- Email: [email protected]
Delivery performance improvement in two-stage supply chain
Accepted Manuscript Delivery performance improvement in two-stage supply chain Maxim A. Bushuev PII:
S0925-5273(17)30322-5
DOI:
10.1016/j.ijpe.2017.10.007
Reference:
PROECO 6839
To appear in:
International Journal of Production Economics
Received Date: 20 April 2017 Revised Date:
23 September 2017
Accepted Date: 7 October 2017
Please cite this article as: Bushuev, M.A., Delivery performance improvement in two-stage supply chain, International Journal of Production Economics (2017), doi: 10.1016/j.ijpe.2017.10.007. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Delivery Performance Improvement in Two-stage Supply Chain
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Maxim A. Bushuev Assistant Professor Department of Information Science and Systems Earl Graves School of Business and Management Morgan State University Baltimore MD 21251 Phone: (330) 389-4967 [email protected]
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Please direct all correspondence to Maxim A. Bushuev Email: [email protected]
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Delivery Performance Improvement in Two-stage Supply Chain
Abstract
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This paper investigates strategies of supply chain delivery performance improvement. The performance is measured using a cost-based analytical model which evaluates the expected penalty cost for early and late delivery. The results demonstrate strategies for improving delivery performance when a supplier uses an optimally positioned delivery window to minimize the expected penalty cost. The effect of the width of the delivery window and penalty costs for early and late deliveries on the optimal position of the delivery window and the expected penalty cost are explored in general case. In addition to that, the effect of a mean and variance parameters of a normal delivery time distribution are revised. Theoretical and managerial implications of the findings are discussed.
Keywords: Supply chain Management, Delivery performance improvement, Delivery window
1. Introduction
The need for performance measurement and evaluation in supply chain management is
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well recognized in the literature (Melnyk et al., 2014; Estampe et al., 2013). Within the hierarchy of supply chain performance metrics delivery performance, as characterized by the timeliness and dependability of product delivery to the final customer in the supply chain, is acknowledged
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as a key metric for supporting supply chain operations (Cirtita and Glaser-Segura, 2012; Forslund et al., 2009; Olhager and Selldin, 2004). Delivery performance is classified as a strategic level performance measure by Gunasekaran et al. (2004) and is also a major component
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of the Supply Chain Operations Reference (SCOR) model (Huan et al., 2004). Delivery windows are often incorporated into performance measurement systems when
performance with respect to deadlines and/or due dates is measured in integrated productiondistribution systems. A delivery window is a time interval within which a delivery can be received. A supplier and a buyer contractually specify allowable deviations (earliness and lateness) from an agreed upon delivery date which are used to classify deliveries as being early, on-time, and late. The on-time portion of the delivery window is between the earliest acceptable delivery time and date and the latest acceptable delivery time and date. When a delivery is made
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within the on-time portion of the delivery window, no penalty cost is incurred. Early and late deliveries introduce waste in the form of excess cost into the supply chain. In supply chain delivery models the costs associated with untimely (early and late) delivery are referred to as penalty costs which are paid by the supplier to the buyer (Guiffrida and Nagi, 2006a, 2006b).
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Therefore, it is in the best interest of a supplier to reduce the penalty costs. Penalty costs for late delivery have been reported in per-unit-time functional forms in the literature from real-world examples in the automotive and aerospace industries (Russell and Taylor, 1998; Slotnick and Sobel, 2005).
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Models for evaluating delivery performance within supply chains can be categorized into two groups: i) index based models (Hsu et al., 2013; Nabhani and Shokri, 2009; Wang and Du,
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2007), and ii) cost based models (Chen et al., 2015; Bushuev and Guiffrida, 2012; Bushuev et al., 2011). These models are applicable to both serial and network supply chains (Safaei and Thoben, 2014; Rezapour et al., 2015). Both the index and cost based categories of models are similar in that delivery timeliness to the final customer is analyzed with regard to the customer’s specification of an on-time delivery window. The models differ in how they report delivery performance in terms of an overall metric. Index based models translate the probability of
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untimely (early and late delivery) into a “delivery capability index” measure. Cost based models use partial expectations to directly translate the probability of untimely delivery into an expected cost measure.
Most of the papers published in the area are focused on delivery performance evaluation.
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Once the currently level of delivery performance has been evaluated, the next natural step is to find ways to improve delivery performance. Evaluating supply chain delivery performance
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models provides additional opportunities for the delivery performance improvement using supply contract negotiation and delivery time distribution improvement. The present paper extends research in the area investigating delivery performance
improvement. The papers focused on delivery performance improvement (Ngniatedema et al., 2016; Tanai and Guiffrida, 2014; Guiffrida and Jaber, 2008; Garg et al., 2006; Guiffrida and Nagi, 2006a, 2006b) fail to address the optimal positioning of the delivery window when evaluating delivery performance. The failure to optimally position the delivery window has been shown by Bushuev and Guiffrida (2012) to lead to suboptimal delivery cost performance and may also impede the coordination of the delivery process within a supply chain. Bushuev and 2
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Guiffrida (2012) demonstrate that optimally positioning of the delivery window for the asymmetric Laplace delivery time distribution lead to a 75% reduction in the expected penalty cost for untimely delivery. The paper herein continues research in the area of delivery performance improvement revising the results of (Guiffrida and Nagi, 2006a) and differs from
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other similar papers in two ways: it applies the concept of the optimal position of the delivery window and analyzes the effect of parameters defined in a contract for general form of delivery time distribution instead of normally distributed delivery time.
In this paper, we address strategies for improving delivery performance using a cost-
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based delivery performance model. Similar to all previous supply chain delivery performance models, the model is limited in that improving delivery performance from the supplier’s
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perspective might lead to sub-optimal decisions from the buyer’s point of view. The effect of delivery performance improvement is analyzed in financial terms thus allowing an estimate of the savings in the expected penalty cost associated with implementation of a delivery performance improvement project. With this information in hand, a supplier then decide in financial terms if it is beneficial to invest in the delivery improvement project. In this paper we employ the delivery cost model of Guiffrida and Nagi (2006a) and build
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on the concept of the optimal position of the delivery window (OPDW) proposed by Bushuev and Guiffrida (2012) to address strategies for improving delivery performance using a cost-based delivery performance model. A set of supporting propositions is presented to provide an analytical analysis of the delivery model that can be used to improve delivery performance. The
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propositions examine the robustness of the optimal positioning of the delivery window and the expected penalty cost of untimely delivery in terms of the width of the on-time portion of the
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delivery window, penalty costs per unit time early and late, mean, and variance of the normally distributed delivery time.
The normal probability distribution is the most widely used probability model for
defining the delivery distribution of supply chain delivery times and was found in 17 of the 38 delivery performance models reviewed by Guiffrida (2014). The widespread use of normal probability density function to model the distribution of delivery time in supply chain delivery performance models suggests an immediate limitation since the delivery time distribution under these models is always symmetric. Also, negative delivery times are possible under the normal distribution. 3
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After analyzing the effect of parameters of the delivery time distribution on delivery performance, we investigate and compare strategies for improving delivery performance. Modeling cost reduction allows managers to compare investments and saving and it should be planned in advance how the parameters will change over time. In the paper herein we analyze
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how the lead time reduction affects the expected penalty costs. Two models of cost reduction are used: hyperbolic and exponential. These forms have been widely adopted in several process improvement studies including delivery performance improvement (Guiffrida and Nagi, 2006a; Tubino and Suri, 2000; Choi, 1994).
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Our model and the supporting propositions derived herein contribute to the literature along the following two dimensions. First, the model represents a generalized modeling approach
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to the evaluation and improvement of supply chain delivery performance. The approach applied to the model with Gaussian distributed delivery time can be used for other delivery time distributions. Second, the analytical analysis of the key attributes of the model that are presented in the set of supporting model propositions provide managerial insight into the dynamics of integrating the model into the long term continuous improvement of delivery performance within the supply chain. The results presented in the paper herein are novel to theory and practice in two
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ways. Firstly, the manuscript shows that the mean of the normally distributed delivery time has no effect on the expected penalty cost. Thus, from this perspective a supplier has no interest in reducing the average delivery time. Secondly, although it’s well-known that delivery variance reduction is beneficial for a supplier, there is no research that shows how much a supplier can
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save by reducing the variance of a delivery time distribution. The research herein provides the results that help to estimate cost savings and justify the investments in delivery variance
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reduction.
This paper is organized as follows. In section 2, a cost-based model for evaluating
delivery performance is reviewed. In sections 3 and 4, a set of propositions which highlight the key analytical characteristics of the model are introduced. Section 3 evaluates the effect of parameters defined in a contract (the width of the on-time portion of the delivery window and penalty costs per unit time early and late) on the optimal position of the delivery window and the expected penalty cost. Section 4 revises the effect of normal delivery time distribution parameters (mean and variance) on the optimal position of the delivery window and the expected
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penalty cost. In section 5, theoretical and managerial implications of the findings are discussed. Section 6 provides a summary and discusses future research directions.
2. Modeling supply chain delivery performance
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A common feature of the supply chain delivery performance models is the use of the delivery window. Under the concept of a delivery window, contractually agreed upon
benchmarks in time are used to classify deliveries as being early, on-time, and late (see Figure 1). Early and late deliveries introduce waste in the form of excess cost into the supply chain;
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early deliveries contribute to excess inventory holding costs while late deliveries may contribute to production stoppage costs, lost sales and loss of goodwill. These costs for untimely (early and
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late) delivery are referred to in the literature as penalty costs (Guiffrida and Nagi, 2006a, 2006b). When a delivery is within the on-time portion of the delivery window, no penalty cost is incurred.
