, denoted by tc(Q). We further assume the transportation cost is in a general form of the tapering (or proportional) function; for example, tc(Q)=a+bln⁡Q, for a, b≥0, where a and b represent fixed and variable transportation cost. Intuitively, high volume corresponds to lower per unit rate of transportation, reflecting that the inequality [tc(Q)/Q]′≤0 holds true. That is, [tc(Q)/Q]′=(b-a-bln⁡Q)/Q2≤0 or equivalently Q≥exp⁡(1-a/b), where the regulated minimal quantity level of delivery is Qs=exp⁡(1-a/b) and Q≥Qs.The assumption is based on the following observations from the existing works and UPS’s on-site data set. First off, economic trade-off for the optimal transportation cost lies between provided service level and shipped quantity [17]. Secondly, in the shipment more weight signifies larger delivery quantity and higher shipment cost [19]. Thirdly, the transportation management models proposed by Swenseth and Godfrey [16] and Toptal et al. [18] indicated that optimal shipping quantity renders minimum of the transportation cost. Finally, we display the on-site shipping data set collected from the UPS worldwide expedited service at zone 7 shown in Figure 1.Figure 1
The fitted regression model for the data set of UPS worldwide expedited service at zone 7.Now, we are ready to combat the DFNP in presence of the transportation cost. Our purpose is to decide an optimal stock quantity in a single-period inventory system for a vendor to sustain its least maximum-expected profits when encountering fierce and diverse market circumstances. First, we construct the vendor’s expected profitE(Q):
(1)E(Q)=pE(min⁡{Q,D})+sE(Q-D)+-cQ-{a+bln⁡[E(min⁡{Q,D})]}-{a+bln⁡[E(Q-D)+]}=pE(min⁡{Q,D})+sE(Q-D)+-cQ-2a-bln⁡{E(min⁡{Q,D})E(Q-D)+}.Then, according to the relationships ofmin⁡{Q,D}=D-(D-Q)+ and (Q-D)+=(Q-D)+(D-Q)+, we further rewrite (1):
(2)E(Q)=(p-s)μ-(p-s)E(D-Q)+-(c-s)Q-2a-bln⁡{[μ-E(D-Q)+][Q-μ+E(D-Q)+]}.
For developing an optimal order quantity for the vendor to sustain its lower bound of maximized expected profit E(Q), we consider G, the distribution of D, to be under the worst possible distribution ϑ. Therefore, based on Gallego and Moon’s Lemma 1 in [7], we have the lower bound of expected profit E(Q) for the vendor:
(3)E(Q)≥(p-s)μ-(p-s)×[σ2+(Q-μ)2]1/2-(Q-μ)2-(c-s)Q-2a+2bln⁡2-bln⁡{-μ2-1+Q2+2μ[σ2+(Q-μ)2]1/2}.Lemma 1 (see [7]).
Under the worst possible distributionϑ, the upper bound of expected value for the positive part of Q-D is
(4)E(D-Q)+≤[σ2+(Q-μ)2]1/2-(Q-μ)2.
Let the right-hand side term of (3) be a continuous function with respect to Q; then, first and second derivatives of E(Q) are elaborately derived as follows:
(5)dE(Q)dQ=p+s-2c2-(p-s)(Q-μ)2[σ2+(Q-μ)2]1/2-b2Q+2μ(Q-μ)[σ2+(Q-μ)2]-1/2-1-μ2+Q2+4μ[σ2+(Q-μ)2]1/2,(6)d2E(Q)dQ2=-(p-s)σ22[σ2+(Q-μ)2]3/2iii-b{(Q-μ)2(2μ3+2μ-2μQ2)[σ2+(Q-μ)2]3/2-2+2μ2-2Q2iiiiiiiiiii+4μ[σ2+(Q-μ)2]1/2iiiiiiiiiii+2μ(-1-μ2+4μQ-3Q2)[σ2+(Q-μ)2]1/2iiiiiiiiiii-8μ2(Q-μ)2σ2+(Q-μ)2iiiiiiiiiii+(Q-μ)2(2μ3+2μ-2μQ2)[σ2+(Q-μ)2]3/2}iii·{-1-μ2+Q2+4μ[σ2+(Q-μ)2]1/2}-2.
Obviously, d2E(Q)/dQ2 in (6) is not necessarily being negative. It implies that the generally explicit and analytical close form for the optimal order quantity max⁡{Q*,Qs} with the least of maximized expected profits is not available. Therefore, there is a need to develop an efficient search procedure to obtain the optimal order quantity Q* and its corresponding lower bound of maximized expected profit E(Q*).


