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Managerial and economic impacts of reducing delivery variance in the supply chain
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Applied Mathematical Modelling 32 (2008) 2149–2161 www.elsevier.com/locate/apm
Managerial and economic impacts of reducing delivery variance in the supply chain Alfred L. Guiffrida
a,*
, Mohamad Y. Jaber
b
a
b
Department of Management and Information Systems, Kent State University, Kent, OH 44242, USA Department of Mechanical and Industrial Engineering, Ryerson University, Toronto, Ontario, Canada M5B 2K3 Received 1 December 2005; received in revised form 1 December 2006; accepted 4 July 2007 Available online 19 July 2007
Abstract This paper addresses the managerial and economic impacts of improving delivery performance in a serial supply chain when delivery performance is evaluated with respect to a delivery window. Building on contemporary management theories that advocate variance reduction as the critical step in improving the overall performance of a system, we model the variance of delivery time to the final customer as a function of the investment to reduce delivery variance and the costs associated with untimely delivery (expected earliness and lateness). A logarithmic investment function is used and the model solution involves the minimization of a convex–concave total cost function. A numerical example is provided to illustrate the model and the solution procedure. The model presented provides guidelines for determining the optimal level of financial investment for reducing delivery variance. The managerial implications as well as the economic aspects of delivery variance reduction in supply chain management are discussed. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Improving delivery performance; Justifying variance reduction; Supply chain management
1. Introduction Modern markets are competitive business environments where customers require their suppliers to be dependable in delivering on-time shipments. One of the important goals of supply chain management is to improve delivery performance [1,2]. Results from recently published empirical studies have identified delivery performance as a key management concern among supply chain practitioners (see for example [3–5]). A conceptual framework for defining delivery performance in supply chain management is found in Gunasekaran et al. [6]. This suggested framework classifies delivery performance as a strategic level supply chain performance measure while delivery reliability is viewed as a tactical level supply chain performance measure. The framework of Gunasekaran et al. [6] also advocates that to be effective, supply chain management tools, delivery performance and delivery reliability need to be measured in financial (as well as non-financial) terms. *
Corresponding author. Tel.: +1 716 954 3504; fax: +1 716 631 2635. E-mail address: aguiff[email protected] (A.L. Guiffrida).
0307-904X/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2007.07.006
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The failure to financially quantify delivery performance in a supply chain presents both short-term and long-term difficulties. In the short term, the buyer–supplier relationship may be negatively impacted. A norm value of ‘‘presumed’’ performance is established by default when delivery performance is not formally measured [7]. This norm value stays constant with time and is generally higher than the organization’s actual delivery performance. It has been demonstrated that supplier evaluation systems have a positive impact on the buyer–supplier relationships, with these relationships ultimately have a positive impact on financial performance [8]. In the long term, failure to measure supplier delivery performance in financial terms may impede the capital budgeting process, which is necessary in order to support the improvement of supplier operations within a supply chain. Delivery lead time is defined to be the elapsed time from the receipt of an order by the originating supplier in the supply chain to the receipt of the product ordered by the final customer in the supply chain. Delivery lead time is composed of a series of internal (manufacturing and processing) lead times at each stage plus the external (distribution and transportation) lead times found at various stages of the supply chain. Fig. 1 illustrates the delivery lead time components of a serial supply chain. Early and late deliveries introduce waste in the form of excess cost into the supply chain; early deliveries contribute to excess inventory holding costs, while late deliveries may contribute to production stoppages costs, lost sales and loss of goodwill. To protect against untimely deliveries, supply chain managers often inflate in process inventory levels and production flow buffers. These actions can contribute to excess operating and administrative costs [9]. An extensive review of available delivery evaluation models by Guiffrida [10] identified several shortcomings in modeling delivery performance. These concerns are threefold. First, delivery performance measures are not cost-based. Second, delivery performance measures ignore variability. Third, delivery performance measures often fail to take into account the penalties associated with both early and late deliveries. The inability to translate delivery performance into financial terms which incorporates uncertainty as well as realistically quantifying delivery timeliness (early as well as late delivery) hinders management’s ability to justify capital investment for continuous improvement programs, which are designed to improve delivery performance. The current research presented herein attempts to overcome these limitations through the development of a cost-based model that incorporates delivery variability and assigns penalties for both early and late deliveries. In this paper, we develop a cost-based performance metric for evaluating delivery performance and reliability to the final customer in a serial supply chain that is operating under a centralized management structure. Contemporary management theories advocate the reduction of variance as a key step in improving the performance of a system [11–13]. In union with these prevailing theories, delivery performance is modeled as a cost-based function of the delivery variance. The financial benefit of investing to reduce variability in delivery performance is demonstrated. This paper is organized as follows. In Section 2, an analytical model based on the expected costs associated with untimely delivery is developed and propositions are introduced to analyze the model in terms of the variance. In Section 3, improvement in delivery performance is modeled using a logarithmic investment function. The reduction of deliver variance is formulated and solved as a convex mathematical optimization problem. In Section 4, a heuristic solution algorithm is introduced and the two solution algorithms are compared. Managerial implications of investing to reduce delivery variance are addressed in Section 5. In the concluding section, we summarize the findings of this research and present directions for future research.
Supplier 1
Supplier 2
…
Supplier n
Delivery Lead Time
Fig. 1. An n-stage serial supply chain.
Final Customer
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2. Model development Consider an n-stage serial supply chain operating under a centralized management structure where an activity at each stage contributes to the overall delivery time to the final customer. The activity duration of stage i, Wi, is defined by probability density function fW(w, h) that Pn is reproductive under addition with respect to parameter set h. Delivery time to the final customer, X ¼ i¼1 W i , is defined by the probability density func P tion fX x; ni¼1 hi . We assume that activity durations at each stage of the supply chain are normally distributed and independent. Normality and independence among stages is often assumed in the literature [14,15]. When activity times at stage i are normally distributed and independent h = {l, v}. Delivery time Pn to the Pnfinal customer is defined as a result of the n-fold convolution of fW(w, h) which yields X N x; i¼1 li ; i¼1 vi where li and vi are the delivery mean and variance for stage i, i 2 [1, n] . Delivery to the final customer is analyzed with regard to the customer’s specification of delivery timeliness as defined by a delivery window. Delivery windows are an effective tool for modeling the expected costs associated with untimely delivery. Jaruphongsa et al. [16], Lee et al. [17], and Corbett [18] advocate the use of delivery windows in time-based manufacturing systems. Johnson and Davis [19] note that metrics based on delivery (order) windows capture the most important aspect of the delivery process, which is reliability. Under the concept of a delivery window, the customer supplies an earliest allowable delivery date (a) and a latest allowable delivery date (b). A delivery window is defined as the difference between the earliest acceptable delivery date and the latest acceptable delivery date. The customer and the end supplier responsible for product delivery contractually agree that no deliveries will arrive before a nor after b, thereby establishing a mutually agreeable framework for managing delivery performance. If a shipment is delivered after b or before a it will be rejected and therefore returned to the supplier. Within the delivery window, a delivery may be classified as early, on-time, or late (see Fig. 2). Delivery lead time, X, is a random variable with probability density function fX(x). The on-time portion of the delivery
f X (x)
x a
c1
Early
b
c2
On-Time Delivery Scenario
Legend: a = earliest delivery time c1 = beginning of on-time delivery c 2 = end of on-time delivery b = latest acceptable delivery time Fig. 2. Illustration of delivery window.