--------------------------------- Insert Figure 1 about here ---------------------------------
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Guiffrida and Nagi (2006a) proposed the following expected penalty cost per period when deliveries are classified as early and late according a delivery window c1
Y = Yearly + Ylate = QH ∫ (c1 − x) f ( x)dx + K 0
∞
∫ ( x − (c + ∆c)) f ( x)dx , 1
(1)
c1 + ∆c
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where Y = expected penalty cost of untimely delivery, c1
Yearly = QH ∫ (c1 − x ) f ( x ) dx is the expected penalty cost of early delivery,
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0
Ylate = K
∞
∫ ( x − (c
1
+ ∆c )) f ( x ) dx is the expected penalty cost of late delivery,
c1 + ∆c
f(x)= the probability density function (pdf) of delivery time x,
QH = penalty cost per time unit early (levied by the buyer), K = penalty cost per time unit late (levied by the buyer),
c1 = difference between the time the delivery process is initiated and the earliest acceptable delivery time, ∆c = the width of the on-time portion of the delivery window. 5
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Although the earliest acceptable delivery time is predefined by the contract, the supplier can define the time when delivery begins, therefore changing the value of c1. For example, if the supplier decides to ship the product 10 hours before the earliest accepted delivery time, c1 is equal to 10 hours.
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As demonstrated in Bushuev and Guiffrida (2012), Y is a convex function of c1 and the optimal value of c1 (which is defined as c1*) that minimizes Y can be determined by evaluating
K ⋅ Plate = QH ⋅ Pearly , ∞
∫
f ( x)dx and Pearly =
c1 + ∆c
c1
∫ f ( x)dx
are the probabilities of late and early deliveries.
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where Plate =
(2)
In the paper herein we assume that the supplier uses the concept of the optimal position
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of the delivery window (OPDW) to define the optimal value of c1 and, accordingly, the time when the product should be shipped to the buyer.
There are several key parameters of the expected penalty cost function that can reduce the expected penalty cost and be used to determine the potential for improvement in delivery performance. The parameters are: the width of the on time portion of the delivery window, the penalty cost for early deliveries, the penalty cost for late delivery, and the parameters of delivery
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time distribution. The width of the on time portion of the delivery window and the penalty costs for early and late delivery are defined in a contract. Thus, they are chosen during a negotiation process between a buyer and a supplier. The delivery time distribution and its parameters can be estimated based on statistics of the past deliveries. These parameters can be changed by a
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supplier alone, but it requires time and resources.
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3. Effect of parameters defined in a contract on expected penalty cost In this section, we present propositions which can be used for examining how the optimal
value of c1 which defines when the product should be shipped with respect to the earliest acceptable delivery date and the expected penalty cost behave as a function of the parameters defined in a contract, i.e. the width of the on time portion of the delivery window and the penalty costs for early and late delivery. The width of the on-time portion of the delivery window and the accompanying penalties costs for early and late delivery are defined by a contract between a buyer and a supplier. Understanding the robustness of Y as a function of the width of the on-time portion of the 6
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delivery window and the penalty costs for untimely delivery is of interest since changes to the width impact the buyer and seller differently. Typically, a buyer would like to decrease the width of the on-time portion of the delivery window because it will decrease delivery time uncertainty. Alternatively, a supplier would like to increase the width of on-time portion of the delivery
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window because it decreases the expected penalty cost for untimely delivery.
3.1. Width of the on-time portion of the delivery window
Proposition 1. For the fixed delivery time distribution parameters and the penalty costs
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per unit per time early and late, increasing the width of the on-time portion of the delivery
window by one unit of time reduces the total expected penalty cost by K·Plate and decreases the
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optimal position of the delivery window.
Proof. The optimal position of the delivery window is
Plate =
QH Pearly or K ⋅ Plate = QH ⋅ Pearly . K
Using the derivatives
dPearly d∆c
dc1* dc1* f (c1* ) − 0 + 0 = f (c1* ) , d (∆c ) d ( ∆c )
(4)
(5)
dc1* QH dc1* − + 1 f (c1* + ∆c ) = f (c1* ) ; d ( ∆ c ) K d ( ∆ c )
(6)
dc1* QH * f (c1* ) = − f (c1* + ∆c) ; f (c1 + ∆c) + d ( ∆c ) K
(7)
K f (c1* + ∆c) dc1* K f (c1* + ∆c) 1 + . =− = − d (∆c) K f (c1* + ∆c) + QH f (c1* ) QH f (c1* )
(8)
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we have
=
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dc1* dc1* dPlate = 0 − + 1 f (c1* + ∆c) + 0 = − + 1 f (c1* + ∆c) , d∆ c d ( ∆c ) d ( ∆c )
(3)
Because the derivative (8) is always negative, increasing the width of the on-time portion
of the delivery window will require delivering earlier to keep the position of the delivery window optimal.
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Using envelope theorem
dy * ∂y x = x* ( a ) the derivative of the expected penalty cost = da ∂a
{
}
is
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b ∂Y = 0 + K 0 − 1 ⋅ ((c1 + ∆c − (c1 + ∆c)) f (c1 + ∆c )) + ∫ (0 − 1) f ( x) dx ; ∂ ( ∆c ) c1 + ∆c
dY = − K ⋅ Plate . d ( ∆c )
(9)
(10)
cost. ■
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3.2. Penalty costs for early and late deliveries
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Thus, increasing the width of the delivery window will decrease the expected penalty
Proposition 2. For the fixed delivery time distribution parameters, the width of the
delivery window, and the penalty cost for late delivery, increasing penalty costs per unit time early by 1 unit increases the total expected penalty cost by Yearly and decreases the optimal position of the delivery window. Proof. Using the derivatives
dPearly
we have
dc1* dc1* f (c1* ) − 0 + 0 = f (c1* ) , d (QH ) d (QH )
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d (QH )
=
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dPlate dc1* dc1* * =0− f (c1 + ∆c) + 0 = − f (c1* + ∆c) , d (QH ) d (QH ) d (QH )
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dc1* dc1* f (c1* + ∆c) = Pearly + QH K − f (c1* ) ; d ( QH ) d ( QH )
(
)
(11)
(12)
(13)
dc1* K f (c1* + ∆c) + QH f (c1* ) = − Pearly ; d (QH )
(14)
Pearly dc1* . =− * d (QH ) K f (c1 + ∆c) + QH f (c1* )
(15)
The derivative (15) is always negative, hence increasing QH decreases the optimal value of c1. Using envelope theorem the derivative of the expected penalty cost is
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dY = Yearly + 0 = Yearly . d (QH )
(16)
The derivative (16) is always positive and increasing QH increases Y. ■ Proposition 3. For the fixed delivery time distribution parameters, the width of the
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delivery window, and the penalty cost for early delivery increasing penalty costs per unit per time late by 1 unit raises the total expected penalty cost by Ylate and increases the optimal position of the delivery window. Proof. Using the derivatives
dc1* dc1* * = f (c1* ) , f (c1 ) − 0 + 0 = dK dK dK
we have
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dPearly
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dPlate dc* dc* = 0 − 1 f (c1* + ∆c ) + 0 = − 1 f (c1* + ∆c ) , dK dK dK
dc1* dc1* * Plate + K − f (c1 + ∆c) = QH f (c1* ) ; dK dK
(
)
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dc1* K f (c1* + ∆c) + QH f (c1* ) = Plate ; dK dc1* Plate = . * dK K f (c1 + ∆c) + QH f (c1* )
(17) (18)
(19)
(20) (21)
The derivative (21) is always negative, thus increasing K increases the optimal value of c1.
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Using envelope theorem the derivative of the expected penalty cost is dY = 0 + Ylate = Ylate . dK
(22)
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The derivative (22) is always positive, thus increasing K increases Y. ■
It can be interesting to estimate the effect of penalties for earliness and lateness together.
From (2) we can see that the optimal value of c1 is a function of QH and K, but if the proportion of QH to K does not change the optimal value of c1 does not change also. So we could use a coefficient which will combine both values of QH and K. We suppose to use the next coefficients K = K 0 ⋅ K1 ; QH = (1 − K 0 ) K1 ; QH + K = K 0 ⋅ K1 .
(23)
And we have 9
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QH (1 − K 0 ) K1 1 − K 0 1 = = = −1. K K 0 ⋅ K1 K0 K0
(24)
Expected penalty cost is Y = K1 ((1 − K 0 )Yearly + K 0Ylate ) ,
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(25)
and the optimal position of the delivery window is K 0 ⋅ Plate = (1 − K 0 ) ⋅ Pearly .
(26)
The coefficient K1 is a common part of early and late penalties per unit per time.
Increasing (decreasing) K1 means that both QH and K increasing (decreasing) but the proportion
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QH/K stays the same.
Proposition 4. For the fixed delivery time distribution parameters and the width of the
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delivery window proportionally increasing penalty costs per unit per time early and late increase the total expected penalty cost by (1 − K 0 )Yearly + K 0Ylate and has no effect on the optimal position of the delivery window.
Proof. It can be concluded from (26) that K1 is not a part of the equation that defines the optimal position of the delivery window. Thus, changing K1 will have no effect on the optimal value of c1.
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The first derivative of Y by K1 is dY = (1 − K 0 )Yearly + K 0Ylate . dK1
(27)
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The derivative (27) is always positive, because 0 < K0 < 1. Thus increasing K1 will increase Y. ■
Proposition 5. For the fixed delivery time distribution parameters and the width of the delivery window
increasing K0 will increase the optimal position of delivery window;
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and the total expected penalty cost is concave by K0 with maximum at Yearly = Ylate.