## 3. An Efficient Solution Procedure forQ* and E(Q*)

Step
1. Start from i=0, let initial order quantity Q0=μ, and set the allowable tolerance ε, for example, the acceptable “precision” or “accuracy” selected by the decision maker for the optimal decision policy.Step
2. Perform Newton’s approach (see Hillier and Lieberman [24, pp. 555–557]) to seeking the optimal order quantity of Q.LetQi+1=Qi-(f′(Qi)/f′′(Qi)). According to (5), we set
(7)f′(Qi)=p+s-2c2-(p-s)(Qi-μ)2[σ2+(Qi-μ)2]1/2-b2Qi+2μ(Qi-μ)[σ2+(Qi-μ)2]-1/2Qi2-μ2-1+2μ[σ2+(Qi-μ)2]1/2.From (6), we set
(8)f′′(Qi)=-(p-s)σ22[σ2+(Qi-μ)2]3/2iii-b{8μ2(Qi-μ)2σ2+(Qi-μ)2-2Qi2-2+2μ2+4μ[σ2+(Qi-μ)2]1/2iiiiiiiiiii+2μ(4μQi-3Qi2-μ2-1)[σ2+(Qi-μ)2]1/2iiiiiiiiiii+(Qi-μ)2(2μ3+2μ-2μQi3)[σ2+(Qi-μ)2]3/2iiiiiiiiiii-8μ2(Qi-μ)2σ2+(Qi-μ)2}.
Stop the search when |Qi+1-Qi|≤ε, so the optimal order quantity Q* can be found at the value Qi+1.Step
3. For verifying adequacy of Newton’s method, substitute Q* into (6); if d2E(Q*)/dQ*2<0 meaning Newton’s method is satisfactory, then the final solution is Q* whose E(Q*) is the vendor’s lower bound of maximized expected profit; otherwise, go to Step 4 to perform the bisection optimization method.Step
4. Select l, a quantifiable order quantity. Start i=0 and let [Q0s,Q0*] be the initial searching interval, where Q0s=exp⁡(1-a/b) is the regulated minimal quantity level of delivery for the transportation cost tc(Q)=a+bln⁡Q and Q0*=Q*.Step
5. If |Qi*-Qis|<l, then stop; the optimal order quantity is Qi**=(Qis+Qi*)/2 along with the lower bound of maximal expected profit E(Qi**); otherwise, let Qib=(Qis+Qi*)/2.Step
6. If E(Qib)≥E(Qi*), then Qi+1*=Qib and Qi+1s=Qi*; otherwise, Qi+1*=Qi* and Qi+1s=Qib. Go back Step 5 with i=i+1.To demonstrate the efficient solution procedure for the DFNP incorporating the explicit transportation cost, a numerical example is illustrated.


### 3.1. FindingQ* and E(Q*)

A chosen product has demand meanμ=9 kg and standard deviation σ=0.5. Its unit cost is c= $3.5/kg, unit selling price p= $5/kg, and unit salvage value s= $2.5/kg. Including fuel and handling charges, on-site data of the transportation cost collected from UPS worldwide expedited service at zone 7, from Europe to Taiwan, are 0.58, 0.69, 0.77, 0.85, 0.93, 1.00, 1.06, 1.12, 1.18, 1.24, 1.31, 1.37, 1.43, 1.49, 1.55, 1.61, 1.64, 1.65, 1.66, and 1.66 for shipment weight of 1,2,…,20 kg, respectively. For clarity of description, the costs considered here are all rounded down to a 45-hundred US dollar-scale. By fitting the data through the nonlinear regression model, we have an empirical tampering function tc(Q)=0.36+0.42ln⁡Q shown in Figure 1 with R2=0.926. We conclude that the fitted function provides high fidelity to represent the actual data.Then, we follow the proposed search procedure.Step
1. From n=0 and i=0, set Q0=μ=9 and ε=10-3.Step
2. When n=1, we have Q1=Q0-(f′(Q0)/f′′(Q0))= 9.211. In this case, |Q1-Q0|> 0.001, so continue Newton’s search until reaching |Qi+1-Qi|≤0.001. Then the optimal order quantity Q*=Qi+1. The searching details are listed in Table 1.Table 1
The optimal order quantity using Newton’s optimization approach.
Iterationi
Q
i
f
′
(
Q
i
)
f
′
′
(
Q
i
)
f
′
(
Q
i
)
/
f
′
′
(
Q
i
)
Q
i
+
1
0
9
−0.695
−2.471
0.281
8.719
1
8.719
0.030
−1.729
−0.017
8.736
2
8.736
−0.014
−1.804
0.008
8.728
3
8.728
0.005
−1.772
−0.003
8.731
4
8.731
−0.000
−1.785
0.000
8.731Step
3. The optimal order quantity Q* = 8.731 (the condition d2E(Q*)/dQ*2=-1.783<0 holds true). Substituting Q* = 8.731 and known parameters into (5), we obtain lower bound of maximized expected profit E(Q*), which is $11.899. Figure 2 concavely exhibits E(Q*) with respect to a wide range of Q*.Figure 2
Illustration of the expected profit with respect to order quantityQ.