Late
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window is defined by c2 c1. Ideally, c2 c1 = 0. However, the extent to which c2 c1 > 0 may be measured in hours, days, or weeks depending on the industrial situation. It is assumed in this paper that: (1) delivery performance is stable enough so that the modal delivery time is within the on-time portion of the delivery window, (2) the mean and on-time portion of the delivery window remain fixed, and (3) the coefficient of variation of X is less than 0.25 thereby limiting the probability of a negative delivery time under the normal density to essentially zero. An alternative to assumption (3) would be to truncate the normal density to prevent nonnegative delivery times or select symmetrical and nonsymmetrical density functions defined for only positive values of delivery time as was done in Guiffrida [10,20]. Consider an n-stage serial supply chain in operation over a time horizon of length T years, where a demand requirement of the final customer for a single product of D units will be met with a constant delivery lot size Q. The expected penalty cost per delivery period for untimely delivery, Y, is Z c1 Z b Y ¼ QH ðc1 xÞfX ðxÞdx þ K ðx c2 ÞfX ðxÞdx; ð1Þ a
c2
where Q constant delivery lot size per cycle H inventory holding cost per unit per time K penalty cost per time unit late (levied by the final customer) a, b, c1, c2 parameters defining the delivery window duration of delivery lead time component j ðj ¼ 1; 2; . . . ; nÞ Wj fX ðxÞ ¼ fW 1 þW 2 þþW n density function of delivery time X It is a common purchasing agreement practice to allow the buyer (final customer) to charge suppliers for untimely deliveries (see for example [21,22]). Reductions in early deliveries reduced inventory holding costs at Hewlett-Packard by $9 million [23]. It has been reported in the automotive industry Saturn levies fines of $500 per minute against suppliers who cause production line stoppages [24] and that Chrysler fines suppliers $32,000 per hour when an order is late [25]. The penalty cost in these cases is an opportunity cost due to lost production. Purchasing managers often view the production disruptions caused by delivery stockouts to be more widespread and more costly than the lost sales that stockouts cause [26]. Hence, K has been defined as an opportunity cost due to lost production as described by Frame [24] and Russell and Taylor [25]. Evaluating (1) under the defined assumptions yields the total expected penalty cost (see Appendix A for derivation) pffiffiffi c1 l pffiffiffi c2 l c1 l c2 l Y ¼ QH v/ pffiffiffi þ ðc1 lÞ U pffiffiffi þ K v/ pffiffiffi ðc2 lÞ 1 U pffiffiffi : v v v v ð2Þ Proposition 1. The expected penalty cost is a monotonically increasing non-convex function of the variance. Proof. The first and second derivatives of Y with respect to the variance are respectively
ffiv þ K/ c2pl ffiv QH / c1pl pffiffiffi Y 0 ðvÞ ¼ 2 v and
ð3Þ
QH c1 l K c2 l 2 2 p ffiffi ffi p ffiffi ffi / lÞ vÞ þ / lÞ vÞ : ð4Þ ððc ððc 1 2 4v5=2 4v5=2 v v Examining (3), the expected penalty cost is an increasing function of the variance since Y 0 (v) > 0 for positive 2 2 values of Q, H, K and v. Examining (4), we note that Y00 (v) > 0 when v 6 minfðc1 lÞ ; ðc2 lÞ g, but 2 2 Y00 (v) < 0 when v P maxfðc1 lÞ ; ðc2 lÞ g, hence the expected penalty cost is not a convex function of the variance. Y 00 ðvÞ ¼
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Proposition 2. The expected penalty cost is a monotonically increasing locally convex function of the variance provided v < minfðc1 lÞ2 ; ðc2 lÞ2 g. Proof. Examining (4), Y00 (v) > 0 when 0 < v < min{(c1 l)2, (c2 l)2}.
h
3. Modeling improvement in delivery performance Ideally the expected costs incurred for untimely delivery should be equal to zero. This implies that, for the currently defined delivery window, all deliveries are within the on-time portion of the delivery window, and that waste in the form of early and late deliveries has been eliminated from the system. For a fixed mean and delivery window, the expected penalty cost can be reduced by reducing the variance of the delivery distribution. The expected penalty cost (1) can be expressed as a function of the delivery period variance, as pffiffiffi c1 l pffiffiffi c2 l c1 l c2 l Y ðvÞ ¼ QH v/ pffiffiffi þ ðc1 lÞU pffiffiffi þ K v/ pffiffiffi ðc2 lÞ 1 U pffiffiffi : v v v v ð5Þ Reducing the variance of delivery can be achieved by initiating improvements such as: (1) gaining tighter control over process flow times at downstream stages of the supply chain, (2) enhanced coordination of freight transport, (3) more efficient material handling of inbound and outbound stock, and (4) improved communications among stages in the supply chain such as implementing electronic data interchange (EDI). Initiating such improvements requires capital investment (see for example, [27,28]). An optimization model which considers the delivery variance as a decision variable is defined. The objective of the model is to determine the variance level that minimizes the costs (expected earliness and lateness) associated with untimely delivery and the investment cost required for reducing the delivery variance. The optimization problem is Minimize
GðvÞ ¼ Y ðvÞ þ CðvÞ;
ð6Þ
where v Y(v) C(v)
variance of delivery distribution expected penalty cost due to untimely delivery investment required for a delivery variance of v
A logarithmic investment cost function is used to model the cost of reducing the delivery variance [29]. Under this investment function, reducing the delivery variance by a fixed percentage requires a fixed amount of investment. This functional form is appealing in that each additional reduction in the delivery variance is more costly than the previous reduction. The logarithmic investment function has been widely adopted in the literature (see for example [30–32]). Let v0 equal the current value of delivery variance and k represent the cost of reducing the delivery variance by h percent. The investment function is then CðvÞ ¼
k ½lnðv0 Þ lnðvÞ lnð1=1 hÞ
for 0 6 v 6 v0 :
ð7Þ
Proposition 3. The investment function is a decreasing convex function of the delivery variance. Proof. The first and second derivatives of (7) are C 0 ðvÞ ¼
k lnð1=1 hÞv
ð8Þ
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and C 00 ðvÞ ¼
k : lnð1=1 hÞv2
ð9Þ
We observe that C 0 (v) < 0 and C00 (v) > 0 for k > 0 and h > 0. h Substituting (5) and (7) into (6), yields the optimization model pffiffiffi c1 l c1 l Minimize GðvÞ ¼ QH v/ pffiffiffi þ ðc1 lÞ U pffiffiffi v v pffiffiffi c2 l c2 l k ½lnðv0 Þ lnðvÞ: þ K v/ pffiffiffi ðc2 lÞ 1 U pffiffiffi þ lnð1=1 hÞ v v ð10Þ 2
2
As per Propositions 1 and 2, Y(v) is a convex function of the variance for v 6 minfðc1 lÞ ; ðc2 lÞ g and a concave function when v P maxfðc1 lÞ2 ; ðc2 lÞ2 g. For minfðc1 lÞ2 ; ðc2 lÞ2 g < v < maxfðc1 lÞ2 ; ðc2 lÞ2 g, the convexity–concavity of Y(v) is parameter specific. Hence, G(v) is the sum of a convex–concave function Y(v) and a convex function C(v). Theorem 1. G(v) is a convex function of the delivery variance provided v 6 minfðc1 lÞ2 ; ðc2 lÞ2 g.
Proof. The first and second derivatives of (10) with respect to the variance are
ffiv þ K/ c2pl ffiv QH / c1pl k pffiffiffi G0 ðvÞ ¼ lnð1=1 hÞv 2 v
ð11Þ
and QH c1 l K c2 l k 2 2 G ðvÞ ¼ 5=2 / pffiffiffi ððc1 lÞ vÞ þ 5=2 / pffiffiffi ððc2 lÞ vÞ þ : 4v 4v lnð1=1 hÞv2 v v 00
2
ð12Þ
2
By direct substitution G00 (v) > 0 for v 6 minfðc1 lÞ ; ðc2 lÞ g hence G(v) is convex when v 6 2 2 minfðc1 lÞ ; ðc2 lÞ g. h Corollary 1. G(v) is minimized at G*(v*) provided v* 6 vm. 2
2
Proof. Let vm ¼ v 6 minfðc1 lÞ ; ðc2 lÞ g. As per Theorem 1, G(v) is a convex function of the delivery variance for v 6 vm, hence the solution of G 0 (v) = 0 yields an optimal solution if v* 6 vm. h The value of v* that minimizes G(v) at G*(v*) does not exist in closed form and must be found by numerically solving G 0 (v) = 0 for v = v*. Solving (10) for v takes the general form of pffiffiffi c1 l c2 l 2k ¼ 0: v QH / pffiffiffi þ K/ pffiffiffi lnð1=ð1 hÞÞ v v
ffi exp z22 gives Introducing /ðzÞ ¼ p1ffiffiffi 2p " ! !# pffiffiffiffiffiffi 2 2 pffiffiffi ðc1 lÞ ðc2 lÞ 2k 2p ¼ 0: v QH exp þ K exp lnð1=ð1 hÞÞ 2v 2v
ð13Þ
ð14Þ
If the on-time portion of the delivery window is symmetric then d = c2 l = (c1 l) and (14) simplifies to pffiffiffiffiffiffi 2 pffiffiffi d 2k 2p ¼ 0: ð15Þ v exp lnð1=ð1 hÞÞðQH þ KÞ 2v
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Fig. 3. Optimal solution to numerical illustration with v < vm.