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Proof. Using the derivatives dPlate dc* dc* = 0 − 1 f (c1* + ∆c) + 0 = − 1 f (c1* + ∆c) , dK 0 dK 0 dK 0 dPearly dK 0
=
dc1* dc* f (c1* ) − 0 + 0 = 1 f (c1* ) , dK 0 dK 0
(28)
(29)
we have
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dc* dc* Plate + K 0 − 1 f (c1* + ∆c ) = − Pearly + (1 − K 0 ) 1 f (c1* ) ; dK 0 dK 0
(
)
dc1* K 0 f (c1* + ∆c) + (1 − K 0 ) f (c1* ) = Plate + Pearly ; dK 0
(30)
(31)
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Plate + Pearly dc1* = . * dK 0 K 0 f (c1 + ∆c) + (1 − K 0 ) f (c1* )
(32)
The derivative (32) is always positive, hence increasing K0 increases the optimal value of c1. The first derivative of the total cost is
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dY = K1 (Ylate − Yearly ) . dK 0
dYearly dK 0
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Using that
(33)
c*
1 dc1* * * dc1* dc1* * = (c1 − c1 ) f (c1 ) − 0 + ∫ f ( x) dx = Pearly , dK 0 dK dK 0 0 a
(34)
b
dYlate dc* dc* dc* = 0 − 1 (c1* + ∆c − (c1* + ∆c)) f (c1* + ∆c ) + ∫ − 1 f ( x )dx = − 1 Plate , (35) dK 0 dK 0 dK 0 dK 0 c * + ∆c 1
the second derivative is
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d 2Y dc1* (Pearly + Plate ). = − K 1 d ( K 0 )2 dK 0
(36)
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The second derivative is always negative and the cost function Y is concave by K0. ■ 4. Effect of normal delivery time distribution parameters on expected penalty cost
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This section revises the results presented in (Guiffrida and Nagi, 2006a) assuming that a
supplier uses the optimal position of the delivery window. For normal distribution, the total expected penalty cost per period for normally distributed delivery time is (Guiffrida and Nagi, 2006a)
c* − µ + c1* − µ Y = QH σ φ 1 σ
(
* 1
c* + ∆c − µ − c1* + ∆c − µ + K σ φ 1 σ
(
) Φ c σ− µ + c* + ∆c − µ 1 − Φ 1 , σ
)
(37)
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where ϕ ( X ) =
X
1 1 exp( X ) , Φ ( X ) = ∫ exp( x )dx . 2π 2π −∞
The optimal position of delivery window is when
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c* − µ c* + ∆c − µ = K ⋅ 1 − Φ 1 . QH ⋅ Φ 1 σ σ
(38)
Proposition 6. Increasing the mean of normally distributed delivery time by one time
unit increases the optimal position of the delivery window by one time unit and has no effect on
Proof. Using the derivative X
1 exp( X ) − 0 + ∫ 0dx = ϕ ( X ) . 2π −∞
(39)
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Φ( X ) = dX
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the expected penalty cost.
the derivative for the optimal position of the delivery window is
(40)
c1* − µ c1* + ∆c − µ c1* − µ c1* + ∆c − µ dc1* ; QH + K = QH + K ϕ ϕ ϕ ϕ σ σ dµ σ σ
(41)
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QH dc1* c1* − µ K dc1* c1* + ∆c − µ = ⋅ − ; − 1ϕ − 1ϕ σ σ dµ σ σ dµ
dc1* =1. dµ
(42)
From (Guiffrida and Nagi, 2006a)
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c* − µ c* + ∆c − µ dY + K 1 − Φ 1 , = QH − Φ 1 dµ σ σ
(43)
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Knowing that the optimal position of the delivery window is used, the equation (38) is
applied and dY dµ = 0 . Thus the mean of the delivery time has no effect on the expected
penalty cost function. ■
Proposition 7. Increasing the standard deviation of normally distributed delivery time
increases the optimal position of the delivery window if −1
c* + ∆c − µ QH c1* − µ c* + ∆c − µ + ϕ 1 > 0 , A = c − µ + ∆c ⋅ ϕ 1 ⋅ ϕ σ σ K σ * 1
(44)
and decrease it otherwise. 12
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Proof. Using the derivative (39), the derivative for the optimal position of the delivery window is
c1* − µ c* + ∆c − µ dc1* + Kσ ⋅ ϕ 1 = QHσ ⋅ ϕ σ dσ σ
(45)
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c1* − µ QH dc1* * ϕ = σ µ c − ( − ) 1 σ 2 dσ σ c* + ∆c − µ K dc* ; = 2 ⋅ − σ 1 − (c1* + ∆c − µ ) ϕ 1 σ dσ σ
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c* − µ c* + ∆c − µ + K (c1* + ∆c − µ )ϕ 1 ; = QH (c − µ )ϕ 1 σ σ * 1
−1
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c* + ∆c − µ dc1* c1* − µ ∆c c1* + ∆c − µ QH c1* − µ + ϕ 1 . ■ = + ϕ ⋅ ϕ dσ σ σ σ σ K σ Remark 1. The value of A is positive for QH/K < 1,
•
negative for QH/K > 1,
•
and equal to 0 for QH/K = 1 (see Appendix A for proof).
(47)
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•
(46)
Proposition 8. Increasing the standard deviation of normally distributed delivery time increases the expected penalty cost.
Proof. Using envelope theorem and the results from (Guiffrida and Nagi, 2006a; eq. 7)
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c1* − µ c1* + ∆c − µ dY ⋅ (2σ ) −1 . ■ = QH ⋅ ϕ + K ⋅ ϕ σ dσ σ
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As it is shown in Proposition 6, the mean of the normally distributed delivery time has no effect on the expected penalty cost; hence, there is no need to change the mean for delivery performance improvement.
A supplier’s strategy on delivery performance improvement should focus on delivery
variance reduction. This reduction should take into account that the optimal position of the delivery window will change with variance. Therefore, the results from (Guiffrida and Nagi, 2006a; eq. 13 and 14) should be rewritten. For hyperbolic delivery time variance reduction the expected penalty cost is
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M t * ( c* − µ ) 2 + c1* − µ Φ Y (ν , t ) = QH exp 1 (c1 − µ ) + 2M M 2π t
)
M (c* + ∆c − µ ) 2 − c1* + ∆c − µ + K exp 1 2M 2π t
(
1 − Φ t (c1* + ∆c − µ ) , M
)
(48)
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(
where t = time,
M = initial level of variance.
The expected penalty cost assuming exponential delivery time variance reduction is
(
)
P rt P + e rt (c1* + ∆c − µ ) 2 − c1* + ∆c − µ + K exp − 2 2 P π where P = initial level of variance, r = variance decay rate.
5. Discussion
rt 1 − Φ e (c* + ∆c − µ ) , P 1
(49)
)
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(
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P e rt * rt P + e rt (c1* − µ ) 2 + c1* − µ Φ Y = QH exp − (c1 − µ ) + P 2P 2π
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As shown in Proposition 1, a supplier would prefer to increase the width of the on-time portion of the delivery window thereby reducing the expected penalty cost for untimely delivery. However, from the buyer’s perspective, a wider width for the on-time portion of the delivery window increases delivery time uncertainty. Higher levels of delivery uncertainty could
with the supplier.
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contribute to unwillingness by the buyer to enter into a contractually defined delivery window
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The effect of penalties for untimely delivery is evaluated in Propositions 2-5. As it is shown in Propositions 2, 3, and 4, increasing the penalties for early or late deliveries will increase the expected penalty cost. Proposition 5 shows that the balance between penalties for early and late deliveries will increase the expected penalty cost if Ylate > Yearly and decrease the cost if Ylate < Yearly. Thus, the supplier prefers a wide on-time portion of the delivery window, low penalties for untimely delivery with a balance between penalties such that Ylate significantly bigger or smaller than Yearly.
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Unlike the width of the on-time portion of the delivery window and penalties for untimely delivery, the parameters of delivery time distribution (mean and variance) can be changed by a supplier alone but the changes require time and investment. Thus the supplier should compare benefits of the delivery performance improvement and expenses needed for that.
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There are several practices that can improve delivery performance, among them are Just-in-time (Mackelprang and Nair, 2010) and communication and information sharing (Carr and Kaynak, 2007), design for manufacturability (Ketokivi and Schroeder, 2004).
Propositions 6, 7 and Remark 1 demonstrate how a supplier should change the parameters
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of the distributed delivery time to decrease the expected penalty cost. Proposition 6 shows that the mean of normally distributed delivery time has no effect on the expected penalty cost when a
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supplier applies optimal position of the delivery window. Hence, in this case a supplier should focus solely on decreasing a variance of delivery time distribution to reduce the expected penalty cost for untimely delivery. Many researches focus on delivery time reduction and show its negative effect on a buyer (Glock, 2012; Hayya et al., 2011; De Treville et al., 2004). Although, in many cases a supplier is responsible for delivery and makes decision about delivery performance improvement. Therefore, a buyer should motivate its suppliers to improve delivery
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performance. The results herein show that penalizing suppliers for untimely delivery makes them interested in delivery time variance reduction, but does not motivate them to reduce the expected delivery time.
At the same time, it is shown how the width of the on-time portion of the delivery
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window, the penalties for untimely delivery, and the delivery time distribution parameters will affect the optimal value of c1 and the time a supplier should ship a product to minimize the if • • • •
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expected penalty cost. It is shown that the optimal position of the delivery window will decrease
the width of on-time portion of the delivery window increases,
the penalty cost for late delivery increases, the penalty cost for early delivery decreases, and the mean of normally distributed delivery time decreases.
Moreover, changing the mean by one time unit will change the optimal position of the delivery window by one time unit. For example, if the average delivery time is decreased by 10 hours, the product will need be shipped exactly 10 hours earlier to maintain the optimal position of the 15
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delivery window. The effect of the standard deviation of normally distributed delivery time on the optimal position of the delivery window depends on the value of QH/K. Increasing the standard deviation will increase the optimal position of the delivery window if QH/K < 1,
•
increase the optimal position of the delivery window if QH/K > 1,
•
and keep the optimal position c1* = µ − ∆c / 2 if QH/K = 1.
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•
That suggests that the supplier should coordinate the shipment of the product so that the product arrives on the buyer’s requested delivery date as dictated by the optimal value of c1. This
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action may require the supplier to adjust their production schedule to avoid increases in
inventory levels and related costs and if needed, adjust delaying the shipping of the product to
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avoid incurring higher in-house inventory levels and related costs.