### 3.2. Models Comparison

For models comparison, we implement the DFNP based on Gallego and Moon [7] whose model does not reckon the transportation cost and perform the similar searching procedure described in Section 3. Their model obtains the optimal order quantity Q* = 8.731 with the lower bound of maximized expected profit E(Q*) = $11.752. In this case, our proposed model in consideration of the transportation cost has manifested (11.899-11.752)/11.899 = 1.2% of gains in E(Q*).


## 3.1. FindingQ* and E(Q*)

A chosen product has demand meanμ=9 kg and standard deviation σ=0.5. Its unit cost is c= $3.5/kg, unit selling price p= $5/kg, and unit salvage value s= $2.5/kg. Including fuel and handling charges, on-site data of the transportation cost collected from UPS worldwide expedited service at zone 7, from Europe to Taiwan, are 0.58, 0.69, 0.77, 0.85, 0.93, 1.00, 1.06, 1.12, 1.18, 1.24, 1.31, 1.37, 1.43, 1.49, 1.55, 1.61, 1.64, 1.65, 1.66, and 1.66 for shipment weight of 1,2,…,20 kg, respectively. For clarity of description, the costs considered here are all rounded down to a 45-hundred US dollar-scale. By fitting the data through the nonlinear regression model, we have an empirical tampering function tc(Q)=0.36+0.42ln⁡Q shown in Figure 1 with R2=0.926. We conclude that the fitted function provides high fidelity to represent the actual data.Then, we follow the proposed search procedure.Step
1. From n=0 and i=0, set Q0=μ=9 and ε=10-3.Step
2. When n=1, we have Q1=Q0-(f′(Q0)/f′′(Q0))= 9.211. In this case, |Q1-Q0|> 0.001, so continue Newton’s search until reaching |Qi+1-Qi|≤0.001. Then the optimal order quantity Q*=Qi+1. The searching details are listed in Table 1.Table 1
The optimal order quantity using Newton’s optimization approach.
Iterationi
Q
i
f
′
(
Q
i
)
f
′
′
(
Q
i
)
f
′
(
Q
i
)
/
f
′
′
(
Q
i
)
Q
i
+
1
0
9
−0.695
−2.471
0.281
8.719
1
8.719
0.030
−1.729
−0.017
8.736
2
8.736
−0.014
−1.804
0.008
8.728
3
8.728
0.005
−1.772
−0.003
8.731
4
8.731
−0.000
−1.785
0.000
8.731Step
3. The optimal order quantity Q* = 8.731 (the condition d2E(Q*)/dQ*2=-1.783<0 holds true). Substituting Q* = 8.731 and known parameters into (5), we obtain lower bound of maximized expected profit E(Q*), which is $11.899. Figure 2 concavely exhibits E(Q*) with respect to a wide range of Q*.Figure 2
Illustration of the expected profit with respect to order quantityQ.


## 3.2. Models Comparison

For models comparison, we implement the DFNP based on Gallego and Moon [7] whose model does not reckon the transportation cost and perform the similar searching procedure described in Section 3. Their model obtains the optimal order quantity Q* = 8.731 with the lower bound of maximized expected profit E(Q*) = $11.752. In this case, our proposed model in consideration of the transportation cost has manifested (11.899-11.752)/11.899 = 1.2% of gains in E(Q*).


