Time and quantity dependent waiting costs in a newsvendor problem with backlogged shortages

Mathematical and Computer Modelling 47 (2008) 60–71 www.elsevier.com/locate/mcm

Time and quantity dependent waiting costs in a newsvendor problem with backlogged shortages Emmett J. Lodree Jr. a,∗ , Yeonjung Kim b , Wooseung Jang b a Department of Industrial and Systems Engineering, Auburn University, Auburn, AL 36849, USA b Department of Industrial and Manufacturing Systems Engineering, University of Missouri-Columbia, Columbia, MO 65211, USA

Received 22 December 2006; accepted 7 February 2007

Abstract Upon demand realization in the newsvendor problem, it is often assumed that shortages result in lost sales penalties. However, in some practical situations, shortages are backlogged and the inventory manager is penalized based on the magnitude and duration of the shortage. In this paper, we investigate a variation of the newsvendor problem in which all shortages are backlogged and replenished through an emergency procurement process. Costs incurred during emergency procurement include a variable emergency ordering cost and a non-linear customer waiting cost. We derive closed form expressions for the optimal order quantity and emergency procurement rate for two special cases, and we propose a heuristic approximation for the general case. The effectiveness of the approximate solution is assessed through numerical experiments and sensitivity analysis. c 2007 Elsevier Ltd. All rights reserved.

Keywords: Inventory; Customer responsiveness; Variable lead time; Direct shipping

1. Introduction Consider the classic newsvendor facing random demand X with distribution and density functions F(x) and f (x), respectively, who orders Q items from his supplier at unit cost c. Upon demand realization X = x, the newsvendor incurs unit cost h for excess inventory and unit cost ψ for shortages. The cost function K (Q) that minimizes expected costs due to ordering, overages, and shortages is commonly expressed as follows in the literature (for example, see [1]) Z Q Z ∞ K (Q) = Qc + h (Q − x) f (x)dx + ψ (x − Q) f (x)dx. (1) 0

Q

Lodree et al. [2] introduced a variation of (1) in which all shortages are backlogged and the newsvendor resolves shortages through an emergency procurement at unit cost g( p) > c, where p is the amount of time required to replenish exactly one backlogged item and g( p) is a convex and decreasing function for all p > 0. The procurement ∗ Corresponding author. Tel.: +1 334 844 1433; fax: +1 334 844 1381.

E-mail address: [email protected] (E.J. Lodree Jr.). c 2007 Elsevier Ltd. All rights reserved. 0895-7177/$ - see front matter doi:10.1016/j.mcm.2007.02.018

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lead time, p, can be attributed to the time associated with producing a new item, locating and retrieving an existing item from a warehouse, or packaging an item for shipment. While it is intuitive that the unit cost function g( p) should be decreasing, the convexity requirement is a technical assumption that enables some analysis presented later in this paper. The emergency procurement process is also characterized by an impatient customer who imposes a unit cost ω upon the newsvendor for each time unit he has to wait for backorders to be resolved. Lodree et al. [2] treat both Q and p as decision variables resulting in the following expected cost function. K (Q, p) = Ordering cost + expected overage cost + expected emergency procurement cost + expected waiting cost Z ∞ Z ∞ Z Q ωp(x − Q) f (x)dx. (x − Q)g( p) f (x)dx + h(Q − x) f (x)dx + = Qc + 0

Q

(2)