Although the effect (positive or negative) of several parameters discussed in the manuscript is obvious, the magnitude of the effect had to be estimated. Understanding the magnitudes of the effect of the width of the on-time portion of the delivery window and penalties for untimely delivery on the expected penalty cost, a supplier and a buyer can better work in partnership to find the value of the width of the on-time portion of the delivery window which
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will satisfy the desired levels of expected cost and delivery time uncertainty valued by the supplier and the buyer. Also the results of the paper herein show a supplier and a buyer how changing the parameters defined in the contract (the width of the on-time portion of the delivery window and penalties for untimely delivery) affect the expected penalties which can help a
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supplier and a buyer during the contract negotiation process. Propositions 1 through 5 can help a supplier during a contract negotiation process. A
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supplier needs to know by how much parameters of a contract increase or decrease the expected penalties paid by the supplier. In this case, a supplier can justify how the product price should be changed to cover the penalty costs and either raise the price for the buyer accordingly or decline to sign the contract.
The effect of distribution parameters on the expected penalty cost analyzed herein can be
used as a tool for supporting and justifying financial investment needed to improve delivery performance. The analysis lets a manager estimate the expected savings that can accrue for temporal changes to the parameters of the delivery time distribution. These expected cost savings can be used in the decision making process of committing financial investment to improvements 16
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in on-time delivery performance. Thus the supplier should compare the delivery performance improvement benefits with the associated costs to determine a long-term return on investment. A company can use the information presented herein to perform a cost benefit analysis to determine the best delivery performance strategy. Firstly, our observations map out the direction
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in which the distribution parameters should be changed to reduce the expected penalty cost. Secondly, implementation of delivery performance improvement practices requires investment. Therefore, before a company invests in a delivery improvement project, a decision maker should compare the investments with the expected savings that can be estimated using the analysis
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presented in this paper. Thirdly, for companies implementing continuous delivery performance improvement, a cost reduction related to untimely delivery can be modeled so that the decision
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maker knows when and how much will be saved in the future. Also, the analysis can show which parameters will have the greatest effect on penalty costs thereby supporting the delivery improvement program.
5. Conclusion
The paper herein investigates the effect of different parameters on the optimal position of
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the delivery window and the expected penalty cost. The results allow developing strategies for improving delivery performance from a supplier’s perspective and answer the question what supplier should do to decrease the expected penalty cost of untimely delivery. The research uses the expected penalty cost function developed by Guiffrida and Nagi (2006a) and assumes that the
Guiffida (2012).
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supplier will choose the optimal position of delivery window as proposed by Bushuev and
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Proposition 1 suggests increasing the width of on-time portion of the delivery window to decrease the expected penalty cost. Results from Propositions 2-5 demonstrate that decreasing the penalties for early and late deliveries reduces the expected penalty cost. The results in Propositions 1-5 are provided in general form and can be used for any delivery time distribution. Although a supplier would obviously prefer low penalties for untimely delivery and wider ontime portion of the delivery window, the propositions clearly define the effect of the parameters on the expected penalty cost. For example, a supplier can easily estimate the expected penalty cost increase for $1 increase in penalties for unit time early or late.
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In addition to that, the results for the normal distribution from Guiffrida and Nagi (2006a) were revised using the concept of the optimal position of the delivery window. Propositions 6-7, which focus on the mean and the variance of normally distributed delivery time, recommend decreasing the variance parameter to reduce the expected penalty cost and show that the mean
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has no effect on the expected penalty cost if the optimal position of the delivery window is
chosen. This is a surprising result, since expected delivery time is considered to have negative effect on costs.
At the same time, changing the parameters (the width of the on-time portion of the
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delivery window, the penalties for early and late deliveries, the mean, and the variance of the delivery time distribution) has an effect on the optimal position of the delivery window and as a
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result on supplier’s costs such as inventory and production costs. Thus delivery performance improvement should be integrated with production and inventory functions within the company to keep costs low.
The results presented in the paper can be used by both researchers and practitioners. The managerial implication of the paper is that it can serve as guidance for practitioners undertaking a program to improve delivery performance. A supplier can use the knowledge about the effect
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of the width of on-time portion of the delivery window and penalties for early and late deliveries on the expected penalty cost during the negotiation process with a buyer. By understanding the effect of delivery time distribution parameters (mean and variance) on the expected penalty cost, a supplier can develop a strategy for continuous improvement of delivery performance.
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Although, a buyer should keep in mind that penalties for untimely delivery do not motivate suppliers to reduce the expected delivery time.
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A major research contribution of this paper is the proposal of a general approach to modeling delivery performance improvement. The model was demonstrated for normally distributed delivery time, but can be applied to other distribution forms which can be done in future publications. Moreover, for other delivery time distributions, the correlation between delivery time distribution parameters can be assumed. The results derived in the paper herein can be used by a supplier to see potential delivery cost savings in order to evaluate alternatives for delivery performance improvement.
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Appendix A. We will evaluate the effect of the penalty cost ratio on A. Using envelope theorem: −1
c* + ∆c − µ c1* − µ dA ⋅ ϕ ≥ 0 . = ∆c ⋅ ϕ 1 d (QH / K ) σ σ
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(A.1)
Thus, A is a monotonically increasing function and for any given values of ∆c, µ, and σ there is only one value of QH/K which turns A into 0. As it will be shown below, A = 0 when QH/K = 1. For QH/K = 1, the optimal c1 is when Pearly = Plate. Because normal distribution is symmetric, that is possible only if the mean is placed right in the middle of the on-time portion
∆c 2
c1* + ∆c − µ c1* − µ and ϕ = ϕ σ . σ Hence, for QH/K = 1,
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c1* = µ −
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of the delivery window and µ – c1 = c1 + ∆c – µ. In this case,
(A.2)
(A.3)
−1
c* − µ c1* − µ c* − µ ∆c ϕ + ϕ 1 = 0 A=µ− − µ + ∆c ⋅ ϕ 1 2 σ σ σ
(A.4)
References
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for any values of ∆c, µ, and σ.
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Bushuev, M. A. and Guiffrida, A. L. (2012). Optimal position of supply chain delivery window: Concepts and general conditions. International Journal of Production Economics, 137(2), 226–234.
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Bushuev, M.A., Guiffrida, A.L., and Shanker, M. (2011). A generalized model for evaluating supply chain delivery performance. Proceedings of the 47 MBAA International Conference, Chicago, pp. 4 - 8 Carr, A. S. and Kaynak, H. (2007). Communication methods, information sharing, supplier development and performance: An empirical study of their relationships. International Journal of Operations & Production Management, 27(4), 346 –370. Chen, L., Guiffrida, A. L., and Datta, P. (2015). Capacity-delivery coordination in supply chains: A cost-based approach, International Journal of Operational Research (forthcoming). Choi, J. W. (1994). Investment in the reduction of uncertainties in just-in-time purchasing systems. Naval Research Logistics, 41(2), 257-272.
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Cirtita, H. and Glaser-Segura, D. A. (2012). Measuring downstream supply chain performance. Journal of Manufacturing and Technology Management 23(3): 299-314.
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De Treville, S., Shapiro, R. D., and Hameri, A. P. (2004). From supply chain to demand chain: the role of lead time reduction in improving demand chain performance. Journal of Operations Management, 21(6), 613-627. Estampe, D., Lamouri, S., Paris, J.-L., and Brahim-Djelloul, S. (2013). A framework for analyzing supply chain performance evaluation models. International Journal of Production Economics, 142(2), 247-258.
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Forslund, H., Jonsson, P., and Mattsson, S. (2009). Order-to-delivery process performance in delivery scheduling environments. International Journal of Productivity and Performance Management 58(1): 41-53.
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Garg, D., Naraharai, Y. and Viswanadham, N. (2006). Achieving sharp deliveries in supply chains through variance reduction. European Journal of Operational Research, 171(1), 227254. Glock, C. H. (2012). Lead time reduction strategies in a single-vendor–single-buyer integrated inventory model with lot size-dependent lead times and stochastic demand. International Journal of Production Economics, 136(1), 37-44. Guiffrida, A. L. (2014). Recent trends in supply chain delivery models. International Journal of Social, Behavioral, Educational, Economic and Management Engineering 8(6): 1801-104.
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Guiffrida, A. L. and Nagi, R. (2006a). Cost characterizations of supply chain delivery performance. International Journal of Production Economics, 102(1), 22-36. Guiffrida, A. L. and Nagi, R. (2006b). Economics of managerial neglect in supply chain delivery performance. The Engineering Economist, 51(1), 1-17.
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Guiffrida, A. L. and Jaber, M. Y. (2008). Managerial and economic impacts of reducing delivery variance in the supply chain. Applied Mathematical Modeling, 32(10), 2149-2161.
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Gunasekaran, A., Patel, C., and McGaughey, X. (2004). A framework for supply chain performance measurement. International Journal of Production Economics, 87(3), 333-347. Hayya, J. C., Harrison, T. P., and He, X. J. (2011). The impact of stochastic lead time reduction on inventory cost under order crossover. European Journal of Operational Research, 211(2), 274-281. Hsu, B.-M., Hsu, L.-Y., and Shu, M.-H. (2013). Evaluation of supply chain performance using delivery-time performance analysis chart approach. Journal of Statistics and Management Systems, Vol. 16 No. 1, pp. 73 – 87 Huan, S. H., Sheoran, S. K. and Wang, G. (2004). A review and analysis of supply chain operations reference (SCOR) model. Supply Chain Management: An International Journal, 9(1), 23-29. 20
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Ketokivi, M. and Schroeder, R. (2004). Manufacturing practices, strategic fit and performance: A routine-based view. International Journal of Operations & Production Management, 24(2), 171–191.
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Mackelprang, A. W. and Nair, A. (2010). Relationship between just-in-time manufacturing practices and performance: A meta-analytic investigation. Journal of Operations Management, 28(4), 283-302.