## 4. Sensitivity Analyses and Comparative Studies

Furthermore, we apply a 24 factorial design to investigate sensitivity of parameters. They are set as follows. Let the unit selling price be p= $5/kg and the unit salvage value be s = $2.5/kg; two levels are selected for each of the four parameters; that is, mean μ∈[7,11], standard deviation σ∈[0.1,1], unit product cost c∈[3,4], and the transportation cost tc(Q)∈[0.36+0.42ln⁡Q,0.31+0.56ln⁡Q] whose selected levels are based on fitting another data set gathered from UPS’s transportation cost (worldwide express saver at zone 7, from Europe to Taiwan), US$ 0.66, 0.78, 0.88, 0.98, 1.08, 1.15, 1.23, 1.31, 1.39, 1.47, 1.55, 1.62, 1.70, 1.79, 1.87, 1.94, 2.01, 2.09, 2.17, and 2.25, respectively, for shipment weight of 1,2,3,…,20 kg.Table2 lists E(Q*) along with Q* for our proposed model in the 6th column and Gallego and Moon’s model [7] in the 7th column. First, this sensitivity analysis demonstrates significant correlations among the parameters whose simultaneous consideration is imperative for the proposed optimal policy. Moreover, in contrast to Gallego and Moon’s model, the percentages of the profit gain obtained from our proposed model are listed in the 8th column. Apparently, our proposed model outperforms Gallego and Moon’s model in every policy; especially, in the ordering policies 13 and 15, the profit advance can be more than 9.4%; on average, our proposed policy provides the return gain as much as 4% as opposed to that of the Gallego and Moon’s model.Table 2
The computational results with fixed values ofp=5 and s=2.5.
Policy
Parameters setting
Our proposed policy
Gallego and Moon [7]
Profit gain
μ
σ
c
tc
(
Q
)
E
(
Q
*
)
E
(
Q
*
)
(%)
1
7
0.4
3
0.36 + 0.42ln⁡Q
12.659(6.824)
12.454(7.300)
1.62
2
11
0.4
3
0.36 + 0.42ln⁡Q
20.470(10.863)
20.262(11.300)
1.02
3
7
0.6
3
0.36 + 0.42ln⁡Q
12.236(6.98)
12.086(7.450)
1.22
4
11
0.6
3
0.36 + 0.42ln⁡Q
20.040(10.700)
19.893(11.450)
0.73
5
7
0.4
4
0.36 + 0.42ln⁡Q
5.939(6.503)
5.533(7.082)
6.84
6
11
0.4
4
0.36 + 0.42ln⁡Q
9.736(10.516)
9.339(11.082)
4.08
7
7
0.6
4
0.36 + 0.42ln⁡Q
5.476(6.509)
5.102(7.122)
6.82
8
11
0.6
4
0.36 + 0.42ln⁡Q
9.271(10.509)
8.907(11.122)
3.93
9
7
0.4
3
0.31 + 0.56ln⁡Q
12.764(6.705)
12.411(7.300)
2.76
10
11
0.4
3
0.31 + 0.56ln⁡Q
20.511(10.754)
20.155(11.300)
1.73
11
7
0.6
3
0.31 + 0.56ln⁡Q
12.250(6.858)
11.988(7.450)
2.13
12
11
0.6
3
0.31 + 0.56ln⁡Q
19.987(10.881)
19.731(11.450)
1.28
13
7
0.4
4
0.31 + 0.56ln⁡Q
6.160(6.444)
5.561(7.082)
9.74
14
11
0.4
4
0.31 + 0.56ln⁡Q
9.889(10.435)
9.303(11.082)
5.93
15
7
0.6
4
0.31 + 0.56ln⁡Q
5.605(6.428)
5.075(7.122)
9.46
16
11
0.6
4
0.31 + 0.56ln⁡Q
9.331(10.414)
8.814(11.122)
5.54
Average
4.05In views of the impact of transportation cost on the DFNP as well as the gains elicited from our proposed policies, we then extend contemplation of the transportation cost into several practical inventory cases such as fixed ordering cost, random yield, and a multiproduct case.