Q

The expected cost function K (Q, p) given by (2) is based on a linear waiting cost that is directly proportional to the number of shortages, x − Q, in which all backlogged items arrive p(x − Q) time units after demand realization. In this paper, we introduce a non-linear waiting cost function based on an impatient customer who wishes to resolve all outstanding backorders immediately after demand realization and the newsvendor has the option of replenishing outstanding backorders through partial shipments. The expected cost function (2) and the model introduced later in this paper can be classified as a newsvendor problem with opportunity for a secondary replenishment and variable backorder lead time (the reader is referred to [3] and [4] for several other variations of the newsvendor problem). The secondary replenishment newsboy problem (referred to as “mid-period replenishment” in the literature) seems to have been first introduced in [5]. Other early models of this type are mentioned in [6]. More recently, [7] determined the profit maximizing order quantity for a single period model with an emergency supply option and showed that this quantity is smaller than the solution to the newsvendor model. Lau and Lau [8,9] considered a variation of the newsvendor model with two ordering opportunities during a single season and determined the profit maximizing order quantities for both orders. Other than [2], it seems that a variable backorder lead time has not been addressed in the newsvendor setting. However, several researchers have studied inventory models in which both order quantity and lead time are treated as decision variables in continuous review stochastic EOQ (Economic Order Quantity) environments. The first seems to be [10], who developed a procedure for determining the optimal order quantity and lead time that minimize expected costs when demand is normally distributed. The framework introduced by [10] has been extended to include mixtures of backorders and lost sales [11,12], service level constraints [13,14], variable reorder point [15] quantity discounts [16,17], general demand distribution [18], and autoregressive demand [19]. Other variations of the framework introduced in [10] include Refs. [20–32]. The remainder of this paper is arranged as follows. The mathematical model and its properties are introduced in Section 2. In Section 3, closed form solutions are derived for two special cases. A heuristic approximation for the general case is described in Section 4 along with sensitivity analysis. Section 5 briefly explores the effect of varying the number of shipments during the backorder fulfillment process and Section 6 closes the paper with a summary and some concluding remarks. 2. Model formulation To motivate our non-linear, time and quantity dependent waiting cost function, consider again our impatient customer who wishes to resolve all backlogged shortages immediately after demand realization. If the number of shortages is x − Q = 10, for example, an appropriate waiting cost function should distinguish between the following two scenarios: (a) 2 items delivered to the customer 1 time unit after demand realization and the remaining 8 items delivered 15 time units after demand realization; (b) 8 items delivered 1 time unit after demand realization and the remaining 2 items delivered 15 time units after demand realization. In particular, scenario (b) is preferable to the impatient customer because more backlogged shortages are resolved soon after demand realization compared to scenario (a). To quantify the difference between the two above scenarios, let tk represent the number of time units after demand realization that the customer receives the k-th partial shipment and Vk the number of items received by the customer

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at time tk . Then if s is the total number of partial shipments, an appropriate waiting cost function has the following form for some well-defined function W . Waiting cost function = W (t1 V1 + t2 V2 + · · · + ts Vs ).

(3)

For illustrative purposes, let W (t1 V1 + t2 V2 + · · · + ts Vs ) = ω · (t1 V1 + t2 V2 + · · · + ts Vs ) .

(4)

Assuming ω = $1 per unit time, the waiting costs for each scenario of our above numerical example are W(a) = $1(1 × 2 + 15 × 8) = $122 W(b) = $1(1 × 8 + 15 × 2) = $38. Since p represents the amount of time required to procure or prepare an item for shipment, the k-th shipment can be dispatched no earlier than p(V1 + · · · + Vk ) time units after demand realization. Assuming that the k-th shipment is dispatched immediately after Vk items have been prepared for shipment, we have tk = p · (V1 + · · · + Vk ) .

(5)

Substituting (5) into (4) and simplifying, we obtain W = ωp

s X

(V1 + · · · + Vk ) Vk = ωp

k=1

s X k=1

Vk2

+

s−1 X s X

! Vi V j .

(6)

i=1 j=i+1

Since the emergency procurement process occurs after demand realization, the shipment sizes Vk that minimize W given by (6) can be determined by solving the following integer non-linear programming problem. INLP Minimize: W

(7)

Subject to: V1 + · · · + Vs = x − Q

(8)

Vk ≥ 0 and integer for k = 1, . . . , s.

(9)

Note that the constraint (8) ensures that the total number of items delivered to the customer during emergency procurement is the same as the number of shortages and constraint (9) requires shipment sizes be non-negative integers. In this paper, we use shipment sizes associated with solving a relaxation of INLP in which the shipment sizes Vk are not restricted to integer values. Theorem 1. The solution to the relaxation of INLP such that Vk for k =1, . . . , s are not restricted to integer values is Vk =

x−Q , s

for k = 1, . . . , s.