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Melnyk, S. A., Bititci, U., Platts, K., Tobias, J., & Andersen, B. (2014). Is performance measurement and management fit for the future? Management Accounting Research, 25(2), 173-186. Nabhani, F. and Shokri, A. (2009). Reducing the delivery lead time in a food distribution SME through the implementation of six sigma methodology. Journal of Manufacturing Technology Management, 20(7), 957-974 Ngniatedema, T., Chen, L., and Guiffrida, A. L. (2016).A modelling framework for improving supply chain delivery performance. International Journal of Business Performance and Supply Chain Modelling, 8(2), 79-96. Olhager, J., and Selldin, E. (2004). Supply chain management survey of Swedish manufacturing firms. International Journal of Production Economics, 89(3), 353-361.
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Rezapour, S., Allen, J. K., and Mistree, F. (2015). Uncertainty propagation in a supply chain or supply network. Transportation Research Part E: Logistics and Transportation Review, 73, 185-206. Russell, R. S. and Taylor, B. W. (1998). Operations management: Focusing on quality and competitiveness. Upper Saddle River, NJ: Prentis Hall, p.733. Safaei, M., and Thoben, K. D. (2014). Measuring and evaluating of the network type impact on time uncertainty in the supply networks with three nodes. Measurement, 56, 121-127.
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Slotnick, S. A. and Sobel, M. J. (2005). Manufacturing lead-time rules: Customer retention versus tardiness costs. European Journal of Operational Research, 163(3), 825-856.
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Tanai, Y. and Guiffrida, A. L. (2015). Reducing the cost of untimely supply chain delivery performance for asymmetric Laplace distributed delivery. Applied Mathematical Modelling, 39(13), 3758-3770. Tubino, F., and Suri, R. (2000). What kind of “numbers” can a company expect after implementing quick response manufacturing. Empirical data from several projects on lead time reduction. Quick Response Manufacturing 2000 Conference Proceedings, 943-972. Dearborn, MI: Society of Manufacturing Engineers Press. Wang, F. K. and Du, T. (2007). Appling capability index to the supply chain network analysis. Total Quality Management, 18(4), 425-434.
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Figure 1. Illustration of delivery window.
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Legend: f(x) is the probability density function (pdf) of delivery time x; c1 is difference between the time the delivery process is initiated and the earliest acceptable delivery time; ∆c is
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the width of the on-time portion of the delivery window.
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• •
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The effect of parameters defined in a contract on the expected penalty cost function is investigated Effect of normal delivery time distribution parameters on the expected penalty cost is analized Delivery performance improvement is explored Theoretical and managerial implications of the findings are discussed
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S0925-5273(17)30322-5
DOI:
10.1016/j.ijpe.2017.10.007
Reference:
PROECO 6839
To appear in:
International Journal of Production Economics
Received Date: 20 April 2017 Revised Date:
23 September 2017
Accepted Date: 7 October 2017
Please cite this article as: Bushuev, M.A., Delivery performance improvement in two-stage supply chain, International Journal of Production Economics (2017), doi: 10.1016/j.ijpe.2017.10.007. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Delivery Performance Improvement in Two-stage Supply Chain
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Maxim A. Bushuev Assistant Professor Department of Information Science and Systems Earl Graves School of Business and Management Morgan State University Baltimore MD 21251 Phone: (330) 389-4967 [email protected]
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Please direct all correspondence to Maxim A. Bushuev Email: [email protected]
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Delivery Performance Improvement in Two-stage Supply Chain
Abstract
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This paper investigates strategies of supply chain delivery performance improvement. The performance is measured using a cost-based analytical model which evaluates the expected penalty cost for early and late delivery. The results demonstrate strategies for improving delivery performance when a supplier uses an optimally positioned delivery window to minimize the expected penalty cost. The effect of the width of the delivery window and penalty costs for early and late deliveries on the optimal position of the delivery window and the expected penalty cost are explored in general case. In addition to that, the effect of a mean and variance parameters of a normal delivery time distribution are revised. Theoretical and managerial implications of the findings are discussed.
Keywords: Supply chain Management, Delivery performance improvement, Delivery window
1. Introduction
The need for performance measurement and evaluation in supply chain management is
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well recognized in the literature (Melnyk et al., 2014; Estampe et al., 2013). Within the hierarchy of supply chain performance metrics delivery performance, as characterized by the timeliness and dependability of product delivery to the final customer in the supply chain, is acknowledged
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as a key metric for supporting supply chain operations (Cirtita and Glaser-Segura, 2012; Forslund et al., 2009; Olhager and Selldin, 2004). Delivery performance is classified as a strategic level performance measure by Gunasekaran et al. (2004) and is also a major component
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of the Supply Chain Operations Reference (SCOR) model (Huan et al., 2004). Delivery windows are often incorporated into performance measurement systems when
performance with respect to deadlines and/or due dates is measured in integrated productiondistribution systems. A delivery window is a time interval within which a delivery can be received. A supplier and a buyer contractually specify allowable deviations (earliness and lateness) from an agreed upon delivery date which are used to classify deliveries as being early, on-time, and late. The on-time portion of the delivery window is between the earliest acceptable delivery time and date and the latest acceptable delivery time and date. When a delivery is made
1
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within the on-time portion of the delivery window, no penalty cost is incurred. Early and late deliveries introduce waste in the form of excess cost into the supply chain. In supply chain delivery models the costs associated with untimely (early and late) delivery are referred to as penalty costs which are paid by the supplier to the buyer (Guiffrida and Nagi, 2006a, 2006b).
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Therefore, it is in the best interest of a supplier to reduce the penalty costs. Penalty costs for late delivery have been reported in per-unit-time functional forms in the literature from real-world examples in the automotive and aerospace industries (Russell and Taylor, 1998; Slotnick and Sobel, 2005).
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Models for evaluating delivery performance within supply chains can be categorized into two groups: i) index based models (Hsu et al., 2013; Nabhani and Shokri, 2009; Wang and Du,
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2007), and ii) cost based models (Chen et al., 2015; Bushuev and Guiffrida, 2012; Bushuev et al., 2011). These models are applicable to both serial and network supply chains (Safaei and Thoben, 2014; Rezapour et al., 2015). Both the index and cost based categories of models are similar in that delivery timeliness to the final customer is analyzed with regard to the customer’s specification of an on-time delivery window. The models differ in how they report delivery performance in terms of an overall metric. Index based models translate the probability of
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untimely (early and late delivery) into a “delivery capability index” measure. Cost based models use partial expectations to directly translate the probability of untimely delivery into an expected cost measure.
Most of the papers published in the area are focused on delivery performance evaluation.
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Once the currently level of delivery performance has been evaluated, the next natural step is to find ways to improve delivery performance. Evaluating supply chain delivery performance
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models provides additional opportunities for the delivery performance improvement using supply contract negotiation and delivery time distribution improvement. The present paper extends research in the area investigating delivery performance
improvement. The papers focused on delivery performance improvement (Ngniatedema et al., 2016; Tanai and Guiffrida, 2014; Guiffrida and Jaber, 2008; Garg et al., 2006; Guiffrida and Nagi, 2006a, 2006b) fail to address the optimal positioning of the delivery window when evaluating delivery performance. The failure to optimally position the delivery window has been shown by Bushuev and Guiffrida (2012) to lead to suboptimal delivery cost performance and may also impede the coordination of the delivery process within a supply chain. Bushuev and 2
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Guiffrida (2012) demonstrate that optimally positioning of the delivery window for the asymmetric Laplace delivery time distribution lead to a 75% reduction in the expected penalty cost for untimely delivery. The paper herein continues research in the area of delivery performance improvement revising the results of (Guiffrida and Nagi, 2006a) and differs from
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other similar papers in two ways: it applies the concept of the optimal position of the delivery window and analyzes the effect of parameters defined in a contract for general form of delivery time distribution instead of normally distributed delivery time.
In this paper, we address strategies for improving delivery performance using a cost-
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based delivery performance model. Similar to all previous supply chain delivery performance models, the model is limited in that improving delivery performance from the supplier’s
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perspective might lead to sub-optimal decisions from the buyer’s point of view. The effect of delivery performance improvement is analyzed in financial terms thus allowing an estimate of the savings in the expected penalty cost associated with implementation of a delivery performance improvement project. With this information in hand, a supplier then decide in financial terms if it is beneficial to invest in the delivery improvement project. In this paper we employ the delivery cost model of Guiffrida and Nagi (2006a) and build
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on the concept of the optimal position of the delivery window (OPDW) proposed by Bushuev and Guiffrida (2012) to address strategies for improving delivery performance using a cost-based delivery performance model. A set of supporting propositions is presented to provide an analytical analysis of the delivery model that can be used to improve delivery performance. The
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propositions examine the robustness of the optimal positioning of the delivery window and the expected penalty cost of untimely delivery in terms of the width of the on-time portion of the
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delivery window, penalty costs per unit time early and late, mean, and variance of the normally distributed delivery time.
The normal probability distribution is the most widely used probability model for
defining the delivery distribution of supply chain delivery times and was found in 17 of the 38 delivery performance models reviewed by Guiffrida (2014). The widespread use of normal probability density function to model the distribution of delivery time in supply chain delivery performance models suggests an immediate limitation since the delivery time distribution under these models is always symmetric. Also, negative delivery times are possible under the normal distribution. 3
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After analyzing the effect of parameters of the delivery time distribution on delivery performance, we investigate and compare strategies for improving delivery performance. Modeling cost reduction allows managers to compare investments and saving and it should be planned in advance how the parameters will change over time. In the paper herein we analyze
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how the lead time reduction affects the expected penalty costs. Two models of cost reduction are used: hyperbolic and exponential. These forms have been widely adopted in several process improvement studies including delivery performance improvement (Guiffrida and Nagi, 2006a; Tubino and Suri, 2000; Choi, 1994).