And a closed form solution for (15) is v¼
d2 2 ; d lnð1=ð1 hÞÞðQH þ KÞ p ffiffiffiffiffi ffi WL 2k 2p
ð16Þ
where WL is the Lambert W function [33]. 3.1. Numerical example Consider a supply chain where delivery time (in days) to the final customer is normally distributed with a mean of 50 and variance of 10. The on-time portion of the delivery window is defined by c1 = 48 and c2 = 53. For these parameters vm = 4 and v0 = 10. Additional parameters for the expected penalty cost model are: Q = 500, H = $10 and K = $5000. Under the logarithmic form of the investment function, a cost of k = $150 is incurred for every h = 10% reduction in the delivery variance. Solving (10) for v yields v = 3.253. As per Corollary 1 of Theorem 1, v = 3.253 < vm = 4, hence v = v* = 3.253 and G*(v*) = $2387 (see Fig. 3). 4. Heuristic solution algorithm 2
2
When v P maxfðc1 lÞ ; ðc2 lÞ g, minimizing G(v) requires the minimization of the sum of a concave function Y(v) and a convex function C(v). There is no guarantee that solving G 0 (v) = 0 will yield the optimal value of the variance. A lower bound on the optimal value of G(v) can be obtained by approximating Y(v) as a 2 2 linear function for minfðc1 lÞ ; ðc2 lÞ g < v < v0 . The range of v for which the linear approximation has been defined includes values of the variance where the sign of G00 (v) is parameter specific. Extending the range of the variance in this fashion eliminates the need to conduct parameter specific analyses when adopting the linear approximation procedure. Let vL and vU represent values of the delivery variance such that vm 6 vL < vU 6 v0 . A linear approximation of Y(v) is defined as Y ðvU Þ Y ðvL Þ Ye ðvÞ ¼ Y ðvL Þ þ ð17Þ ðv vL Þ: vU vL When Y00 (v) < 0, a lower bound on G(v) is then defined by e GðvÞ ¼ Ye ðvÞ þ CðvÞ:
ð18Þ
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e Theorem 2. GðvÞ is a convex function of the delivery variance for vm < v < v0. Proof. As per Proposition 3, C00 (v) > 0 thus C(v) is a convex function of the variance. By inspection (17) is a e linear (and hence convex function) of the variance. GðvÞ is the sum of two convex functions and is therefore convex. h e e ð~v Þ. Corollary 2. GðvÞ is minimized at G e Proof. As per Theorem 2, GðvÞ is a convex function of the delivery variance for vm < v < v0 and is minimized at value of the variance, ~v , that solves f G0 ðvÞ ¼ 0. e GðvÞ is minimized at ~v ¼
kðvU vL Þ . lnð1=1 hÞ½Y ðvU Þ Y ðvL Þ
h
A heuristic procedure is defined for determining the variance level that minimizes the costs (expected earliness and lateness) associated with untimely delivery and the investment cost required for reducing the delivery variance. STEP 1. Input model parameter values: c1 ; c2 ; l; Q; H ; K; h; k; STEP 2. Initialize search indices: j = 0 (iteration counter); e = 0.01 (accuracy tolerance); v0 ; vm ¼ minfðc1 lÞ2 ; ðc2 lÞ2 g. STEP 3. 1. If v0 6 vm solve G 0 (v) = 0 for v = v* where the minimum cost is G*(v*), THEN GOTO STEP 8. 2. ELSE GOTO STEP 4. STEP 4. e vÞ < minfGðvL Þ; GðvU Þg, THEN SET 1. Set vL ¼ vm ; vU ¼ v0 and solve f G0 ðvÞ ¼ 0 for v ¼ ~v. If Gð~ L L U L U U vj;1 ¼ v ; vj;1 ¼ vj;2 ¼ ~v; vj;2 ¼ v , compute the minimum cost as G ðv Þ ¼ minfGðvL Þ; GðvU Þg; GOTO STEP 5. 2. ELSE GOTO STEP 8. STEP 5. Solve f G0 ðvÞ for v ¼ ~v and f G0 ðvÞ for v ¼ ~v ; GOTO STEP 6. j;1
j;1
j;2
j;2
STEP 6. e vÞ minf G e j;1 ð~vj;1 Þ; G e j;2 ð~vj;2 Þgj < e (no improvement is possible), the minimum cost is G ðv Þ ¼ 1. If j Gð~ e Gð~vÞ; GOTO STEP 8. 2. ELSE GOTO STEP 7. STEP 7. e j;1 ð~vj;1 Þ < G e j;2 ð~vj;2 Þ set Gð~ e vÞ ¼ G e j;1 ð~vj;1 Þ, j ¼ j þ 1; vL ¼ vL ; vU ¼ vL ¼ ~vj1;1 ; vU ¼ vU . 1. If G j;1 j1;1 j;1 j;2 j;2 j1;1 L U e e 2. ELSE set Gð~vÞ ¼ G j;2 ð~vj;2 Þ; j ¼ j þ 1; vLj;1 ¼ vLj1;2 ; vU vj1;2 ; vU j;1 ¼ vj;2 ¼ ~ j;2 ¼ vj1;2 . 3. GOTO STEP 5. STEP 8. STOP e j;1 ð~vj;1 Þ < G e j;2 ð~vj;2 Þ set Gð~ e vÞ ¼ G e j;1 ð~vj;1 Þ; j ¼ j þ 1; vL ¼ vL ; vU ¼ vL ¼ ~vj1;1 ; vU ¼ vU ; and Step 6. If G j;1 j1;1 j;1 j;2 j;2 j1;1 e e go to step 4; else set Gð~vÞ ¼ G j;2 ð~vj;2 Þ, j = j + 1.
4.1. Numerical example Consider a supply chain where delivery time (in days) to the final customer is normally distributed with a mean of 50 and variance of 20. The on-time portion of the delivery window is defined by c1 = 46 and c2 = 53.
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For these parameters vm = 9 and v0 = 20. Additional parameters for the expected penalty cost model are: Q = 500, H = $10 and K = $5000. Under the logarithmic form of the investment function, a cost of k = $650 is incurred for every h = 10% reduction in the delivery variance. The solution to this numerical example is summarized in Table 1. e vÞ ¼ $5610:49. Reducing The algorithm terminated after two iterations at v ¼ ~v ¼ 18:551 and G ðv Þ ¼ Gð~ the delivery variance from v0 = 20 to v ¼ ~v ¼ 18:551 improved the probability of an early delivery improved from 0.2148 to 0.1762 and the probability of late delivery from 0.2514 to 0.2420. For the parameter values selected in this numerical example, the cost function G(v) = Y(v) + C(v) appears to be flat in the neighborhood of v* = 18.551. This suggests that the non-convexity in the expected penalty cost term Y(v) may be slight and e the linear approximating Ye ðvÞ provides a close fit between G(v) = Y(v) + C(v) and GðvÞ ¼ Ye ðvÞ þ CðvÞ. 0 This assertion can be tested by solving G (v) = 0 despite the fact that Theorem 5.1 is violated for the parameters used in this numerical example. Solving, G 0 (v) = 0 yields v* = 18.580 and G*(v*) = $5610.49 which corresponds to a zero percentage cost penalty in comparison to the heuristic solution procedure. Solving e 0 ðvÞ ¼ 0 can be solved by directly by a G 0 (v) = 0 requires use of nonlinear programming software, while G 0 e closed form expression. Solving G ðvÞ ¼ 0 directly (as is done in Step 1 of the heuristic algorithm) yields e vÞ ¼ $5611:88. Hence, continuing the solution heuristic until termination v ¼ ~v ¼ 18:151 and G ðv Þ ¼ Gð~ led to a 0.02% improvement in the total cost. Alternatively, the easily solved closed form expression for
Table 1 Solution to numerical example using logarithmic investment heuristic algorithm Step Iteration j = 0 3
Description 1. Is v0 = 20 < vm = 9? 2. No, go to Step 4
4
1. vL ¼ 9 vU ¼ 20; ~v ¼ 18:151 GðvL Þ ¼ $6811:89 GðvU Þ ¼ $5624:29 Gð~vÞ ¼ $5611:88 Is Gð~vÞ ¼ $5611:88 minfGðvL Þ ¼ $6811:89; GðvU Þ ¼ $5624:29g < 0:01? v ¼ 18:151; vL0;2 ¼ ~v ¼ 18:151; vU No, set vL0;1 ¼ vL ¼ 9; vU 0;1 ¼ ~ 0;2 ¼ v0 ¼ 20 and go to Step 5
5
~v0;1 ¼ 18:050 ~v0;2 ¼ 18:667
6
e 0;2 ð~v0;2 Þ ¼ $5610:54gj < 0:01? e vÞ ¼ $5611:88 minf G e 0;1 ð~v0;1 Þ ¼ $5611:89; G 1. Is j Gð~ 2. No, go to Step 7
7
e vÞ ¼ G e 0;2 ð~v0;2 Þ ¼ $5610:54; j ¼ 0 þ 1 ¼ 1 2. Set Gð~ vL1;1 ¼ vL0;2 ¼ 18:151; vU v0;2 ¼ 18:667 1;1 ¼ ~ U vL1;2 ¼ ~v0;2 ¼ 18:667; vU 1;2 ¼ v0;2 ¼ 20 3. Go to Step 5
Iteration j = 1 5
~v11 ¼ 18:551 ~v12 ¼ 18:713
6
e 1;2 ð~v1;2 Þ ¼ $5610:62gj < 0:01? e vÞ ¼ $5610:54 minf G e 1;1 ð~v1;1 Þ ¼ $5610:49; G 1. Is j Gð~ 2. No, go to Step 7
7
e vÞ ¼ G e 1;1 ð~v1;1 Þ ¼ $5610:49; j ¼ 1 þ 1 ¼ 2 2. Set Gð~ vL2;1 ¼ vL1;1 ¼ 18:151; vU v1;1 ¼ 18:551 2;1 ¼ ~ U vL2;2 ¼ ~v1;1 ¼ 18:551; vU 2;2 ¼ v1;1 ¼ 18:667 3. Go to Step 5
Iteration j = 2 5
~v2;1 ¼ 18:541 ~v2;2 ¼ 18:585
6
e 2;2 ð~v2;2 Þ ¼ $5610:49gj < 0:01? e vÞ ¼ $5610:49 minf G e 2;1 ð~v2;1 Þ ¼ $5610:50; G 1. Is j Gð~ Yes, go to Step 8
8
STOP, The minimum cost solution is G*(v*) = $5610.49
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e 0 ðvÞ ¼ 0 resulted in a solution that differed by 0.02% in comparison to the heuristic and nonlinear programG ming software solutions. 5. Managerial implications of delivery variance reduction In this section, we address the managerial implications of improving delivery performance through the reduction of delivery variance. An optimization model has been introduced to demonstrate the tradeoffs between investment in reducing the variance of delivery and the expected costs that accrue as a result of untimely delivery. The modeling effort herein has provided a conceptual framework for linking financial investment and delivery improvement. This framework is similar in characteristic to process improvement espoused in the quality management literature where process improvement is predicated on the reduction of process variability. We contend that logistics and supply chain managers can use the delivery variance reduction framework to improve delivery performance in similar fashion to how quality assurance managers have historically used the reduction in process variation to improve product quality. Improvement in delivery performance has both immediate and long term benefits. For a given level of investment, the optimal amount of delivery variance reduction can be estimated. For a fixed mean and on-time portion of the delivery window the most obvious effect of delivery variance reduction is the improvement of delivery performance. As the delivery variance is reduced the probability of on-time shipments increases and the probability of early and late shipments decreases. The improved delivery performance can lead to increased business from existing customers. Also, improved delivery performance can be used a basis of competitive advantage to gain new customers whose delivery standards could not be met prior to the reduction of delivery variance. The model can be used to identify the investment cost required to attract new customers based on the customers’ requirements of delivery timeliness. The optimization model presented in this paper determines the optimal variance level that minimizes the costs due to untimely delivery and the investment cost required for reducing the variance. This modeling framework can be adapted to integrate with the capital budgeting process for justifying funds for process improvement. The static optimization problem defined in (6) can be stated as a temporal model of the form Gðv; tÞ ¼ Y ðv; tÞ þ Cðv; tÞ over time horizon T, 0 6 t 6 T. The present worth of the cost stream G(v, t) provides as estimate in current dollars of cost to be incurred over time budgeting horizon T. The present worth of this cost stream provides a benchmark that can be used within the capital budgeting process for justifying the investment required to improve delivery performance.