## 5. The Fixed Ordering Cost Case with Transportation Cost

Let a vendor have an initial inventoryI (I≥0) prior to placing an order Q>0, where ordering cost A is fixed for any size of order. Let r denote the reorder point, known as an inventory level when the order is submitted. Let S=I+Q be end inventory level, an inventory level after receiving the order.Similarly,min⁡{S,D} units are sold; S-D units are salvaged. For an (r,S) inventory replenishment policy in consideration of the transportation cost, expected profit E(S) is constructed as
(9)E(S)=pE(min⁡{S,D})+sE(S-D)+-c(S-I)-A1[S>I]-{a+bln⁡[E(min⁡{S,D})]}-{a+bln⁡[E(S-D)+]},E(S)=(p-s)μ-(p-s)E(D-S)+-(c-s)S+cI-AI[S>I]-2a-bln⁡{[μ-E(D-S)+][S-μ+E(D-S)+]},
where I[S>I]={1ifS>I,0otherwise.According to Lemma1, the expression can be simplified as min⁡S≥I⁡{AI[S>I]+J(S)}, where
(10)J(S)=-(p-s)μ+(p-s)[σ2+(S-μ)2]1/2-(S-μ)2+(c-s)S-cI+2a-2bln⁡2+bln⁡{-μ2-1+S2+2μ[σ2+(S-μ)2]1/2}.The relationship ofS=I+Q implies that acquiring the optimal end inventory level of S for the fixed ordering cost model is equivalent to having optimal order quantity of Q for the single-product model. Clearly, because I<S, J(I)>A+J(S). For determining the optimal reorder point of r, J(r)=A+J(S) is set. Then, we have
(11)p-s2{[σ2+(r-μ)2]1/2-r}+(c-s)r+bln⁡{-μ2-1+r2+2μ[σ2+(r-μ)2]1/2}-A-p-s2{[σ2+(S-μ)2]1/2-S}-(c-s)S-bln⁡{-μ2-1+S2+2μ[σ2+(S-μ)2]1/2}=0.Furthermore, we develop a solution procedure to determine the optimal reorder point.Step
1. By performing the solution procedure for the optimal order quantity in Section 3, we first obtain Q*. Then let Q* be the end inventory level S, where I is set to be 0 for brevity.Step
2. Start i=0, set the initial reorder point r0 to be S, and determine the allowable tolerance ε for accuracy of the final result.Step
3. Perform Newton’s search (see Grossman [25, pp. 228]) to compute the optimal reorder level of r. That is, ri+1=ri-(f(ri)/f′(ri)), where
(12)f(ri)=p-s2{[σ2+(ri-μ)2]1/2-ri}+(c-s)ri+bln⁡{-μ2-1+ri2+2μ[σ2+(ri-μ)2]1/2}-A-p-s2{[σ2+(S-μ)2]1/2-S}-(c-s)S-bln⁡{-μ2-1+S2+2μ[σ2+(S-μ)2]1/2},f′(ri)=2c-p-s2+(p-s)(ri-μ)2[σ2+(ri-μ)2]1/2+b2ri+2μ(ri-μ)[σ2+(ri-μ)2]-1/2-μ2-1+ri2+2μ[σ2+(ri-μ)2]1/2.
Stop the search when |ri+1-ri|≤ε. Then, the optimal order quantity is ri+1.Step
4. The optimal policy is to order up to S units if the initial inventory is less than r and not to order otherwise.


### 5.1. An Example

Continuing the numerical example in Section3, we assume that the ordering cost is given by A= $0.3. Using the above solution procedure, we find that the optimal reorder level of r is 8.210 and the end inventory level S= 8.731.


## 5.1. An Example

Continuing the numerical example in Section3, we assume that the ordering cost is given by A= $0.3. Using the above solution procedure, we find that the optimal reorder level of r is 8.210 and the end inventory level S= 8.731.