(10)

The proof to Theorem 1 is presented in [33] and is based on the Karush–Kuhn–Tucker conditions. Now substituting (10) into (6) and simplifying yields   s+1 W = · ωp(x − Q)2 . (11) 2s By replacing ωp(x − Q) in Eq. (2) with the expression for waiting costs given by (11), the new expected cost function becomes Z Q ωp(s + 1) K (Q, p) = cQ + h (Q − x) f (x)dx + g( p)I1 (Q) + I2 (Q) (12) 2s 0 R∞ R∞ where I1 (Q) = Q (x − Q) f (x)dx and I2 (Q) = Q (x − Q)2 f (x)dx. We consider the following notes and assumptions regarding (12) and the subsequent analysis:

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1. All variables and parameters are non-negative. 2. While a general (unit) expedite procurement cost g( p) is used whenever possible, we often use g( p) = M/ p as a special case, where M is proportional to the item value. Note that g( p) = M/ p satisfies the convex and decreasing requirement discussed in Section 1. 3. We assume that s is fixed and appropriately determined by managers, considering the vehicle capacity and backorder quantity. However, at the end of this paper we present a methodology for computing the optimal value of s. 4. We assume that each shipment size is equal to (x − Q)/s from Theorem 1 when expedite production is necessary. That is, Eq. (12) does not rigorously consider the effect of requiring integer valued shipment sizes. 2.1. Analysis of the expected cost function We wish to determine the values Q ∗ and p ∗ that minimize the total expected cost function (12). The standard procedure is to examine the convexity of (12) to determine whether or not a unique minimum exists and then to solve the system of equations resulting from the first order conditions (provided the function is indeed convex). We present an approach that establishes the convexity for a single variable at a time rather than the Hessian matrix of two variables simultaneously, as described below. 1. First, determine the optimal p ∗ in terms of Q assuming that p is the only decision variable and Q is a fixed given value. 2. Second, substitute p ∗ , a function of Q, into the original cost function and optimize it, assuming now Q is the only decision variable. If both functions are convex, the resulting solution is optimal, provided an optimal solution exists. Applying this method, first assume that K (Q, p) in (12) is a single variable function in terms of p. Observe that ∂ K (Q, p) ω(s + 1) = g 0 ( p)I1 (Q) + I2 (Q) ∂p 2s and ∂ 2 K (Q, p) = g 00 ( p)I1 (Q). ∂ p2 Because g( p) is a convex function, K (Q, p) is a convex function in terms of p for a fixed Q value. Hence, the optimal value p ∗ satisfies g0 ( p∗ ) = −

ω(s + 1) I2 (Q) · . 2s I1 (Q)

Define G( p) ≡ g 0 ( p) and G −1 ( p) as the inverse function of G( p). Then the optimal emergency procurement lead time for a given order quantity Q is given as follows.   ω(s + 1) I2 (Q) · . (13) p ∗ = G −1 − 2s I1 (Q) For our procurement cost function g( p) = M/ p, the optimal value is given as   2Ms I1 (Q) 1/2 ∗ p = · . ω(s + 1) I2 (Q) We obtain the expected cost function in terms of Q by substituting p ∗ given by Eq. (14) into Eq. (12) to obtain   Z Q Mω(s + 1) I2 (Q) 1/2 I1 (Q) + h (Q − x) f (x)dx K (Q, p ∗ ) = cQ + · 2s I1 (Q) 0   ω(s + 1) 2Ms I1 (Q) 1/2 + · I2 (Q) 2s ω(s + 1) I2 (Q)

(14)

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Z = cQ + h

Q

(Q − x) f (x)dx + ξ

p

I1 (Q)I2 (Q)

(15)

0

√ √ where ξ = 2Mω(s + 1)/s. Unfortunately, the second derivative of I1 (Q)I2 (Q) involves the subtraction of two non-negative functions, and hence, the convexity of (15) cannot be analytically determined for a general demand distribution. However, this function is often convex for common distributions such as exponential and uniform distributions as shown below. We later develop a heuristic method to estimate the optimal Q ∗ when the analytic solution is difficult to obtain. 3. Special cases In this section, we utilize the above results to derive the closed form optimal solution p ∗ and Q ∗ for exponential and uniform demand. 3.1. Exponential demand Using the probability density function of an exponential distribution f (x) = λe−λx , straightforward integration yields Z ∞ 1 (x − Q) f (x)dx = e−λQ I1 (Q) = λ Q and I2 (Q) =

Z

∞

(x − Q)2 f (x)dx = Q

2 −λQ e . λ2

Hence, the optimal expedite processing time in (13) becomes   ω(s + 1) ∗ −1 p =G − . λs

(16)