6. Conclusions This paper addressed the economic and managerial implications of improving delivery performance to the final customer in a serial supply chain from the contemporary perspective of reducing delivery variability. A cost-based model has been presented that financially evaluates the effects of reducing delivery variability on overall delivery performance. The model was demonstrated under a logarithmic investment function and a framework for determining optimal and near optimal levels of delivery variance was developed. As demonstrated through a numerical example, the results of this modeling may prove useful to justifying the investment required to reduce delivery variance to a targeted goal as a part of an overall continuous improvement program to improve supply chain delivery performance. An efficient heuristic solution algorithm for determining the variance level that minimizes the costs was also developed in this paper. Logistic and supply chain managers can utilize the variance reduction framework set forth in this paper to gain both short and long term benefits. Reducing the variability of shipment delivery times will improve overall delivery performance in the short term which will increase customer satisfaction among the existing customer base. In the long term, less variability in delivery can lead to new customers whose delivery requirements can now be met. There are several aspects of this research that could be expanded. An optimization model could be used to optimally allocate the reduction in delivery variance to the downstream stages of the supply chain. Second, an industrial case study utilizing the model developed herein could be conducted. Third, disruptions in the var-
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iance reduction process could be investigated. Lastly, the assumption of independence of the activity times among the stages could be investigated. Acknowledgements The authors are grateful to the two anonymous referees and to the Editor of the Journal for their helpful comments and suggestions. Appendix A. Derivation of expected penalty cost (Eq. (2)) The density function of delivery time X with mean l and variance v, is the convolution of the normally distributed lead time components W i ði ¼ 1; 2; . . . ; nÞ; fX ðxÞ ¼ fW 1 þW 2 þþW n . If an earliest delivery time (a) and latest acceptable delivery time (b) are imposed on fX(x), then fX ðxÞ hX ðxÞ ¼ R b f ðxÞ a X
ðA:1Þ
and (1) is Y ¼ QH
Z
c1
ðc1 xÞhX ðxÞdx þ K a
Z
b
ðx c2 ÞhX ðxÞdx:
ðA:2Þ
c2
Examining (A.2), we observe that Y is separable in terms of the expected earliness cost and the expected lateness cost. A.1. Expected lateness cost The expected penalty cost for late delivery is Z K b Y late ¼ ðx c2 ÞfX ðxÞdx; p c2
ðA:3Þ
where p¼
Z
b
fX ðxÞdx; "Z ( ) ( ) # Z b 2 2 b K x ðx lÞ c2 ðx lÞ pffiffiffiffiffiffiffiffi exp pffiffiffiffiffiffiffiffi exp Y late ¼ dx dx : p c2 2pv 2v 2v 2pv c2 pffiffiffi pffiffiffi pffi ; x ¼ vz þ l, and dx ¼ v dz into (A.5) and simplifying yields Substituting z ¼ xl v " Z bl # Z bl pffi v K pffiffiffi pffiv z 1 v c l pffiffiffiffiffiffi expfz2 =2gdz þ ðl c2 Þ c l pffiffiffiffiffiffi expfz2 =2gdz : Y late ¼ 2pffi 2pffi p 2p 2p
ðA:4Þ
a
v
ðA:5Þ
ðA:6Þ
v
Introducing /( Æ ) and U( Æ ) as the standard normal density (ordinate) R 1 and cumulative distribution functions, respectively, and recognizing that for the standard normal that w zf ðzÞ ¼ /ðwÞ, gives 2 3 pffiffiffi K c2 l bl bl c2 l 4 5
Y late ¼ / pffiffiffi ðc2 lÞ U pffiffiffi U pffiffiffi : v / pffiffiffi v v v v pffi pffi U bl U al v
v
ðA:7Þ
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A.2. Expected earliness cost Repeating the steps outlined in (A.3)–(A.7) for the expected earliness cost Z QH c1 Y early ¼ ðc1 xÞfX ðxÞdx p a yields
2
ðA:8Þ
3
pffiffiffi QH c1 l al al c1 l 5 Y early ¼ 4
/ pffiffiffi ðc1 lÞ U pffiffiffi U pffiffiffi : v / pffiffiffi al v v v v pffi pffi U bl U v v ðA:9Þ pffiffiffi pffiffiffi Negative values for the normal distribution
are negligible provided l > 4 v, hence we set a ¼ l 4 v and pffiffiffi al bl ffi ffi p p pffi pffi b ¼ l þ 4 v. This implies / v ¼ / v ffi 0, U al ffi 0:0 and U bl ffi 1:0. v v Combining (A.7) and (A.9) and simplifying gives pffiffiffi c1 l pffiffiffi c2 l c1 l c2 l Y ¼ QH v/ pffiffiffi þ ðc1 lÞ U pffiffiffi þ K v/ pffiffiffi ðc2 lÞ 1 U pffiffiffi : v v v v ðA:10Þ References [1] D. Garg, Y. Narahari, N. Viswanadham, Achieving sharp deliveries in supply chains through variance pool allocation, Eur. J. Oper. Res. 171 (1) (2006) 227–254. [2] F.-K. Wang, T. Du, Applying capability index to the supply network analysis, Total Qual. Manage. 18 (4) (2007) 425–434. [3] A. Lockamy, K. McCormack, Linking SCOR planning practices to supply chain performance, Int. J. Oper. Prod. Manage. 24 (12) (2004) 1192–1218. [4] S. Vachon, R.D. Klassen, An exploratory investigation of the effects of supply chain complexity on delivery performance, IEEE Trans. Eng. Manage. 49 (3) (2002) 218–230. [5] R. Verma, M.E. Pullman, An analysis of the supplier certification process, Omega 26 (6) (1998) 739–750. [6] A.C. Gunasekaran, C. Patel, E. Tirtiroglu, Performance measures and metrics in the supply chain environment, Int. J. Oper. Prod. Manage. 21 (1–2) (2001) 71–87. [7] C.C. New, T.M. Sweeney, Delivery performance and throughput efficiency in UK manufacturing, Int. J. Phys. Distrib. Mater. Manage. 14 (7) (1984) 3–48. [8] A.S. Carr, J.N. Pearson, Strategically managed buyer–supplier relationships and performance outcomes, J. Oper. Manage. 17 (5) (1999) 497–519. [9] A.L. Guiffrida, R. Nagi, Cost characterizations of supply chain delivery performance, Int. J. Prod. Econ. 102 (1) (2006) 22–36. [10] A.L. Guiffrida, A Cost-based Model for Evaluating Vendor Delivery Performance, MS thesis, Department of Industrial Engineering, State University of New York at Buffalo, Buffalo, NY, 1999. [11] J. Blackhurst, T. Wu, P. O’Grady, Network-based approach to modeling uncertainty in a supply chain, Int. J. Prod. Res. 42 (8) (2004) 1639–1658. [12] E.H. Sabri, B.M. Beamon, A multi-objective approach to simultaneous strategic and operational planning in supply chain design, Omega 28 (5) (2000) 581–590. [13] W.J. Hopp, M.L. Spearman, Factory Physics: Foundations of Manufacturing Management, McGraw-Hill, Boston, 1996. [14] S.J. Erlebacher, M.R. Singh, Optimal variance structures and performance improvement of synchronous assembly lines, Oper. Res. 47 (4) (1999) 601–618. [15] J.E. Tyworth, L. O’Neill, Robustness of the normal approximation of lead-time demand in a distribution setting, Nav. Res. Logist. 44 (2) (1997) 165–186. [16] W. Jaruphongsa, S. Cetinkaya, C.-H. Lee, Warehouse space capacity and delivery time window considerations in dynamic lot-sizing for a simple supply chain, Int. J. Prod. Econ. 92 (2) (2004) 169–180. [17] C.-H. Lee, S. Cetinkaya, A.P.M. Wagelmans, A dynamic lot-sizing model with demand time windows, Manage. Sci. 47 (10) (2001) 1384–1395. [18] L.M. Corbett, Delivery windows – a new way on improving manufacturing flexibility and on-time delivery performance, Prod. Invent. Manage. 33 (3) (1992) 74–79. [19] M.E. Johnson, T. Davis, Improving supply chain performance by using order fulfillment metrics, Natl. Prod. Rev. 17 (3) (1998) 3–16.