## 6. The Random Yield Case with Transportation Cost

Suppose random variableG(Q) expresses the number of good units produced from ordered quantity Q, where each good unit being ordered or produced has an equal probability of ρ. Thus, G(Q) is a binomial random variable with mean Qρ and variance Qρq, where q=1-ρ. Let m be the price markup rate and d the discount rate, so unit selling price p=(1+m)c/ρ, and salvage value s=(1-d)c/ρ. Thus, the expected profit in (1) can be rewritten as
(13)E(Q)=pE(min⁡{G(Q),D})+sE(G(Q)-D)+-cQ-2a-bln⁡{E(min⁡{G(Q),D})E(G(Q)-D)+}=cρ{(m+d)μ-(m+d)E[D-G(Q)]+-(ρ+d-1)Q[D-G(Q)]+}-2a-bln⁡{[μ-E[D-G(Q)]+]×[Q-μ+E[D-G(Q)]+]}.
Applying Lemma 1 to this case, we have
(14)E[D-G(Q)]+≤[σ2+ρqQ+(ρQ-μ)2]1/2-(ρQ-μ)2.
Substituting the above relationship into (13), we have lower bound of the expected profit in this case. Consider
(15)E(Q)≥cρ{[σ2+ρqQ+(ρQ-μ)2]1/2-(ρQ-μ)2(m+d)μ-(m+d)×[σ2+ρqQ+(ρQ-μ)2]1/2-(ρQ-μ)2-(ρ+d-1)Q[σ2+ρqQ+(ρQ-μ)2]1/2-(ρQ-μ)2}-2a+2bln⁡2-bln⁡{[σ2+ρqQ+(ρQ-μ)2]1/22(ρQ+μ)(1-ρ)Q-σ2-ρqQ+2(μ+ρQ-Q)×[σ2+ρqQ+(ρQ-μ)2]1/2}.The right-hand side of (15) is a continuous function in terms of Q. Then, first and second derivatives of E(Q) can be derived as
(16)dE(Q)dQ=-c(m+d)2[12X-1/2(q-2μ+2ρQ)-1]-cρ(ρ+d-1)-b(2(1-ρ)(μ+2ρQ)-ρq-2(1-ρ)X1/2+ρ(μ+ρQ-Q)(q-2μ+2ρQ)X-1/2)×(2(ρQ+μ)(1-ρ)Q-σ2-ρqQ+2(μ+ρQ-Q)X1/2)-1,
where X=σ2+ρqQ+(ρQ-μ)2(17)d2E(Q)dQ2=-c(m+d)2[-ρ4(q-2μ+2ρQ)2X-3/2+ρX-1/2]-bY′Z-YZ′Z2,
where
(18)Y=2(1-ρ)(μ+2ρQ)-ρq-2(1-ρ)X1/2+ρ(μ+ρQ-Q)(q-2μ+2ρQ)X-1/2,Z=2(ρQ+μ)(1-ρ)Q-σ2-ρqQ+2(μ+ρQ-Q)X1/2,Y′=4ρ(1-ρ)-2ρ×[(1-ρ)(q-2μ+2ρQ)-ρ(μ+ρQ-Q)]X-1/2-ρ22(μ+ρQ-Q)(q-2μ+2ρQ)2X-3/2,Z′=2(1-ρ)(2ρQ+μ)-ρq-2(1-ρ)X1/2+ρ4(μ+ρQ-Q)×(q-2μ+2Q)X-1/2.
Obviously, d2E(Q)/dQ2 is not necessarily being negative.Similarly, we develop a solution procedure to find the optimal order quantity in this random yield case.Step
1. Start i=0 and Q0=μ. Set the allowable tolerance ε.Step
2. Perform Newton’s search (see Hillier and Lieberman [24], pp.555–557) to compute the optimal order quantity Q. That is, Qi+1=Qi-(f′(Qi)/f′′(Qi)), where f′(Qi) and f′′(Qi) stand for (16) and (17), respectively. Stop the search when |Qi+1-Qi|≤ε. The optimal order quantity is Qi+1.Step
3. For verifying adequacy of Newton’s method, substitute Qi+1 into (19); if d2E(Qi+1)/dQi+12<0, representing Newton’s method, is satisfactory, then the final solution is Q* = Qi+1 whose E(Q*) is the vendor’s lower bound of the maximized expected profit; otherwise, go to Step 4 to perform the bisection optimization method.Step
4. Select l, a quantifiable order quantity. Start i=0 and let [Q0s,Q0*] be the initial searching interval, where Q0s=exp⁡(1-a/b) is the regulated minimal quantity level of delivery for the transportation cost tc(Q)=a+bln⁡Q and Q0*=Q*.Step
5. If |Qi*-Qis|<l, then stop; the optimal order quantity is Qi**=(Qis+Qi*)/2 along with E(Qi**), the lower bound of maximal expected profit; otherwise, let Qib=(Qis+Qi*)/2.Step
6. If E(Qib)≥E(Qi*), then Qi+1*=Qib and Qi+1s=Qi*; otherwise, Qi+1*=Qi* and Qi+1s=Qib. Go back Step 5 with i=i+1.


### 6.1. An Example

We continue Section3. We assume that, for each unit of Q, the probability of being good is ρ=0.9. We find the optimal order quantity Q*=10.403, and the lower bound of the maximum expected profit E(Q*) is 14.573. The condition d2E(Qi+1)/dQi+12=-0.916<0 is satisfactory. In contrast, the order quantity placed on the product with perfect quality can be computed as much as 8.731, which is smaller than Q*= 10.403. Apparently, in therandom yield case the order quantity is increased to provide safeguard against a possible shortage.