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Our model and the supporting propositions derived herein contribute to the literature along the following two dimensions. First, the model represents a generalized modeling approach
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to the evaluation and improvement of supply chain delivery performance. The approach applied to the model with Gaussian distributed delivery time can be used for other delivery time distributions. Second, the analytical analysis of the key attributes of the model that are presented in the set of supporting model propositions provide managerial insight into the dynamics of integrating the model into the long term continuous improvement of delivery performance within the supply chain. The results presented in the paper herein are novel to theory and practice in two
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ways. Firstly, the manuscript shows that the mean of the normally distributed delivery time has no effect on the expected penalty cost. Thus, from this perspective a supplier has no interest in reducing the average delivery time. Secondly, although it’s well-known that delivery variance reduction is beneficial for a supplier, there is no research that shows how much a supplier can
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save by reducing the variance of a delivery time distribution. The research herein provides the results that help to estimate cost savings and justify the investments in delivery variance
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reduction.
This paper is organized as follows. In section 2, a cost-based model for evaluating
delivery performance is reviewed. In sections 3 and 4, a set of propositions which highlight the key analytical characteristics of the model are introduced. Section 3 evaluates the effect of parameters defined in a contract (the width of the on-time portion of the delivery window and penalty costs per unit time early and late) on the optimal position of the delivery window and the expected penalty cost. Section 4 revises the effect of normal delivery time distribution parameters (mean and variance) on the optimal position of the delivery window and the expected
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penalty cost. In section 5, theoretical and managerial implications of the findings are discussed. Section 6 provides a summary and discusses future research directions.
2. Modeling supply chain delivery performance
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A common feature of the supply chain delivery performance models is the use of the delivery window. Under the concept of a delivery window, contractually agreed upon
benchmarks in time are used to classify deliveries as being early, on-time, and late (see Figure 1). Early and late deliveries introduce waste in the form of excess cost into the supply chain;
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early deliveries contribute to excess inventory holding costs while late deliveries may contribute to production stoppage costs, lost sales and loss of goodwill. These costs for untimely (early and
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late) delivery are referred to in the literature as penalty costs (Guiffrida and Nagi, 2006a, 2006b). When a delivery is within the on-time portion of the delivery window, no penalty cost is incurred.
--------------------------------- Insert Figure 1 about here ---------------------------------
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Guiffrida and Nagi (2006a) proposed the following expected penalty cost per period when deliveries are classified as early and late according a delivery window c1
Y = Yearly + Ylate = QH ∫ (c1 − x) f ( x)dx + K 0
∞
∫ ( x − (c + ∆c)) f ( x)dx , 1
(1)
c1 + ∆c
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where Y = expected penalty cost of untimely delivery, c1
Yearly = QH ∫ (c1 − x ) f ( x ) dx is the expected penalty cost of early delivery,
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0
Ylate = K
∞
∫ ( x − (c
1
+ ∆c )) f ( x ) dx is the expected penalty cost of late delivery,
c1 + ∆c
f(x)= the probability density function (pdf) of delivery time x,
QH = penalty cost per time unit early (levied by the buyer), K = penalty cost per time unit late (levied by the buyer),
c1 = difference between the time the delivery process is initiated and the earliest acceptable delivery time, ∆c = the width of the on-time portion of the delivery window. 5
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Although the earliest acceptable delivery time is predefined by the contract, the supplier can define the time when delivery begins, therefore changing the value of c1. For example, if the supplier decides to ship the product 10 hours before the earliest accepted delivery time, c1 is equal to 10 hours.
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As demonstrated in Bushuev and Guiffrida (2012), Y is a convex function of c1 and the optimal value of c1 (which is defined as c1*) that minimizes Y can be determined by evaluating
K ⋅ Plate = QH ⋅ Pearly , ∞
∫
f ( x)dx and Pearly =
c1 + ∆c
c1
∫ f ( x)dx
are the probabilities of late and early deliveries.
0
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where Plate =
(2)
In the paper herein we assume that the supplier uses the concept of the optimal position
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of the delivery window (OPDW) to define the optimal value of c1 and, accordingly, the time when the product should be shipped to the buyer.
There are several key parameters of the expected penalty cost function that can reduce the expected penalty cost and be used to determine the potential for improvement in delivery performance. The parameters are: the width of the on time portion of the delivery window, the penalty cost for early deliveries, the penalty cost for late delivery, and the parameters of delivery
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time distribution. The width of the on time portion of the delivery window and the penalty costs for early and late delivery are defined in a contract. Thus, they are chosen during a negotiation process between a buyer and a supplier. The delivery time distribution and its parameters can be estimated based on statistics of the past deliveries. These parameters can be changed by a
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supplier alone, but it requires time and resources.
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3. Effect of parameters defined in a contract on expected penalty cost In this section, we present propositions which can be used for examining how the optimal
value of c1 which defines when the product should be shipped with respect to the earliest acceptable delivery date and the expected penalty cost behave as a function of the parameters defined in a contract, i.e. the width of the on time portion of the delivery window and the penalty costs for early and late delivery. The width of the on-time portion of the delivery window and the accompanying penalties costs for early and late delivery are defined by a contract between a buyer and a supplier. Understanding the robustness of Y as a function of the width of the on-time portion of the 6
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delivery window and the penalty costs for untimely delivery is of interest since changes to the width impact the buyer and seller differently. Typically, a buyer would like to decrease the width of the on-time portion of the delivery window because it will decrease delivery time uncertainty. Alternatively, a supplier would like to increase the width of on-time portion of the delivery
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window because it decreases the expected penalty cost for untimely delivery.
3.1. Width of the on-time portion of the delivery window
Proposition 1. For the fixed delivery time distribution parameters and the penalty costs
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per unit per time early and late, increasing the width of the on-time portion of the delivery
window by one unit of time reduces the total expected penalty cost by K·Plate and decreases the
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optimal position of the delivery window.
Proof. The optimal position of the delivery window is
Plate =
QH Pearly or K ⋅ Plate = QH ⋅ Pearly . K
Using the derivatives
dPearly d∆c
dc1* dc1* f (c1* ) − 0 + 0 = f (c1* ) , d (∆c ) d ( ∆c )
(4)
(5)
dc1* QH dc1* − + 1 f (c1* + ∆c ) = f (c1* ) ; d ( ∆ c ) K d ( ∆ c )
(6)
dc1* QH * f (c1* ) = − f (c1* + ∆c) ; f (c1 + ∆c) + d ( ∆c ) K
(7)
K f (c1* + ∆c) dc1* K f (c1* + ∆c) 1 + . =− = − d (∆c) K f (c1* + ∆c) + QH f (c1* ) QH f (c1* )
(8)
AC C
EP
we have
=
TE D
dc1* dc1* dPlate = 0 − + 1 f (c1* + ∆c) + 0 = − + 1 f (c1* + ∆c) , d∆ c d ( ∆c ) d ( ∆c )
(3)
Because the derivative (8) is always negative, increasing the width of the on-time portion
of the delivery window will require delivering earlier to keep the position of the delivery window optimal.
7
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Using envelope theorem
dy * ∂y x = x* ( a ) the derivative of the expected penalty cost = da ∂a
{
}
is
RI PT
b ∂Y = 0 + K 0 − 1 ⋅ ((c1 + ∆c − (c1 + ∆c)) f (c1 + ∆c )) + ∫ (0 − 1) f ( x) dx ; ∂ ( ∆c ) c1 + ∆c
dY = − K ⋅ Plate . d ( ∆c )
(9)
(10)
cost. ■
M AN U
3.2. Penalty costs for early and late deliveries
SC
Thus, increasing the width of the delivery window will decrease the expected penalty
Proposition 2. For the fixed delivery time distribution parameters, the width of the
delivery window, and the penalty cost for late delivery, increasing penalty costs per unit time early by 1 unit increases the total expected penalty cost by Yearly and decreases the optimal position of the delivery window. Proof. Using the derivatives
dPearly
we have
dc1* dc1* f (c1* ) − 0 + 0 = f (c1* ) , d (QH ) d (QH )
EP
d (QH )
=
TE D
dPlate dc1* dc1* * =0− f (c1 + ∆c) + 0 = − f (c1* + ∆c) , d (QH ) d (QH ) d (QH )
AC C
dc1* dc1* f (c1* + ∆c) = Pearly + QH K − f (c1* ) ; d ( QH ) d ( QH )
(
)
(11)
(12)
(13)
dc1* K f (c1* + ∆c) + QH f (c1* ) = − Pearly ; d (QH )
(14)
Pearly dc1* . =− * d (QH ) K f (c1 + ∆c) + QH f (c1* )
(15)
The derivative (15) is always negative, hence increasing QH decreases the optimal value of c1. Using envelope theorem the derivative of the expected penalty cost is
8
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dY = Yearly + 0 = Yearly . d (QH )
(16)
The derivative (16) is always positive and increasing QH increases Y. ■ Proposition 3. For the fixed delivery time distribution parameters, the width of the
RI PT
delivery window, and the penalty cost for early delivery increasing penalty costs per unit per time late by 1 unit raises the total expected penalty cost by Ylate and increases the optimal position of the delivery window. Proof. Using the derivatives
dc1* dc1* * = f (c1* ) , f (c1 ) − 0 + 0 = dK dK dK
we have
M AN U
dPearly
SC
dPlate dc* dc* = 0 − 1 f (c1* + ∆c ) + 0 = − 1 f (c1* + ∆c ) , dK dK dK
dc1* dc1* * Plate + K − f (c1 + ∆c) = QH f (c1* ) ; dK dK
(
)
TE D
dc1* K f (c1* + ∆c) + QH f (c1* ) = Plate ; dK dc1* Plate = . * dK K f (c1 + ∆c) + QH f (c1* )
(17) (18)
(19)
(20) (21)
The derivative (21) is always negative, thus increasing K increases the optimal value of c1.
EP
Using envelope theorem the derivative of the expected penalty cost is dY = 0 + Ylate = Ylate . dK
(22)
AC C
The derivative (22) is always positive, thus increasing K increases Y. ■
It can be interesting to estimate the effect of penalties for earliness and lateness together.
From (2) we can see that the optimal value of c1 is a function of QH and K, but if the proportion of QH to K does not change the optimal value of c1 does not change also. So we could use a coefficient which will combine both values of QH and K. We suppose to use the next coefficients K = K 0 ⋅ K1 ; QH = (1 − K 0 ) K1 ; QH + K = K 0 ⋅ K1 .