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[20] A.L. Guiffrida, Cost Characterizations of Supply Chain Delivery Performance, Ph.D. thesis, Department of Industrial Engineering, State University of New York at Buffalo, Buffalo, NY, 2005. [21] A.M. Schneiderman, Metrics for the order fulfillment process—part 1, J. Cost. Manage. 10 (2) (1996) 30–42. [22] J.R. Freehand, A survey of just-in-time purchasing practices in the United States, Prod. Invent. Manage. 32 (2) (1991) 43–49. [23] D.N. Burt, Managing suppliers up to speed, Harvard Bus. Rev. 67 (5) (1989) 127–135. [24] P. Frame, Saturn to fine suppliers $500/minute for delays, Automot. News 21 (12) (1992) 36. [25] R. Russell, B. Taylor, Operations Management: Focusing on Quality and Competitiveness, Prentice-Hall, New York, 1998. [26] P.A. Dion, L.M. Hasey, P.C. Dorin, J. Lundin, Consequences of inventory stockouts, Ind. Market. Manage. 20 (2) (1991) 23–27. [27] G.J.C. da Silveira, R. Arkader, The direct and mediated relationships between supply chain coordination investments and delivery performance, Int. J. Oper. Prod. Manage. 27 (2) (2007) 140–158. [28] C.A. Maritan, Capital investment as investing in organizational capabilities: an empirically grounded process model, Acad. Manage. J. 44 (3) (2001) 513–531. [29] E.L. Porteus, Investing in reduced setups in the EOQ model, Manage. Sci. 31 (8) (1985) 998–1010. [30] R.K. Cho, Y. Gerchak, Supply chain coordination with downstream operating costs: coordination and investment to improve downstream operating efficiency, Eur. J. Oper. Res. 162 (3) (2005) 762–772. [31] J.P. Leschke, E.N. Weiss, The multi-item setup-reduction investment allocation problem with continuous investment-cost functions, Manage. Sci. 43 (6) (1997) 890–894. [32] J.-W. Choi, Investment in the reduction of uncertainties in just-in-time purchasing systems, Nav. Res. Logist. 41 (2) (1994) 257–272. [33] R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffery, D.E. Knuth, On the Lambert W function, Adv. Comput. Math. 5 (4) (1996) 329–359.
Applied Mathematical Modelling 32 (2008) 2149–2161 www.elsevier.com/locate/apm
Managerial and economic impacts of reducing delivery variance in the supply chain Alfred L. Guiffrida
a,*
, Mohamad Y. Jaber
b
a
b
Department of Management and Information Systems, Kent State University, Kent, OH 44242, USA Department of Mechanical and Industrial Engineering, Ryerson University, Toronto, Ontario, Canada M5B 2K3 Received 1 December 2005; received in revised form 1 December 2006; accepted 4 July 2007 Available online 19 July 2007
Abstract This paper addresses the managerial and economic impacts of improving delivery performance in a serial supply chain when delivery performance is evaluated with respect to a delivery window. Building on contemporary management theories that advocate variance reduction as the critical step in improving the overall performance of a system, we model the variance of delivery time to the final customer as a function of the investment to reduce delivery variance and the costs associated with untimely delivery (expected earliness and lateness). A logarithmic investment function is used and the model solution involves the minimization of a convex–concave total cost function. A numerical example is provided to illustrate the model and the solution procedure. The model presented provides guidelines for determining the optimal level of financial investment for reducing delivery variance. The managerial implications as well as the economic aspects of delivery variance reduction in supply chain management are discussed. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Improving delivery performance; Justifying variance reduction; Supply chain management
1. Introduction Modern markets are competitive business environments where customers require their suppliers to be dependable in delivering on-time shipments. One of the important goals of supply chain management is to improve delivery performance [1,2]. Results from recently published empirical studies have identified delivery performance as a key management concern among supply chain practitioners (see for example [3–5]). A conceptual framework for defining delivery performance in supply chain management is found in Gunasekaran et al. [6]. This suggested framework classifies delivery performance as a strategic level supply chain performance measure while delivery reliability is viewed as a tactical level supply chain performance measure. The framework of Gunasekaran et al. [6] also advocates that to be effective, supply chain management tools, delivery performance and delivery reliability need to be measured in financial (as well as non-financial) terms. *
Corresponding author. Tel.: +1 716 954 3504; fax: +1 716 631 2635. E-mail address: aguiff[email protected] (A.L. Guiffrida).
0307-904X/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2007.07.006
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The failure to financially quantify delivery performance in a supply chain presents both short-term and long-term difficulties. In the short term, the buyer–supplier relationship may be negatively impacted. A norm value of ‘‘presumed’’ performance is established by default when delivery performance is not formally measured [7]. This norm value stays constant with time and is generally higher than the organization’s actual delivery performance. It has been demonstrated that supplier evaluation systems have a positive impact on the buyer–supplier relationships, with these relationships ultimately have a positive impact on financial performance [8]. In the long term, failure to measure supplier delivery performance in financial terms may impede the capital budgeting process, which is necessary in order to support the improvement of supplier operations within a supply chain. Delivery lead time is defined to be the elapsed time from the receipt of an order by the originating supplier in the supply chain to the receipt of the product ordered by the final customer in the supply chain. Delivery lead time is composed of a series of internal (manufacturing and processing) lead times at each stage plus the external (distribution and transportation) lead times found at various stages of the supply chain. Fig. 1 illustrates the delivery lead time components of a serial supply chain. Early and late deliveries introduce waste in the form of excess cost into the supply chain; early deliveries contribute to excess inventory holding costs, while late deliveries may contribute to production stoppages costs, lost sales and loss of goodwill. To protect against untimely deliveries, supply chain managers often inflate in process inventory levels and production flow buffers. These actions can contribute to excess operating and administrative costs [9]. An extensive review of available delivery evaluation models by Guiffrida [10] identified several shortcomings in modeling delivery performance. These concerns are threefold. First, delivery performance measures are not cost-based. Second, delivery performance measures ignore variability. Third, delivery performance measures often fail to take into account the penalties associated with both early and late deliveries. The inability to translate delivery performance into financial terms which incorporates uncertainty as well as realistically quantifying delivery timeliness (early as well as late delivery) hinders management’s ability to justify capital investment for continuous improvement programs, which are designed to improve delivery performance. The current research presented herein attempts to overcome these limitations through the development of a cost-based model that incorporates delivery variability and assigns penalties for both early and late deliveries. In this paper, we develop a cost-based performance metric for evaluating delivery performance and reliability to the final customer in a serial supply chain that is operating under a centralized management structure. Contemporary management theories advocate the reduction of variance as a key step in improving the performance of a system [11–13]. In union with these prevailing theories, delivery performance is modeled as a cost-based function of the delivery variance. The financial benefit of investing to reduce variability in delivery performance is demonstrated. This paper is organized as follows. In Section 2, an analytical model based on the expected costs associated with untimely delivery is developed and propositions are introduced to analyze the model in terms of the variance. In Section 3, improvement in delivery performance is modeled using a logarithmic investment function. The reduction of deliver variance is formulated and solved as a convex mathematical optimization problem. In Section 4, a heuristic solution algorithm is introduced and the two solution algorithms are compared. Managerial implications of investing to reduce delivery variance are addressed in Section 5. In the concluding section, we summarize the findings of this research and present directions for future research.
Supplier 1
Supplier 2
…
Supplier n
Delivery Lead Time
Fig. 1. An n-stage serial supply chain.