## 6.1. An Example

We continue Section3. We assume that, for each unit of Q, the probability of being good is ρ=0.9. We find the optimal order quantity Q*=10.403, and the lower bound of the maximum expected profit E(Q*) is 14.573. The condition d2E(Qi+1)/dQi+12=-0.916<0 is satisfactory. In contrast, the order quantity placed on the product with perfect quality can be computed as much as 8.731, which is smaller than Q*= 10.403. Apparently, in therandom yield case the order quantity is increased to provide safeguard against a possible shortage.


## 7. The Multiproduct Case with Transportation Cost

We now study a multiproduct newsboy problem in the presence of a budget constraint, also known as the stochastic product-mixed problem [26]. Suppose that each product j, for j=1,…,N, has order quantity Qj received from either purchasing or manufacturing, where a limited budget is allocated due to the limited production capacity in the system. That is, the total purchasing or manufacturing cost for all the N competing products cannot exceed allotted budget B. Denote that each item’s unit cost of the jth product is cj, its unit selling price is pj, and its unit salvage value is sj. For the jth product’s demand, its mean and variance are denoted by μj and σj2, respectively.In the sequel, under the distribution-free demand jointed with the explicit transportation cost, the vendor is in need of deciding the optimal order quantities forN competing products whose total purchasing or manufacturing cost does not exceed the allocated budge B, where he/she guarantees to possess the least of all possible maximum expected profits.For solving this problem, we first extend the single product case in (3) to have lower bound of expected profit E(Q1,…,QN) for the vendor, provided that the individual order quantity of Q1, Q2,…, and QN is affected by the budget constraint B. For the vendor to secure the least amount of the maximum expected profit over various situations of market, we maximize (19) with a budget constraint expressed in (20) to determine the optimal order quantities Q1*, Q2*,…, and QN*:
(19)max⁡Q1,…,QN∑j=1N{(pj-sj)(Qj+μj)-[σj2+(Qj-μj)2]1/22-(cj-sj)Qj-2a+2bln⁡2-bln⁡{[σj2+(Qj-μj)2]1/2-μj2-1+Qj2+2μ[σj2+(Qj-μj)2]1/2}(Qj+μj)-[σj2+(Qj-μj)2]1/22},(20)Subject to∑j=1NcjQj≤B.We further transfer the problem into an unconstrained optimization equation:(21)L(Q1,…,QN,λ)=∑j=1N{(pj-sj)(Qj+μj)-[σj2+(Qj-μj)2]1/22-(cj-sj)Qj-2a+2bln⁡2-bln⁡{[σj2+(Qj-μj)2]1/2-μj2-1+Qj2+2μj[σj2+(Qj-μj)2]1/2}(Qj+μj)-[σj2+(Qj-μj)2]1/22}+λ(∑j=1NcjQj-B),
where λ is the Lagrange multiplier. Hence, we have
(22)∂L(Q1,…,QN,λ)∂Qj=pj+sj-2cj2-(pj-sj)(Qj-μj)2[σj2+(Qj-μj)2]1/2-b2Qj+2μj(Qj-μj)[σj2+(Qj-μj)2]-1/2-1-μ2+Qj2+4μj[σj2+(Qj-μj)2]1/2+λcj.
To find the optimal order quantities Q1*, Q2*,…, and QN* with maximum L, we set ∂L/∂Qj=0. In this case, a line search procedure is developed.Step
1. For multiple products N, let j={1,…,N}.Step
2. Let λ=0 and perform the solution procedure proposed in Section 3 to find Qj*. If (20) is satisfied, go to Step 6; otherwise, go to Step 3.Step
3. Substituting each of Q1*, Q2*,…, and QN* into (22), their corresponding λ can be obtained.Step
4. Start from the smallest nonnegative λ, let its corresponding optimal order quantity be 0 (others are intact), and check the condition of (20).Step
5. If the condition is satisfactory, then we have the final solution Q1*, Q2*,…, and QN*; otherwise, select the next smallest nonnegative λ to perform the same procedure in Step 4 until (20) is satisfied.Step
6. Find the least amount of the maximum expected profit E(Q1*,…,QN*).