Note that the optimal processing time p ∗ is independent of the production quantity Q. The objective function K (Q, p ∗ ) becomes convex in Q after substituting p ∗ , and the optimal order quantity satisfies  ωp ∗ (s + 1)  ∂ K (Q, p ∗ ) = c + g( p ∗ )I10 (Q) + h 1 − e−λQ + I20 (Q) = 0. ∂Q 2s Because I10 (Q) = −e−λQ and I20 (Q) = −2e−λQ /λ, the optimal order quantity is given by   ∗ 
ωp (s + 1)/λs + g ( p ∗ ) + h ωp ∗ (s + 1) 1 ln , if + g p∗ > c ∗ Q = λ c+h λs  0, otherwise

(17)

where p ∗ is the optimal expedite processing time in (16). From (17), the following reasonably intuitive insights can be interpreted. • Because the mean of exponential distribution is 1/λ, the optimal order quantity increases linearly with the average demand. • The optimal order quantity also increases with customer impatience and with emergency procurement costs. • The optimal order quantity decreases with regular ordering cost. • The optimal order quantity decreases with s, but not significantly. • The optimal order quantity decreases with inventory holding cost. • If ωp/λ + g( p) < c, which is unlikely in practice (see [34]), then the optimal policy is strictly make-to-order as opposed to a combination of make-to-order and make-to-stock.

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For the case when g( p) = M/ p, we obtain  1/2 λMs p∗ = . ω (s + 1) So the optimal lead time decreases with the expected demand and customer impatience. Substituting p ∗ and g( p ∗ ) into (8) yields the optimal order quantity Q ∗ for g( p) = M/ p and exponentially distributed demand such that    √ p ξ 2/λ + h 1 , if ξ 2/λ > c ln ∗ Q = λ (18) c+h  0, otherwise √ where ξ = 2Mω(s + 1)/s. The insights gained from observing the optimal order quantity given by Eq. (18) are consistent with the above-mentioned insights associated with the optimal production quantity given by Eq. (17). In addition, Eq. (18) also reveals that the optimal order quantity increases with item value, although not very significantly. 3.2. Uniformly distributed demand In this section, we apply the same approach for determining Q ∗ and p ∗ in closed form when demand is uniformly distributed. As before, we first determine p ∗ and then compute Q ∗ by substituting p ∗ . We then verify that these values are optimal by showing the convexity of the objective function. Using the probability density function of a uniform random variable f (x) = 1/(b − a), straightforward integration yields Z ∞ (b − Q)2 I1 (Q) = (x − Q) f (x)dx = 2(b − a) Q and I2 (Q) =

Z

∞

(x − Q)2 f (x)dx = Q

(b − Q)3 . 3(b − a)

Hence, the optimal expedite processing time in (13) becomes   ω(s + 1)(b − Q) p ∗ = G −1 − . 3s When g( p) = M/ p,  1/2 3Ms p∗ = . ω(s + 1)(b − Q)

(19)

The expected cost function K (Q, p ∗ ) then becomes Z Q q ∗ K (Q, p ) = cQ + h (Q − x) f (x)dx + ξ (b − Q)5 /6(b − a)2 , 0

and it can be shown that this function is convex in Q. Therefore, the optimal order quantity satisfies   ∂ K (Q, p ∗ ) Q−a 5ξ(b − Q)3/2 =c+h − √ = 0. ∂Q b−a 2 6(b − a) After some algebraic simplification, the above equation becomes Q3 + x Q2 + y Q + z = 0 where x = 24h 2 /25ξ 2 − 3b, y = 48h(cb − ca − ha)/25ξ 2 + 3b2 , and z = 24(cb − ca − ha)2 /25ξ 2 − b3 . By solving this equation, the optimal order quantity, in general, becomes q q  p p 3 3 ∗ 3 2 3 2 Q = max v + u + v + v − u + v ,0 (20)

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Fig. 1. Relationship between the expected cost function and its bounds.

where u = (−x 2 + 3y)/9 and v = (−2x 3 + 9x y − 27z)/54. Sensitivity analysis (and some tedious algebra) reveals that the optimal solution given by Eqs. (9) and (20) results in managerial insights that are consistent with those discussed in Section 3.1. 4. A heuristic method Because the optimal order quantity Q ∗ cannot be determined analytically in many cases, it is necessary to develop an appropriate heuristic method to solve the problem for a general demand distribution. The approach we design uses upper and lower bounding functions of K (Q, p ∗ ). Let Z Q K A (Q) = cQ + h (Q − x) f (x)dx + ξ I1 (Q) 0

and K B (Q) = cQ + h

Z

Q

(Q − x) f (x)dx + ξ I2 (Q).