Final Customer
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2. Model development Consider an n-stage serial supply chain operating under a centralized management structure where an activity at each stage contributes to the overall delivery time to the final customer. The activity duration of stage i, Wi, is defined by probability density function fW(w, h) that Pn is reproductive under addition with respect to parameter set h. Delivery time to the final customer, X ¼ i¼1 W i , is defined by the probability density func P tion fX x; ni¼1 hi . We assume that activity durations at each stage of the supply chain are normally distributed and independent. Normality and independence among stages is often assumed in the literature [14,15]. When activity times at stage i are normally distributed and independent h = {l, v}. Delivery time Pn to the Pnfinal customer is defined as a result of the n-fold convolution of fW(w, h) which yields X N x; i¼1 li ; i¼1 vi where li and vi are the delivery mean and variance for stage i, i 2 [1, n] . Delivery to the final customer is analyzed with regard to the customer’s specification of delivery timeliness as defined by a delivery window. Delivery windows are an effective tool for modeling the expected costs associated with untimely delivery. Jaruphongsa et al. [16], Lee et al. [17], and Corbett [18] advocate the use of delivery windows in time-based manufacturing systems. Johnson and Davis [19] note that metrics based on delivery (order) windows capture the most important aspect of the delivery process, which is reliability. Under the concept of a delivery window, the customer supplies an earliest allowable delivery date (a) and a latest allowable delivery date (b). A delivery window is defined as the difference between the earliest acceptable delivery date and the latest acceptable delivery date. The customer and the end supplier responsible for product delivery contractually agree that no deliveries will arrive before a nor after b, thereby establishing a mutually agreeable framework for managing delivery performance. If a shipment is delivered after b or before a it will be rejected and therefore returned to the supplier. Within the delivery window, a delivery may be classified as early, on-time, or late (see Fig. 2). Delivery lead time, X, is a random variable with probability density function fX(x). The on-time portion of the delivery
f X (x)
x a
c1
Early
b
c2
On-Time Delivery Scenario
Legend: a = earliest delivery time c1 = beginning of on-time delivery c 2 = end of on-time delivery b = latest acceptable delivery time Fig. 2. Illustration of delivery window.
Late
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window is defined by c2 c1. Ideally, c2 c1 = 0. However, the extent to which c2 c1 > 0 may be measured in hours, days, or weeks depending on the industrial situation. It is assumed in this paper that: (1) delivery performance is stable enough so that the modal delivery time is within the on-time portion of the delivery window, (2) the mean and on-time portion of the delivery window remain fixed, and (3) the coefficient of variation of X is less than 0.25 thereby limiting the probability of a negative delivery time under the normal density to essentially zero. An alternative to assumption (3) would be to truncate the normal density to prevent nonnegative delivery times or select symmetrical and nonsymmetrical density functions defined for only positive values of delivery time as was done in Guiffrida [10,20]. Consider an n-stage serial supply chain in operation over a time horizon of length T years, where a demand requirement of the final customer for a single product of D units will be met with a constant delivery lot size Q. The expected penalty cost per delivery period for untimely delivery, Y, is Z c1 Z b Y ¼ QH ðc1 xÞfX ðxÞdx þ K ðx c2 ÞfX ðxÞdx; ð1Þ a
c2
where Q constant delivery lot size per cycle H inventory holding cost per unit per time K penalty cost per time unit late (levied by the final customer) a, b, c1, c2 parameters defining the delivery window duration of delivery lead time component j ðj ¼ 1; 2; . . . ; nÞ Wj fX ðxÞ ¼ fW 1 þW 2 þþW n density function of delivery time X It is a common purchasing agreement practice to allow the buyer (final customer) to charge suppliers for untimely deliveries (see for example [21,22]). Reductions in early deliveries reduced inventory holding costs at Hewlett-Packard by $9 million [23]. It has been reported in the automotive industry Saturn levies fines of $500 per minute against suppliers who cause production line stoppages [24] and that Chrysler fines suppliers $32,000 per hour when an order is late [25]. The penalty cost in these cases is an opportunity cost due to lost production. Purchasing managers often view the production disruptions caused by delivery stockouts to be more widespread and more costly than the lost sales that stockouts cause [26]. Hence, K has been defined as an opportunity cost due to lost production as described by Frame [24] and Russell and Taylor [25]. Evaluating (1) under the defined assumptions yields the total expected penalty cost (see Appendix A for derivation) pffiffiffi c1 l pffiffiffi c2 l c1 l c2 l Y ¼ QH v/ pffiffiffi þ ðc1 lÞ U pffiffiffi þ K v/ pffiffiffi ðc2 lÞ 1 U pffiffiffi : v v v v ð2Þ Proposition 1. The expected penalty cost is a monotonically increasing non-convex function of the variance. Proof. The first and second derivatives of Y with respect to the variance are respectively
ffiv þ K/ c2pl ffiv QH / c1pl pffiffiffi Y 0 ðvÞ ¼ 2 v and
ð3Þ
QH c1 l K c2 l 2 2 p ffiffi ffi p ffiffi ffi / lÞ vÞ þ / lÞ vÞ : ð4Þ ððc ððc 1 2 4v5=2 4v5=2 v v Examining (3), the expected penalty cost is an increasing function of the variance since Y 0 (v) > 0 for positive 2 2 values of Q, H, K and v. Examining (4), we note that Y00 (v) > 0 when v 6 minfðc1 lÞ ; ðc2 lÞ g, but 2 2 Y00 (v) < 0 when v P maxfðc1 lÞ ; ðc2 lÞ g, hence the expected penalty cost is not a convex function of the variance. Y 00 ðvÞ ¼
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Proposition 2. The expected penalty cost is a monotonically increasing locally convex function of the variance provided v < minfðc1 lÞ2 ; ðc2 lÞ2 g. Proof. Examining (4), Y00 (v) > 0 when 0 < v < min{(c1 l)2, (c2 l)2}.
h
3. Modeling improvement in delivery performance Ideally the expected costs incurred for untimely delivery should be equal to zero. This implies that, for the currently defined delivery window, all deliveries are within the on-time portion of the delivery window, and that waste in the form of early and late deliveries has been eliminated from the system. For a fixed mean and delivery window, the expected penalty cost can be reduced by reducing the variance of the delivery distribution. The expected penalty cost (1) can be expressed as a function of the delivery period variance, as pffiffiffi c1 l pffiffiffi c2 l c1 l c2 l Y ðvÞ ¼ QH v/ pffiffiffi þ ðc1 lÞU pffiffiffi þ K v/ pffiffiffi ðc2 lÞ 1 U pffiffiffi : v v v v ð5Þ Reducing the variance of delivery can be achieved by initiating improvements such as: (1) gaining tighter control over process flow times at downstream stages of the supply chain, (2) enhanced coordination of freight transport, (3) more efficient material handling of inbound and outbound stock, and (4) improved communications among stages in the supply chain such as implementing electronic data interchange (EDI). Initiating such improvements requires capital investment (see for example, [27,28]). An optimization model which considers the delivery variance as a decision variable is defined. The objective of the model is to determine the variance level that minimizes the costs (expected earliness and lateness) associated with untimely delivery and the investment cost required for reducing the delivery variance. The optimization problem is Minimize
GðvÞ ¼ Y ðvÞ þ CðvÞ;
ð6Þ
where v Y(v) C(v)
variance of delivery distribution expected penalty cost due to untimely delivery investment required for a delivery variance of v
A logarithmic investment cost function is used to model the cost of reducing the delivery variance [29]. Under this investment function, reducing the delivery variance by a fixed percentage requires a fixed amount of investment. This functional form is appealing in that each additional reduction in the delivery variance is more costly than the previous reduction. The logarithmic investment function has been widely adopted in the literature (see for example [30–32]). Let v0 equal the current value of delivery variance and k represent the cost of reducing the delivery variance by h percent. The investment function is then CðvÞ ¼
k ½lnðv0 Þ lnðvÞ lnð1=1 hÞ
for 0 6 v 6 v0 :
ð7Þ
Proposition 3. The investment function is a decreasing convex function of the delivery variance. Proof. The first and second derivatives of (7) are C 0 ðvÞ ¼
k lnð1=1 hÞv
ð8Þ
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and C 00 ðvÞ ¼
k : lnð1=1 hÞv2
ð9Þ
We observe that C 0 (v) < 0 and C00 (v) > 0 for k > 0 and h > 0. h Substituting (5) and (7) into (6), yields the optimization model pffiffiffi c1 l c1 l Minimize GðvÞ ¼ QH v/ pffiffiffi þ ðc1 lÞ U pffiffiffi v v pffiffiffi c2 l c2 l k ½lnðv0 Þ lnðvÞ: þ K v/ pffiffiffi ðc2 lÞ 1 U pffiffiffi þ lnð1=1 hÞ v v ð10Þ 2
2
As per Propositions 1 and 2, Y(v) is a convex function of the variance for v 6 minfðc1 lÞ ; ðc2 lÞ g and a concave function when v P maxfðc1 lÞ2 ; ðc2 lÞ2 g. For minfðc1 lÞ2 ; ðc2 lÞ2 g < v < maxfðc1 lÞ2 ; ðc2 lÞ2 g, the convexity–concavity of Y(v) is parameter specific. Hence, G(v) is the sum of a convex–concave function Y(v) and a convex function C(v). Theorem 1. G(v) is a convex function of the delivery variance provided v 6 minfðc1 lÞ2 ; ðc2 lÞ2 g.