(23)
And we have 9
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QH (1 − K 0 ) K1 1 − K 0 1 = = = −1. K K 0 ⋅ K1 K0 K0
(24)
Expected penalty cost is Y = K1 ((1 − K 0 )Yearly + K 0Ylate ) ,
RI PT
(25)
and the optimal position of the delivery window is K 0 ⋅ Plate = (1 − K 0 ) ⋅ Pearly .
(26)
The coefficient K1 is a common part of early and late penalties per unit per time.
Increasing (decreasing) K1 means that both QH and K increasing (decreasing) but the proportion
SC
QH/K stays the same.
Proposition 4. For the fixed delivery time distribution parameters and the width of the
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delivery window proportionally increasing penalty costs per unit per time early and late increase the total expected penalty cost by (1 − K 0 )Yearly + K 0Ylate and has no effect on the optimal position of the delivery window.
Proof. It can be concluded from (26) that K1 is not a part of the equation that defines the optimal position of the delivery window. Thus, changing K1 will have no effect on the optimal value of c1.
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The first derivative of Y by K1 is dY = (1 − K 0 )Yearly + K 0Ylate . dK1
(27)
EP
The derivative (27) is always positive, because 0 < K0 < 1. Thus increasing K1 will increase Y. ■
Proposition 5. For the fixed delivery time distribution parameters and the width of the delivery window
increasing K0 will increase the optimal position of delivery window;
•
and the total expected penalty cost is concave by K0 with maximum at Yearly = Ylate.
AC C
•
Proof. Using the derivatives dPlate dc* dc* = 0 − 1 f (c1* + ∆c) + 0 = − 1 f (c1* + ∆c) , dK 0 dK 0 dK 0 dPearly dK 0
=
dc1* dc* f (c1* ) − 0 + 0 = 1 f (c1* ) , dK 0 dK 0
(28)
(29)
we have
10
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dc* dc* Plate + K 0 − 1 f (c1* + ∆c ) = − Pearly + (1 − K 0 ) 1 f (c1* ) ; dK 0 dK 0
(
)
dc1* K 0 f (c1* + ∆c) + (1 − K 0 ) f (c1* ) = Plate + Pearly ; dK 0
(30)
(31)
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Plate + Pearly dc1* = . * dK 0 K 0 f (c1 + ∆c) + (1 − K 0 ) f (c1* )
(32)
The derivative (32) is always positive, hence increasing K0 increases the optimal value of c1. The first derivative of the total cost is
SC
dY = K1 (Ylate − Yearly ) . dK 0
dYearly dK 0
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Using that
(33)
c*
1 dc1* * * dc1* dc1* * = (c1 − c1 ) f (c1 ) − 0 + ∫ f ( x) dx = Pearly , dK 0 dK dK 0 0 a
(34)
b
dYlate dc* dc* dc* = 0 − 1 (c1* + ∆c − (c1* + ∆c)) f (c1* + ∆c ) + ∫ − 1 f ( x )dx = − 1 Plate , (35) dK 0 dK 0 dK 0 dK 0 c * + ∆c 1
the second derivative is
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d 2Y dc1* (Pearly + Plate ). = − K 1 d ( K 0 )2 dK 0
(36)
EP
The second derivative is always negative and the cost function Y is concave by K0. ■ 4. Effect of normal delivery time distribution parameters on expected penalty cost
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This section revises the results presented in (Guiffrida and Nagi, 2006a) assuming that a
supplier uses the optimal position of the delivery window. For normal distribution, the total expected penalty cost per period for normally distributed delivery time is (Guiffrida and Nagi, 2006a)
c* − µ + c1* − µ Y = QH σ φ 1 σ
(
* 1
c* + ∆c − µ − c1* + ∆c − µ + K σ φ 1 σ
(
) Φ c σ− µ + c* + ∆c − µ 1 − Φ 1 , σ
)
(37)
11
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where ϕ ( X ) =
X
1 1 exp( X ) , Φ ( X ) = ∫ exp( x )dx . 2π 2π −∞
The optimal position of delivery window is when
RI PT
c* − µ c* + ∆c − µ = K ⋅ 1 − Φ 1 . QH ⋅ Φ 1 σ σ
(38)
Proposition 6. Increasing the mean of normally distributed delivery time by one time
unit increases the optimal position of the delivery window by one time unit and has no effect on
Proof. Using the derivative X
1 exp( X ) − 0 + ∫ 0dx = ϕ ( X ) . 2π −∞
(39)
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Φ( X ) = dX
SC
the expected penalty cost.
the derivative for the optimal position of the delivery window is
(40)
c1* − µ c1* + ∆c − µ c1* − µ c1* + ∆c − µ dc1* ; QH + K = QH + K ϕ ϕ ϕ ϕ σ σ dµ σ σ
(41)
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QH dc1* c1* − µ K dc1* c1* + ∆c − µ = ⋅ − ; − 1ϕ − 1ϕ σ σ dµ σ σ dµ
dc1* =1. dµ
(42)
From (Guiffrida and Nagi, 2006a)
EP
c* − µ c* + ∆c − µ dY + K 1 − Φ 1 , = QH − Φ 1 dµ σ σ
(43)
AC C
Knowing that the optimal position of the delivery window is used, the equation (38) is
applied and dY dµ = 0 . Thus the mean of the delivery time has no effect on the expected
penalty cost function. ■
Proposition 7. Increasing the standard deviation of normally distributed delivery time
increases the optimal position of the delivery window if −1
c* + ∆c − µ QH c1* − µ c* + ∆c − µ + ϕ 1 > 0 , A = c − µ + ∆c ⋅ ϕ 1 ⋅ ϕ σ σ K σ * 1
(44)
and decrease it otherwise. 12
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Proof. Using the derivative (39), the derivative for the optimal position of the delivery window is
c1* − µ c* + ∆c − µ dc1* + Kσ ⋅ ϕ 1 = QHσ ⋅ ϕ σ dσ σ
(45)
RI PT
c1* − µ QH dc1* * ϕ = σ µ c − ( − ) 1 σ 2 dσ σ c* + ∆c − µ K dc* ; = 2 ⋅ − σ 1 − (c1* + ∆c − µ ) ϕ 1 σ dσ σ
SC
c* − µ c* + ∆c − µ + K (c1* + ∆c − µ )ϕ 1 ; = QH (c − µ )ϕ 1 σ σ * 1
−1
M AN U
c* + ∆c − µ dc1* c1* − µ ∆c c1* + ∆c − µ QH c1* − µ + ϕ 1 . ■ = + ϕ ⋅ ϕ dσ σ σ σ σ K σ Remark 1. The value of A is positive for QH/K < 1,
•
negative for QH/K > 1,
•
and equal to 0 for QH/K = 1 (see Appendix A for proof).
(47)
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•
(46)
Proposition 8. Increasing the standard deviation of normally distributed delivery time increases the expected penalty cost.
Proof. Using envelope theorem and the results from (Guiffrida and Nagi, 2006a; eq. 7)
EP
c1* − µ c1* + ∆c − µ dY ⋅ (2σ ) −1 . ■ = QH ⋅ ϕ + K ⋅ ϕ σ dσ σ
AC C
As it is shown in Proposition 6, the mean of the normally distributed delivery time has no effect on the expected penalty cost; hence, there is no need to change the mean for delivery performance improvement.
A supplier’s strategy on delivery performance improvement should focus on delivery
variance reduction. This reduction should take into account that the optimal position of the delivery window will change with variance. Therefore, the results from (Guiffrida and Nagi, 2006a; eq. 13 and 14) should be rewritten. For hyperbolic delivery time variance reduction the expected penalty cost is
13
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M t * ( c* − µ ) 2 + c1* − µ Φ Y (ν , t ) = QH exp 1 (c1 − µ ) + 2M M 2π t
)
M (c* + ∆c − µ ) 2 − c1* + ∆c − µ + K exp 1 2M 2π t
(
1 − Φ t (c1* + ∆c − µ ) , M
)
(48)
RI PT
(
where t = time,
M = initial level of variance.
The expected penalty cost assuming exponential delivery time variance reduction is
(
)
P rt P + e rt (c1* + ∆c − µ ) 2 − c1* + ∆c − µ + K exp − 2 2 P π where P = initial level of variance, r = variance decay rate.
5. Discussion
rt 1 − Φ e (c* + ∆c − µ ) , P 1
(49)
)
M AN U
(
SC
P e rt * rt P + e rt (c1* − µ ) 2 + c1* − µ Φ Y = QH exp − (c1 − µ ) + P 2P 2π
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As shown in Proposition 1, a supplier would prefer to increase the width of the on-time portion of the delivery window thereby reducing the expected penalty cost for untimely delivery. However, from the buyer’s perspective, a wider width for the on-time portion of the delivery window increases delivery time uncertainty. Higher levels of delivery uncertainty could
with the supplier.
EP
contribute to unwillingness by the buyer to enter into a contractually defined delivery window
AC C
The effect of penalties for untimely delivery is evaluated in Propositions 2-5. As it is shown in Propositions 2, 3, and 4, increasing the penalties for early or late deliveries will increase the expected penalty cost. Proposition 5 shows that the balance between penalties for early and late deliveries will increase the expected penalty cost if Ylate > Yearly and decrease the cost if Ylate < Yearly. Thus, the supplier prefers a wide on-time portion of the delivery window, low penalties for untimely delivery with a balance between penalties such that Ylate significantly bigger or smaller than Yearly.
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ACCEPTED MANUSCRIPT
Unlike the width of the on-time portion of the delivery window and penalties for untimely delivery, the parameters of delivery time distribution (mean and variance) can be changed by a supplier alone but the changes require time and investment. Thus the supplier should compare benefits of the delivery performance improvement and expenses needed for that.