0

Lemma 1. The functions K A and K B are lower and upper bounding functions of K (Q, p ∗ ). That is, for any Q, K A (Q) ≤ K (Q, p ∗ ) ≤ K B (Q)

or

K B (Q) ≤ K (Q, p ∗ ) ≤ K A (Q).

Proof. Note first that I1 (Q) ≥ 0 and I2 (Q) ≥ 0. If I1 (Q) ≤ I2 (Q), then p I1 (Q) ≤ I1 (Q)I2 (Q) ≤ I2 (Q) ⇔ K A (Q) ≤ K (Q, p ∗ ) ≤ K B (Q). If I1 (Q) > I2 (Q), then p I1 (Q) ≥ I1 (Q)I2 (Q) ≥ I2 (Q) ⇔ K A (Q) ≥ K (Q, p ∗ ) ≥ K B (Q). This completes the proof.



Using Lemma 1 and the fact that K A (Q) and K B (Q) are indeed convex functions, the relationship among K (Q, p ∗ ), K A , and K B can be depicted as Fig. 1 assuming that the expected cost function K (Q, p ∗ ) may not be convex. In this situation, it is likely that the optimal order quantity Q ∗ is somewhere between Q ∗A and Q ∗B , where Q ∗A and Q ∗B are the values that minimize K A (Q) and K B (Q), respectively. Therefore, the following approximation is proposed for estimating the optimal quantityQ ∗ .  ∗ (Q A + Q ∗B )/2, if Q ∗A + Q ∗B > 0 Q˜ ∗ = (21) 0, otherwise where Q˜ ∗ is our heuristic approximation of Q ∗ .

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We now discuss approaches for computing Q ∗A and Q ∗B . Because both K A (Q) and K B (Q) are convex in Q, Q ∗A and Q ∗B can be computed using the first derivatives. That is, we can determine Q ∗A and Q ∗B by solving the following two equations: Z Q Z ∞ dK A (Q) =c+h f (x)dx − ξ f (x)dx = 0 dQ 0 Q and dK B (Q) =c+h dQ

Q

Z

f (x)dx − 2ξ I1 (Q) = 0.

0

Hence, Q ∗A is determined as follows, leading to the standard newsboy result. Z Q∗ A ξ −c . f (x)dx = ξ +h 0 For a given demand distribution, Q ∗B can be often computed analytically. Even if it is not possible, a bisection procedure can be used to numerically determine Q ∗B thanks to the convexity of the function K B (Q). Once Q˜ ∗ is obtained, the corresponding procurement lead time is determined by Eq. (14). 4.1. Analytic validation We validate our approximation (23) analytically by applying it to exponentially distributed demand. In this case, it is possible to derive closed form expressions for both Q ∗A and Q ∗B as follows.     ξ +h 2ξ/λ + h 1 1 Q ∗A = ln and Q ∗B = ln . λ c+h λ c+h By substituting these values into (21), we obtain  √  p (ξ + h)(2ξ/λ + h) 1 ln , if (ξ + h)(2ξ/λ + h) > c + h Q˜ ∗ = λ c+h  0, otherwise.

(22)

Next in Lemma 2, we establish error bounds for Q˜ ∗ by comparing it to the optimal value Q ∗ in (18). The following property is needed to prove Lemma 2. Property 1. The function θ (y) = y/ ln y is increasing in y when y ≥ e. Lemma 2. Q ∗ ≤ Q˜ ∗ ≤ (1 + h/ξ )Q ∗ when

√ ξ 2/λ+h c+h

≥ e and λ ≤ 2.

Proof. Comparing Q˜ ∗ in (13) to the optimal value Q ∗ in (9), we observe that p p Q ∗ ≤ Q˜ ∗ ⇔ ξ 2/λ + h ≤ (ξ + h)(2ξ/λ + h) p ⇔ 2hξ 2/λ ≤ hξ(1 + 2/λ). (23) √ 2 ∗ ∗ Because (1 − 2/λ) ≥ 0, the last inequality in (23) is always true, which proves that Q ≤ Q˜ . We can also observe that √   √  Q˜ ∗ (ξ + h)(2ξ/λ + h) ξ 2/λ + h = ln ln Q∗ c+h c+h √ √  √ (ξ + h)(2ξ/λ + h) ξ 2/λ + h (ξ + h)(2ξ/λ + h) = ≤ √ c+h c+h ξ 2/λ + h √ (ξ + h)(2ξ/λ + h) ( 2/λ − 1)2 hξ ≤ =1+ √ √ (ξ 2/λ + h)2 (ξ 2/λ + h)2 2hξ/λ h ≤ 1+ 2 =1+ . (24) ξ 2ξ /λ