Proof. The first and second derivatives of (10) with respect to the variance are
ffiv þ K/ c2pl ffiv QH / c1pl k pffiffiffi G0 ðvÞ ¼ lnð1=1 hÞv 2 v
ð11Þ
and QH c1 l K c2 l k 2 2 G ðvÞ ¼ 5=2 / pffiffiffi ððc1 lÞ vÞ þ 5=2 / pffiffiffi ððc2 lÞ vÞ þ : 4v 4v lnð1=1 hÞv2 v v 00
2
ð12Þ
2
By direct substitution G00 (v) > 0 for v 6 minfðc1 lÞ ; ðc2 lÞ g hence G(v) is convex when v 6 2 2 minfðc1 lÞ ; ðc2 lÞ g. h Corollary 1. G(v) is minimized at G*(v*) provided v* 6 vm. 2
2
Proof. Let vm ¼ v 6 minfðc1 lÞ ; ðc2 lÞ g. As per Theorem 1, G(v) is a convex function of the delivery variance for v 6 vm, hence the solution of G 0 (v) = 0 yields an optimal solution if v* 6 vm. h The value of v* that minimizes G(v) at G*(v*) does not exist in closed form and must be found by numerically solving G 0 (v) = 0 for v = v*. Solving (10) for v takes the general form of pffiffiffi c1 l c2 l 2k ¼ 0: v QH / pffiffiffi þ K/ pffiffiffi lnð1=ð1 hÞÞ v v
ffi exp z22 gives Introducing /ðzÞ ¼ p1ffiffiffi 2p " ! !# pffiffiffiffiffiffi 2 2 pffiffiffi ðc1 lÞ ðc2 lÞ 2k 2p ¼ 0: v QH exp þ K exp lnð1=ð1 hÞÞ 2v 2v
ð13Þ
ð14Þ
If the on-time portion of the delivery window is symmetric then d = c2 l = (c1 l) and (14) simplifies to pffiffiffiffiffiffi 2 pffiffiffi d 2k 2p ¼ 0: ð15Þ v exp lnð1=ð1 hÞÞðQH þ KÞ 2v
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Fig. 3. Optimal solution to numerical illustration with v < vm.
And a closed form solution for (15) is v¼
d2 2 ; d lnð1=ð1 hÞÞðQH þ KÞ p ffiffiffiffiffi ffi WL 2k 2p
ð16Þ
where WL is the Lambert W function [33]. 3.1. Numerical example Consider a supply chain where delivery time (in days) to the final customer is normally distributed with a mean of 50 and variance of 10. The on-time portion of the delivery window is defined by c1 = 48 and c2 = 53. For these parameters vm = 4 and v0 = 10. Additional parameters for the expected penalty cost model are: Q = 500, H = $10 and K = $5000. Under the logarithmic form of the investment function, a cost of k = $150 is incurred for every h = 10% reduction in the delivery variance. Solving (10) for v yields v = 3.253. As per Corollary 1 of Theorem 1, v = 3.253 < vm = 4, hence v = v* = 3.253 and G*(v*) = $2387 (see Fig. 3). 4. Heuristic solution algorithm 2
2
When v P maxfðc1 lÞ ; ðc2 lÞ g, minimizing G(v) requires the minimization of the sum of a concave function Y(v) and a convex function C(v). There is no guarantee that solving G 0 (v) = 0 will yield the optimal value of the variance. A lower bound on the optimal value of G(v) can be obtained by approximating Y(v) as a 2 2 linear function for minfðc1 lÞ ; ðc2 lÞ g < v < v0 . The range of v for which the linear approximation has been defined includes values of the variance where the sign of G00 (v) is parameter specific. Extending the range of the variance in this fashion eliminates the need to conduct parameter specific analyses when adopting the linear approximation procedure. Let vL and vU represent values of the delivery variance such that vm 6 vL < vU 6 v0 . A linear approximation of Y(v) is defined as Y ðvU Þ Y ðvL Þ Ye ðvÞ ¼ Y ðvL Þ þ ð17Þ ðv vL Þ: vU vL When Y00 (v) < 0, a lower bound on G(v) is then defined by e GðvÞ ¼ Ye ðvÞ þ CðvÞ:
ð18Þ
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e Theorem 2. GðvÞ is a convex function of the delivery variance for vm < v < v0. Proof. As per Proposition 3, C00 (v) > 0 thus C(v) is a convex function of the variance. By inspection (17) is a e linear (and hence convex function) of the variance. GðvÞ is the sum of two convex functions and is therefore convex. h e e ð~v Þ. Corollary 2. GðvÞ is minimized at G e Proof. As per Theorem 2, GðvÞ is a convex function of the delivery variance for vm < v < v0 and is minimized at value of the variance, ~v , that solves f G0 ðvÞ ¼ 0. e GðvÞ is minimized at ~v ¼
kðvU vL Þ . lnð1=1 hÞ½Y ðvU Þ Y ðvL Þ
h
A heuristic procedure is defined for determining the variance level that minimizes the costs (expected earliness and lateness) associated with untimely delivery and the investment cost required for reducing the delivery variance. STEP 1. Input model parameter values: c1 ; c2 ; l; Q; H ; K; h; k; STEP 2. Initialize search indices: j = 0 (iteration counter); e = 0.01 (accuracy tolerance); v0 ; vm ¼ minfðc1 lÞ2 ; ðc2 lÞ2 g. STEP 3. 1. If v0 6 vm solve G 0 (v) = 0 for v = v* where the minimum cost is G*(v*), THEN GOTO STEP 8. 2. ELSE GOTO STEP 4. STEP 4. e vÞ < minfGðvL Þ; GðvU Þg, THEN SET 1. Set vL ¼ vm ; vU ¼ v0 and solve f G0 ðvÞ ¼ 0 for v ¼ ~v. If Gð~ L L U L U U vj;1 ¼ v ; vj;1 ¼ vj;2 ¼ ~v; vj;2 ¼ v , compute the minimum cost as G ðv Þ ¼ minfGðvL Þ; GðvU Þg; GOTO STEP 5. 2. ELSE GOTO STEP 8. STEP 5. Solve f G0 ðvÞ for v ¼ ~v and f G0 ðvÞ for v ¼ ~v ; GOTO STEP 6. j;1
j;1
j;2
j;2
STEP 6. e vÞ minf G e j;1 ð~vj;1 Þ; G e j;2 ð~vj;2 Þgj < e (no improvement is possible), the minimum cost is G ðv Þ ¼ 1. If j Gð~ e Gð~vÞ; GOTO STEP 8. 2. ELSE GOTO STEP 7. STEP 7. e j;1 ð~vj;1 Þ < G e j;2 ð~vj;2 Þ set Gð~ e vÞ ¼ G e j;1 ð~vj;1 Þ, j ¼ j þ 1; vL ¼ vL ; vU ¼ vL ¼ ~vj1;1 ; vU ¼ vU . 1. If G j;1 j1;1 j;1 j;2 j;2 j1;1 L U e e 2. ELSE set Gð~vÞ ¼ G j;2 ð~vj;2 Þ; j ¼ j þ 1; vLj;1 ¼ vLj1;2 ; vU vj1;2 ; vU j;1 ¼ vj;2 ¼ ~ j;2 ¼ vj1;2 . 3. GOTO STEP 5. STEP 8. STOP e j;1 ð~vj;1 Þ < G e j;2 ð~vj;2 Þ set Gð~ e vÞ ¼ G e j;1 ð~vj;1 Þ; j ¼ j þ 1; vL ¼ vL ; vU ¼ vL ¼ ~vj1;1 ; vU ¼ vU ; and Step 6. If G j;1 j1;1 j;1 j;2 j;2 j1;1 e e go to step 4; else set Gð~vÞ ¼ G j;2 ð~vj;2 Þ, j = j + 1.
4.1. Numerical example Consider a supply chain where delivery time (in days) to the final customer is normally distributed with a mean of 50 and variance of 20. The on-time portion of the delivery window is defined by c1 = 46 and c2 = 53.