### 7.1. An Example

The total budget is $80 for the four items. The relevant data are as follows:c = (3.5, 2.5, 2.8, 0.5), p = (5, 4, 3.2, 0.6), s = (2.5, 1.2, 1.5, 0.2), μ = c(9, 8, 12, 23), and σ = c(0.5, 1, 0.7, 1). Performing the above procedure, we have the following.Step
1. Let j={1,2,3,4}.Step
2. Let λ=0. We solve the four order quantities by using the solution procedure introduced in Section 3. The optimal order quantities Q1* = 8.731, Q2* = 7.762, Q3* = 11.072, and Q4* = 21.243. Check ∑j=14cjQj*= $92 > $80, where (20) is not satisfied, so we go to Step 3.Step
3. Performing a simple line search, we increase the optimal value of the Lagrangian multiplier until λ= 0.147. In this case, its corresponding Q3* is set to 0.Step
4. Since ∑j=14cjQj*= $61 < $80, (20) is satisfied.Step
5. The optimal order quantities are 8.731, 7.762, 0, and 21.243 and the lower bound of the maximum expected profit is $21.667.


## 7.1. An Example

The total budget is $80 for the four items. The relevant data are as follows:c = (3.5, 2.5, 2.8, 0.5), p = (5, 4, 3.2, 0.6), s = (2.5, 1.2, 1.5, 0.2), μ = c(9, 8, 12, 23), and σ = c(0.5, 1, 0.7, 1). Performing the above procedure, we have the following.Step
1. Let j={1,2,3,4}.Step
2. Let λ=0. We solve the four order quantities by using the solution procedure introduced in Section 3. The optimal order quantities Q1* = 8.731, Q2* = 7.762, Q3* = 11.072, and Q4* = 21.243. Check ∑j=14cjQj*= $92 > $80, where (20) is not satisfied, so we go to Step 3.Step
3. Performing a simple line search, we increase the optimal value of the Lagrangian multiplier until λ= 0.147. In this case, its corresponding Q3* is set to 0.Step
4. Since ∑j=14cjQj*= $61 < $80, (20) is satisfied.Step
5. The optimal order quantities are 8.731, 7.762, 0, and 21.243 and the lower bound of the maximum expected profit is $21.667.


## 8. Conclusions and Implications

Models for the distribution-free newsboy problem have been widely introduced over the past two decades to provide the optimal order quantity for securing the vendor with the least amount of the maximum expected profit when facing a variety of situations in modern business environment.Over the past few years, energy prices have risen significantly so that the transportation of goods has become a vital component for the vendor’s logistic-cost function to determine its required purchase quantities. However, impacts of the transportation cost on previous models for the DFNP were inattentive by either overlooking or deeming it as part of implicit components of ordering cost. In this paper, three main contributions along with their managerial implication have been done.First, we develop the DFNP incorporating the explicit transportation cost into the expected profit function. In particular, the transportation cost is modeled based on the economic theory from transportation disciplines and fitted a nonlinear regression via actual rate data collected from the shipper. In practice, this way has implied that (1) economic trade-off for the optimal transportation cost lies between provided service level and shipped quantity; (2) in the shipment, more weight signifies larger delivery quantity and higher shipment cost; and (3) optimal shipping quantity renders minimum of the transportation cost.Secondly, since the expected profit function is neither concave nor convex, the optimization problem underlying this generalization is challenging; therefore, we developed analytical and efficient procedures to acquire the optimal policy. As a result of the computational studies, our proposed optimal ordering rules in comparison with the optimal policy recommended by Gallego and Moon [7] increased the lower bound of the maximal expected profit by as much as 4% on average. This result has demonstrated that the expenditure of the inbound/outbound material transportation has become a critical component of a total annual logistic cost function for determining purchase quantities. Effects of transportation have gained substantial recognition in the DFNP.Thirdly, according to the results of sensitivity analyses, the parameters, such as demand mean and variance, product’s unit cost, and transportation cost, are the key decision variables whose joint reckoning is imperative for the optimal policy proposed. Moreover, we proceed to analyses of several practical inventory cases including fixed ordering cost, random yield, and multiproduct case. These studies further demonstrate the impacts of transportation cost as well as the realized-least profit gains drawn from our recommended policies on the DFNP that explicitly incorporates the transportation cost into consideration. In addition, these numerical findings have implied that joint decision, coordinated operation, or integrated management is crucial in lowering the vendor-and-buyer operating cost as well as balancing a supply-chain operation and structure.Finally, based on the shipping data sets collected from United Parcel Service (UPS), the transportation cost is modeled using a natural logarithm for a nonlinear regression function in this paper. For future studies, other functional forms may be reckoned to model different transportation costs, such as a step function or a logistic function, to validate a wide variety of applications. Besides, using our proposed model as a basis model in a couple of more advanced studies with certain circumstances, such as the multiproduct newsboy under a value-at-risk and the multiple newsvendors with loss-averse preferences, is intriguing.



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*Source: 307935-2014-09-30.xml*