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Table 1 Experimental design factors Factor

Low

High

c ω µ

20 1 100

50 3 1000

Table 2 Results of numerical experiment High factors

p∗

Q∗

Q˜ ∗

Error (%)

None c ω µ cω cµ ωµ cωµ

0.39 0.61 0.22 0.12 0.35 0.19 0.07 0.11

162 117 217 2770 171 2313 3319 2861

165 121 219 2801 174 2362 3337 2890

1.75 3.78 0.77 1.14 1.52 2.12 0.56 1.01

The first inequality is true from the first condition of Lemma 2 and Property 1. The second inequality holds because √ the numerator is larger than the denominator√ (i.e., the value is larger than 1). The third one is true because 2/λ ≥ ( 2/λ − 1)2 in the numerator and 2ξ 2 /λ ≤ (ξ 2/λ + h)2 in the denominator. Note that we need a condition that 1/λ ≥ 1/2, which is obvious because 1/λ represents the mean demand in the exponential distribution. This completes the proof.  From Eq. (23), we can identify conditions that the approximation (22) is the optimal solution. That is, we see that Q˜ ∗ = Q ∗ if and only if λ = 2, h = 0, or ξ = 0 (or equivalently, M = 0 or ω = 0). If h = 0, emergency procurement is not necessary and the problem becomes trivial. If ξ = 0 (i.e., emergency procurement or unit waiting costs are negligible), regular procurement would be unnecessary and the situation would again be trivial. From these observations, we can also claim that our approximation will perform well when the values of such cost parameters are small. When these cost parameters have large values, Lemma 2 provides the worst case error bound. Note the condition √ given in the lemma, (ξ 2/λ+h)/(c+h) ≥ e, often holds because mean demand is large for many practical problems, and is particularly true when the value of ξ (or equivalently, M or ω) is large. In this case, the value of h/ξ becomes small, and hence, we have a tight error bound of our approximation. However, it is very difficult to analytically estimate the magnitude and trend of typical errors because the values of h, M, and ω are often dependent upon the value of c and move in tandem. Hence, some numerical tests are needed for this purpose. 4.2. Numerical experiment The following screening experiment allows us to numerically gauge the effectiveness of our approximation. We consider the three factors shown in Table 1 for the experimental design. In addition, we assume the following. • It is assumed that M = c. • One season is equivalent to a three-month period and the inventory cost for one season is 2.5% of c. • It is assumed that s = 3 and the vehicle capacity is large enough to fulfill the demand with the specified number of vehicles. Our experiment tests eight example problems using the 23 factorial design. The results are presented in Table 2. According to the experiment results, the approximation is quite accurate for the tested eight problems: the average error is 1.58% and the maximum error is less than 4%. However, we can also observe that errors tend to become larger whenever ordering cost is high. In order to verify this phenomenon and identify the potentially worst case scenario, we expand our numerical tests to examples with larger ordering costs. The results, shown in Fig. 2, illustrate the

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69

Fig. 2. Analysis of the worst case scenarios.

deteriorating accuracy of our approximation. As expected, the error becomes larger as c increases and the largest error in our analysis reaches over 35%. Therefore, it is recommended to use an optimal solution rather than a heuristic approximation (if possible) for an expensive item with low order quantity. On the other hand, our heuristic method yields good approximations in many cases, and works particularly well for inexpensive items with large order quantities. In addition, we observe from Table 2 that the optimal and approximate order quantities are very large relative to average demand. The reason for this is that penalty costs for over-ordering include only linear inventory holding costs whereas the costs for under-ordering include both waiting and emergency procurement costs. Because of this and the fact that our waiting cost function is non-linear, it is reasonable to expect that solutions will tend to suggest over-ordering as opposed to under-ordering. 5. Extension to variable shipment frequency In this section, we extend the numerical experiment to determine the optimal number of shipments s ∗ . To do this, the expected cost function (21) is transformed into the following equation, which now includes a linear shipping cost A associated with each of the sshipments. Z Q p K (Q, p ∗ , s) = cQ + h (25) (Q − x) f (x)dx + ξ I1 (Q)I2 (Q) + As. 0