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For these parameters vm = 9 and v0 = 20. Additional parameters for the expected penalty cost model are: Q = 500, H = $10 and K = $5000. Under the logarithmic form of the investment function, a cost of k = $650 is incurred for every h = 10% reduction in the delivery variance. The solution to this numerical example is summarized in Table 1. e vÞ ¼ $5610:49. Reducing The algorithm terminated after two iterations at v ¼ ~v ¼ 18:551 and G ðv Þ ¼ Gð~ the delivery variance from v0 = 20 to v ¼ ~v ¼ 18:551 improved the probability of an early delivery improved from 0.2148 to 0.1762 and the probability of late delivery from 0.2514 to 0.2420. For the parameter values selected in this numerical example, the cost function G(v) = Y(v) + C(v) appears to be flat in the neighborhood of v* = 18.551. This suggests that the non-convexity in the expected penalty cost term Y(v) may be slight and e the linear approximating Ye ðvÞ provides a close fit between G(v) = Y(v) + C(v) and GðvÞ ¼ Ye ðvÞ þ CðvÞ. 0 This assertion can be tested by solving G (v) = 0 despite the fact that Theorem 5.1 is violated for the parameters used in this numerical example. Solving, G 0 (v) = 0 yields v* = 18.580 and G*(v*) = $5610.49 which corresponds to a zero percentage cost penalty in comparison to the heuristic solution procedure. Solving e 0 ðvÞ ¼ 0 can be solved by directly by a G 0 (v) = 0 requires use of nonlinear programming software, while G 0 e closed form expression. Solving G ðvÞ ¼ 0 directly (as is done in Step 1 of the heuristic algorithm) yields e vÞ ¼ $5611:88. Hence, continuing the solution heuristic until termination v ¼ ~v ¼ 18:151 and G ðv Þ ¼ Gð~ led to a 0.02% improvement in the total cost. Alternatively, the easily solved closed form expression for
Table 1 Solution to numerical example using logarithmic investment heuristic algorithm Step Iteration j = 0 3
Description 1. Is v0 = 20 < vm = 9? 2. No, go to Step 4
4
1. vL ¼ 9 vU ¼ 20; ~v ¼ 18:151 GðvL Þ ¼ $6811:89 GðvU Þ ¼ $5624:29 Gð~vÞ ¼ $5611:88 Is Gð~vÞ ¼ $5611:88 minfGðvL Þ ¼ $6811:89; GðvU Þ ¼ $5624:29g < 0:01? v ¼ 18:151; vL0;2 ¼ ~v ¼ 18:151; vU No, set vL0;1 ¼ vL ¼ 9; vU 0;1 ¼ ~ 0;2 ¼ v0 ¼ 20 and go to Step 5
5
~v0;1 ¼ 18:050 ~v0;2 ¼ 18:667
6
e 0;2 ð~v0;2 Þ ¼ $5610:54gj < 0:01? e vÞ ¼ $5611:88 minf G e 0;1 ð~v0;1 Þ ¼ $5611:89; G 1. Is j Gð~ 2. No, go to Step 7
7
e vÞ ¼ G e 0;2 ð~v0;2 Þ ¼ $5610:54; j ¼ 0 þ 1 ¼ 1 2. Set Gð~ vL1;1 ¼ vL0;2 ¼ 18:151; vU v0;2 ¼ 18:667 1;1 ¼ ~ U vL1;2 ¼ ~v0;2 ¼ 18:667; vU 1;2 ¼ v0;2 ¼ 20 3. Go to Step 5
Iteration j = 1 5
~v11 ¼ 18:551 ~v12 ¼ 18:713
6
e 1;2 ð~v1;2 Þ ¼ $5610:62gj < 0:01? e vÞ ¼ $5610:54 minf G e 1;1 ð~v1;1 Þ ¼ $5610:49; G 1. Is j Gð~ 2. No, go to Step 7
7
e vÞ ¼ G e 1;1 ð~v1;1 Þ ¼ $5610:49; j ¼ 1 þ 1 ¼ 2 2. Set Gð~ vL2;1 ¼ vL1;1 ¼ 18:151; vU v1;1 ¼ 18:551 2;1 ¼ ~ U vL2;2 ¼ ~v1;1 ¼ 18:551; vU 2;2 ¼ v1;1 ¼ 18:667 3. Go to Step 5
Iteration j = 2 5
~v2;1 ¼ 18:541 ~v2;2 ¼ 18:585
6
e 2;2 ð~v2;2 Þ ¼ $5610:49gj < 0:01? e vÞ ¼ $5610:49 minf G e 2;1 ð~v2;1 Þ ¼ $5610:50; G 1. Is j Gð~ Yes, go to Step 8
8
STOP, The minimum cost solution is G*(v*) = $5610.49
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e 0 ðvÞ ¼ 0 resulted in a solution that differed by 0.02% in comparison to the heuristic and nonlinear programG ming software solutions. 5. Managerial implications of delivery variance reduction In this section, we address the managerial implications of improving delivery performance through the reduction of delivery variance. An optimization model has been introduced to demonstrate the tradeoffs between investment in reducing the variance of delivery and the expected costs that accrue as a result of untimely delivery. The modeling effort herein has provided a conceptual framework for linking financial investment and delivery improvement. This framework is similar in characteristic to process improvement espoused in the quality management literature where process improvement is predicated on the reduction of process variability. We contend that logistics and supply chain managers can use the delivery variance reduction framework to improve delivery performance in similar fashion to how quality assurance managers have historically used the reduction in process variation to improve product quality. Improvement in delivery performance has both immediate and long term benefits. For a given level of investment, the optimal amount of delivery variance reduction can be estimated. For a fixed mean and on-time portion of the delivery window the most obvious effect of delivery variance reduction is the improvement of delivery performance. As the delivery variance is reduced the probability of on-time shipments increases and the probability of early and late shipments decreases. The improved delivery performance can lead to increased business from existing customers. Also, improved delivery performance can be used a basis of competitive advantage to gain new customers whose delivery standards could not be met prior to the reduction of delivery variance. The model can be used to identify the investment cost required to attract new customers based on the customers’ requirements of delivery timeliness. The optimization model presented in this paper determines the optimal variance level that minimizes the costs due to untimely delivery and the investment cost required for reducing the variance. This modeling framework can be adapted to integrate with the capital budgeting process for justifying funds for process improvement. The static optimization problem defined in (6) can be stated as a temporal model of the form Gðv; tÞ ¼ Y ðv; tÞ þ Cðv; tÞ over time horizon T, 0 6 t 6 T. The present worth of the cost stream G(v, t) provides as estimate in current dollars of cost to be incurred over time budgeting horizon T. The present worth of this cost stream provides a benchmark that can be used within the capital budgeting process for justifying the investment required to improve delivery performance.
6. Conclusions This paper addressed the economic and managerial implications of improving delivery performance to the final customer in a serial supply chain from the contemporary perspective of reducing delivery variability. A cost-based model has been presented that financially evaluates the effects of reducing delivery variability on overall delivery performance. The model was demonstrated under a logarithmic investment function and a framework for determining optimal and near optimal levels of delivery variance was developed. As demonstrated through a numerical example, the results of this modeling may prove useful to justifying the investment required to reduce delivery variance to a targeted goal as a part of an overall continuous improvement program to improve supply chain delivery performance. An efficient heuristic solution algorithm for determining the variance level that minimizes the costs was also developed in this paper. Logistic and supply chain managers can utilize the variance reduction framework set forth in this paper to gain both short and long term benefits. Reducing the variability of shipment delivery times will improve overall delivery performance in the short term which will increase customer satisfaction among the existing customer base. In the long term, less variability in delivery can lead to new customers whose delivery requirements can now be met. There are several aspects of this research that could be expanded. An optimization model could be used to optimally allocate the reduction in delivery variance to the downstream stages of the supply chain. Second, an industrial case study utilizing the model developed herein could be conducted. Third, disruptions in the var-
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iance reduction process could be investigated. Lastly, the assumption of independence of the activity times among the stages could be investigated. Acknowledgements The authors are grateful to the two anonymous referees and to the Editor of the Journal for their helpful comments and suggestions. Appendix A. Derivation of expected penalty cost (Eq. (2)) The density function of delivery time X with mean l and variance v, is the convolution of the normally distributed lead time components W i ði ¼ 1; 2; . . . ; nÞ; fX ðxÞ ¼ fW 1 þW 2 þþW n . If an earliest delivery time (a) and latest acceptable delivery time (b) are imposed on fX(x), then fX ðxÞ hX ðxÞ ¼ R b f ðxÞ a X
ðA:1Þ
and (1) is Y ¼ QH
Z
c1
ðc1 xÞhX ðxÞdx þ K a
Z
b
ðx c2 ÞhX ðxÞdx:
ðA:2Þ
c2
Examining (A.2), we observe that Y is separable in terms of the expected earliness cost and the expected lateness cost. A.1. Expected lateness cost The expected penalty cost for late delivery is Z K b Y late ¼ ðx c2 ÞfX ðxÞdx; p c2
ðA:3Þ
where p¼
Z
b
fX ðxÞdx; "Z ( ) ( ) # Z b 2 2 b K x ðx lÞ c2 ðx lÞ pffiffiffiffiffiffiffiffi exp pffiffiffiffiffiffiffiffi exp Y late ¼ dx dx : p c2 2pv 2v 2v 2pv c2 pffiffiffi pffiffiffi pffi ; x ¼ vz þ l, and dx ¼ v dz into (A.5) and simplifying yields Substituting z ¼ xl v " Z bl # Z bl pffi v K pffiffiffi pffiv z 1 v c l pffiffiffiffiffiffi expfz2 =2gdz þ ðl c2 Þ c l pffiffiffiffiffiffi expfz2 =2gdz : Y late ¼ 2pffi 2pffi p 2p 2p
ðA:4Þ
a
v
ðA:5Þ
ðA:6Þ
v
Introducing /( Æ ) and U( Æ ) as the standard normal density (ordinate) R 1 and cumulative distribution functions, respectively, and recognizing that for the standard normal that w zf ðzÞ ¼ /ðwÞ, gives 2 3 pffiffiffi K c2 l bl bl c2 l 4 5
Y late ¼ / pffiffiffi ðc2 lÞ U pffiffiffi U pffiffiffi : v / pffiffiffi v v v v pffi pffi U bl U al v
v
ðA:7Þ
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A.2. Expected earliness cost Repeating the steps outlined in (A.3)–(A.7) for the expected earliness cost Z QH c1 Y early ¼ ðc1 xÞfX ðxÞdx p a yields
2
ðA:8Þ
3
pffiffiffi QH c1 l al al c1 l 5 Y early ¼ 4
/ pffiffiffi ðc1 lÞ U pffiffiffi U pffiffiffi : v / pffiffiffi al v v v v pffi pffi U bl U v v ðA:9Þ pffiffiffi pffiffiffi Negative values for the normal distribution
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