RI PT
There are several practices that can improve delivery performance, among them are Just-in-time (Mackelprang and Nair, 2010) and communication and information sharing (Carr and Kaynak, 2007), design for manufacturability (Ketokivi and Schroeder, 2004).
Propositions 6, 7 and Remark 1 demonstrate how a supplier should change the parameters
SC
of the distributed delivery time to decrease the expected penalty cost. Proposition 6 shows that the mean of normally distributed delivery time has no effect on the expected penalty cost when a
M AN U
supplier applies optimal position of the delivery window. Hence, in this case a supplier should focus solely on decreasing a variance of delivery time distribution to reduce the expected penalty cost for untimely delivery. Many researches focus on delivery time reduction and show its negative effect on a buyer (Glock, 2012; Hayya et al., 2011; De Treville et al., 2004). Although, in many cases a supplier is responsible for delivery and makes decision about delivery performance improvement. Therefore, a buyer should motivate its suppliers to improve delivery
TE D
performance. The results herein show that penalizing suppliers for untimely delivery makes them interested in delivery time variance reduction, but does not motivate them to reduce the expected delivery time.
At the same time, it is shown how the width of the on-time portion of the delivery
EP
window, the penalties for untimely delivery, and the delivery time distribution parameters will affect the optimal value of c1 and the time a supplier should ship a product to minimize the if • • • •
AC C
expected penalty cost. It is shown that the optimal position of the delivery window will decrease
the width of on-time portion of the delivery window increases,
the penalty cost for late delivery increases, the penalty cost for early delivery decreases, and the mean of normally distributed delivery time decreases.
Moreover, changing the mean by one time unit will change the optimal position of the delivery window by one time unit. For example, if the average delivery time is decreased by 10 hours, the product will need be shipped exactly 10 hours earlier to maintain the optimal position of the 15
ACCEPTED MANUSCRIPT
delivery window. The effect of the standard deviation of normally distributed delivery time on the optimal position of the delivery window depends on the value of QH/K. Increasing the standard deviation will increase the optimal position of the delivery window if QH/K < 1,
•
increase the optimal position of the delivery window if QH/K > 1,
•
and keep the optimal position c1* = µ − ∆c / 2 if QH/K = 1.
RI PT
•
That suggests that the supplier should coordinate the shipment of the product so that the product arrives on the buyer’s requested delivery date as dictated by the optimal value of c1. This
SC
action may require the supplier to adjust their production schedule to avoid increases in
inventory levels and related costs and if needed, adjust delaying the shipping of the product to
M AN U
avoid incurring higher in-house inventory levels and related costs.
Although the effect (positive or negative) of several parameters discussed in the manuscript is obvious, the magnitude of the effect had to be estimated. Understanding the magnitudes of the effect of the width of the on-time portion of the delivery window and penalties for untimely delivery on the expected penalty cost, a supplier and a buyer can better work in partnership to find the value of the width of the on-time portion of the delivery window which
TE D
will satisfy the desired levels of expected cost and delivery time uncertainty valued by the supplier and the buyer. Also the results of the paper herein show a supplier and a buyer how changing the parameters defined in the contract (the width of the on-time portion of the delivery window and penalties for untimely delivery) affect the expected penalties which can help a
EP
supplier and a buyer during the contract negotiation process. Propositions 1 through 5 can help a supplier during a contract negotiation process. A
AC C
supplier needs to know by how much parameters of a contract increase or decrease the expected penalties paid by the supplier. In this case, a supplier can justify how the product price should be changed to cover the penalty costs and either raise the price for the buyer accordingly or decline to sign the contract.
The effect of distribution parameters on the expected penalty cost analyzed herein can be
used as a tool for supporting and justifying financial investment needed to improve delivery performance. The analysis lets a manager estimate the expected savings that can accrue for temporal changes to the parameters of the delivery time distribution. These expected cost savings can be used in the decision making process of committing financial investment to improvements 16
ACCEPTED MANUSCRIPT
in on-time delivery performance. Thus the supplier should compare the delivery performance improvement benefits with the associated costs to determine a long-term return on investment. A company can use the information presented herein to perform a cost benefit analysis to determine the best delivery performance strategy. Firstly, our observations map out the direction
RI PT
in which the distribution parameters should be changed to reduce the expected penalty cost. Secondly, implementation of delivery performance improvement practices requires investment. Therefore, before a company invests in a delivery improvement project, a decision maker should compare the investments with the expected savings that can be estimated using the analysis
SC
presented in this paper. Thirdly, for companies implementing continuous delivery performance improvement, a cost reduction related to untimely delivery can be modeled so that the decision
M AN U
maker knows when and how much will be saved in the future. Also, the analysis can show which parameters will have the greatest effect on penalty costs thereby supporting the delivery improvement program.
5. Conclusion
The paper herein investigates the effect of different parameters on the optimal position of
TE D
the delivery window and the expected penalty cost. The results allow developing strategies for improving delivery performance from a supplier’s perspective and answer the question what supplier should do to decrease the expected penalty cost of untimely delivery. The research uses the expected penalty cost function developed by Guiffrida and Nagi (2006a) and assumes that the
Guiffida (2012).
EP
supplier will choose the optimal position of delivery window as proposed by Bushuev and
AC C
Proposition 1 suggests increasing the width of on-time portion of the delivery window to decrease the expected penalty cost. Results from Propositions 2-5 demonstrate that decreasing the penalties for early and late deliveries reduces the expected penalty cost. The results in Propositions 1-5 are provided in general form and can be used for any delivery time distribution. Although a supplier would obviously prefer low penalties for untimely delivery and wider ontime portion of the delivery window, the propositions clearly define the effect of the parameters on the expected penalty cost. For example, a supplier can easily estimate the expected penalty cost increase for $1 increase in penalties for unit time early or late.
17
ACCEPTED MANUSCRIPT
In addition to that, the results for the normal distribution from Guiffrida and Nagi (2006a) were revised using the concept of the optimal position of the delivery window. Propositions 6-7, which focus on the mean and the variance of normally distributed delivery time, recommend decreasing the variance parameter to reduce the expected penalty cost and show that the mean
RI PT
has no effect on the expected penalty cost if the optimal position of the delivery window is
chosen. This is a surprising result, since expected delivery time is considered to have negative effect on costs.
At the same time, changing the parameters (the width of the on-time portion of the
SC
delivery window, the penalties for early and late deliveries, the mean, and the variance of the delivery time distribution) has an effect on the optimal position of the delivery window and as a
M AN U
result on supplier’s costs such as inventory and production costs. Thus delivery performance improvement should be integrated with production and inventory functions within the company to keep costs low.
The results presented in the paper can be used by both researchers and practitioners. The managerial implication of the paper is that it can serve as guidance for practitioners undertaking a program to improve delivery performance. A supplier can use the knowledge about the effect
TE D
of the width of on-time portion of the delivery window and penalties for early and late deliveries on the expected penalty cost during the negotiation process with a buyer. By understanding the effect of delivery time distribution parameters (mean and variance) on the expected penalty cost, a supplier can develop a strategy for continuous improvement of delivery performance.
EP
Although, a buyer should keep in mind that penalties for untimely delivery do not motivate suppliers to reduce the expected delivery time.
AC C
A major research contribution of this paper is the proposal of a general approach to modeling delivery performance improvement. The model was demonstrated for normally distributed delivery time, but can be applied to other distribution forms which can be done in future publications. Moreover, for other delivery time distributions, the correlation between delivery time distribution parameters can be assumed. The results derived in the paper herein can be used by a supplier to see potential delivery cost savings in order to evaluate alternatives for delivery performance improvement.
18
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Appendix A. We will evaluate the effect of the penalty cost ratio on A. Using envelope theorem: −1
c* + ∆c − µ c1* − µ dA ⋅ ϕ ≥ 0 . = ∆c ⋅ ϕ 1 d (QH / K ) σ σ
RI PT
(A.1)
Thus, A is a monotonically increasing function and for any given values of ∆c, µ, and σ there is only one value of QH/K which turns A into 0. As it will be shown below, A = 0 when QH/K = 1. For QH/K = 1, the optimal c1 is when Pearly = Plate. Because normal distribution is symmetric, that is possible only if the mean is placed right in the middle of the on-time portion
∆c 2
c1* + ∆c − µ c1* − µ and ϕ = ϕ σ . σ Hence, for QH/K = 1,
M AN U
c1* = µ −
SC
of the delivery window and µ – c1 = c1 + ∆c – µ. In this case,
(A.2)
(A.3)
−1
c* − µ c1* − µ c* − µ ∆c ϕ + ϕ 1 = 0 A=µ− − µ + ∆c ⋅ ϕ 1 2 σ σ σ
(A.4)
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for any values of ∆c, µ, and σ.
EP
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Slotnick, S. A. and Sobel, M. J. (2005). Manufacturing lead-time rules: Customer retention versus tardiness costs. European Journal of Operational Research, 163(3), 825-856.
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Tanai, Y. and Guiffrida, A. L. (2015). Reducing the cost of untimely supply chain delivery performance for asymmetric Laplace distributed delivery. Applied Mathematical Modelling, 39(13), 3758-3770. Tubino, F., and Suri, R. (2000). What kind of “numbers” can a company expect after implementing quick response manufacturing. Empirical data from several projects on lead time reduction. Quick Response Manufacturing 2000 Conference Proceedings, 943-972. Dearborn, MI: Society of Manufacturing Engineers Press. Wang, F. K. and Du, T. (2007). Appling capability index to the supply chain network analysis. Total Quality Management, 18(4), 425-434.
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Figure 1. Illustration of delivery window.
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Legend: f(x) is the probability density function (pdf) of delivery time x; c1 is difference between the time the delivery process is initiated and the earliest acceptable delivery time; ∆c is
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the width of the on-time portion of the delivery window.
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The effect of parameters defined in a contract on the expected penalty cost function is investigated Effect of normal delivery time distribution parameters on the expected penalty cost is analized Delivery performance improvement is explored Theoretical and managerial implications of the findings are discussed
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