We perform numerical tests using the data in the fourth row of Table 1, that is, when the only high factor is the mean, and the results are presented in Fig. 3. It suggests that the expected cost function (25) is convex, which is not surprising because the function (12) is convex when demand is an exponential random variable. Using Fig. 3 as a visual aid, an extensive search was performed to determine s ∗ corresponding to A = 10, 50, 100, and 200 as follows: 31, 14, 10, and 7 respectively. It is intuitive to use a smaller number of shipments as the shipping cost increases. This is confirmed in our tests, and we can also observe a non-linear relationship between the shipping cost and frequency. In many applications, it is more realistic to determine the number of shipments first with non-negligible shipping costs. In such cases, the above procedure can be used to determine the optimal number of shipments, which is then used to make other decisions as discussed in the previous sections. 6. Conclusion This paper addresses an extension of the newsvendor problem in which all shortages are backlogged and the inventory manager incurs waiting costs proportional to the time it takes to fulfill backlogged demand. Lead time during the backorder fulfillment process is treated as a variable and results in variable emergency procurement costs. Thus the expected cost function is composed of regular procurement costs that are independent of backorder lead time, a backorder procurement (or production) cost that is dictated by variable lead time, waiting costs associated with

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Fig. 3. Expected cost versus number of shipments.

fulfilling backorders, and holding costs for ordering excess items. A previous model in the literature that addresses this problem is based on a linear waiting cost function in which the optimal order quantity and backorder procurement lead time can be expressed in closed form. This paper introduces a non-linear waiting cost function characterized by multiple shipments during the emergency procurement process. However, it is not possible to obtain the optimal solution in closed form for a general demand distribution. Therefore, we derive closed form solutions for both exponential and uniform demand distributions, and we propose a heuristic approximation that can be applied to any demand distribution. The effectiveness of the approximation is assessed analytically and numerically through sensitivity analysis. Finally, we extend the model by introducing transportation costs that are proportional to the number of shipments used to fulfill backlogged demand and by also considering the number of shipments as a decision parameter. A straightforward procedure is then described that enables the calculation of the optimal number of shipments corresponding to a given order quantity and lead time. References [1] F.S. Hillier, G.J. Lieberman, Introduction to Operations Research, 8th ed., McGraw-Hill, Boston, 2004. [2] E. Lodree, C. Klein, W. Jang, Minimizing customer response time in a two-stage supply chain system with variable lead time and stochastic demand, International Journal of Production Research 42 (11) (2004) 2263–2278. [3] E.A. Silver, D.F. Pyke, R. Peterson, Inventory Management and Production Planning and Scheduling, 3rd ed., Wiley, 1998. [4] M. Khouja, The single period (news-vendor) problem: Literature review and suggestions for future research, Omega 27 (1999) 537–553. [5] E.W. Barankin, A delivery-lag inventory model with emergency provision, Naval Research Logistics Quarterly 8 (1961) 285–311. [6] E.L. Porteus, Stochastic inventory theory, in: Stochastic Models, in: Handbooks in Operations Research, vol. 2, Elsevier, New York, 1990. [7] M. Khouja, A note on the newsboy problem with emergency supply option, Journal of the Operational Research Society 47 (1996) 1530–1534. [8] A.H.-L. Lau, H.-S. Lau, Reordering strategies for a newsboy-type product, European Journal of Operational Research 103 (1997) 557–572. [9] A.H.-L. Lau, H.-S. Lau, A semi-analytic solution for a newsboy problem with mid-period replenishment, Journal of the Operational Research Society 48 (1997) 1245–1253. [10] M. Ben-Daya, A. Raouf, Inventory models involving lead time as a decision variable, Journal of the Operational Research Society 45 (1994) 579–582. [11] L.Y. Ouyang, N.C. Yen, K.S. Wu, Mixture inventory model with backorders and lost sales for variable lead time, Journal of the Operational Research Society 47 (1996) 829–832. [12] S.W. Lan, P. Chu, K.-J. Chung, W.-J. Wan, R. Lo, A simple method to locate the optimal solution of the inventory model with variable lead time, Computers and Operations Research 26 (1999) 599–605. [13] L.Y. Ouyang, K.S. Wu, Mixture inventory model involving variable lead time with a service level constraint, Computers and Industrial Engineering 24 (9) (1997) 875–882.

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