Int. J. Business Performance and Supply Chain Modelling, Vol. 8, No. 2, 2016
A modelling framework for improving supply chain delivery performance Thomas Ngniatedema* Department of Business, Kettering University, 1700 University Avenue, Flint, Michigan 48504-6214, USA Email:
[email protected] *Corresponding author
Lihua Chen Department of Management and Marketing, West Liberty University, West Liberty, West Virginia, USA Email:
[email protected]
Alfred L. Guiffrida Department of Management and Information Systems, Kent State University, Kent, Ohio 44242, USA Email:
[email protected] Abstract: The evaluation of delivery performance is a crucial component in the overall management and control of a supply chain. The main objective of this research is to develop a framework that can be used to improve the delivery performance of a supplier to the end customer in a supply chain. The framework provides a bound on the financial investment required to make improvement to the delivery process. A variance reduction modelling approach is used that directly incorporates the uncertainty in the delivery time distribution into a financial delivery performance metric. The framework is demonstrated using numerical illustrations. Keywords: supply chain delivery performance; variance reduction; process improvement. Reference to this paper should be made as follows: Ngniatedema, T., Chen, L. and Guiffrida, A.L. (2016) ‘A modelling framework for improving supply chain delivery performance’, Int. J. Business Performance and Supply Chain Modelling, Vol. 8, No. 2, pp.79–96. Biographical notes: Thomas Ngniatedema is an Assistant Professor of Computer Information Systems and Analytics at Kettering University, Flint, Michigan. He holds a PhD in Computer Information Systems from Kent State University. His research has been published in European Journal of Operational Research, Journal of Computer Information Systems, Environmental Quality Management, Academy of Information and Copyright © 2016 Inderscience Enterprises Ltd.
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T. Ngniatedema et al. Management Sciences Journal, and IAPQR Transactions. His research interests are in operations and supply chain management, mass customisation strategies, information systems, and business intelligence. Lihua Chen is an Assistant Professor of Management in the Gary E. West College of Business at West Liberty University, West Liberty, West Virginia. She holds a PhD in Operations Management from Kent State University. Her research has been published in Lean Six Sigma Approaches in Manufacturing, Services, and Production and in numerous international and national operations management and supply chain management conferences. Her research interests are in strategic supply chain management and operations management. Alfred L. Guiffrida is an Associate Professor of Management in the Department of Management and Information Systems at Kent State University, Kent, Ohio. He holds a PhD in Industrial Engineering from the State University of New York at Buffalo. His papers have been published in Applied Mathematical Modelling, The Engineering Economist, European Journal of Operational Research, International Journal of Integrated Supply Management, International Journal of Production Economics, International Journal of Logistics and Applications, International Journal of Production Research, and International Transactions in Operations Research. His research interests are in the areas of operations and supply chain management. This paper is a revised and expanded version of a paper entitled ‘A learning-based model for evaluating supply chain delivery performance’ presented at the Proceedings of the Midwest Decision Sciences Conference, Grand Rapids, Michigan, 12–14 April 2012.
1
Introduction
Today’s globally competitive business environment places pressures on firms to continually improve customer service while simultaneously reducing costs and shortening product lifecycles. In response to these pressures, many organisations have adopted the supply chain management (SCM) philosophy as the foundation of their strategic initiatives. The SCM philosophy advocates integrating value-adding activities such as production planning, inventory control, sourcing, physical distribution, vendor relations and customer relationship management. The positive impact of the SCM philosophy on firm performance has been documented by several researchers (see for example, Leuschner et al., 2013; Wagner et al., 2012; Kristal et al., 2010; Kim, 2009). Continuous improvement of supply chain operations is a critical concern to management (Cai et al., 2009; Nabhani and Shokri, 2009; Dasgupta, 2003). The successes achieved by organisations who have implemented continuous improvement programmes for their supply chain operations have been reported in the literature. A short list of firms and their successes is found in Table 1. The entries in Table 1 are not meant to be an exhaustive review of the literature but are summarised to illustrate the scope of the types of improvements in supply chain operations that can be achieved under a continuous improvement programme. The successful implementation of actions to improve performance is highly dependent on having a formal performance measurement system that provides management with meaningful performance metrics to meet day-to-day as well as
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long-term performance goals (Neely et al., 1995). The importance and integration of performance measurement systems within supply chains has been addressed by Martin and Patterson (2009), and Gunasekaran et al. (2004). Comprehensive frameworks for evaluating supply chain performance may be found in Cuthbertson and Piotrowicz (2011), Azevedo et al. (2011), Abd El-Aal et al. (2011) and Chan et al. (2006). Research highlighting specific metrics for use in measuring supply chain performance is found in Gopal and Thakkar (2012), Sambasivan et al. (2009), and Gunasekaran and Kobu (2007). Table 1
Selected industry examples of supply chain continuous improvement
Company
Continuous improvement results
Source
Wal-Mart
Reduced carbon footprint Improved inventory management Enhanced product tractability Reduced costs of product returns Improved environmental sustainability Increased profitability Improved customer satisfaction Increased profitability Improved product quality Improved market competitiveness Reduced inventory/transaction costs Decreased procurement costs Reduced inventory levels Improved customer relations Improved cash to cash cycle Improved productivity Enhanced supplier relations Reduced knowledge transfer costs
Bose and Yan (2011)
Cisco Systems The Samsung Group Bose
FedEx
Toyota
Van Hoek and Johnson (2010) Yang et al. (2007)
Segars et al. (2001)
Williams and Frolick (2001) Dyer and Nobeoka (2000)
In this paper, we concentrate on one aspect of overall supply chain performance, delivery timeliness to the final customer. The delivery process in a supply chain is a key component of overall supply chain operations. As summarised in Bushuev and Guiffrida (2012), the delivery process within a supply chain is a key concern to supply chain and logistics managers since delivery performance directly impacts customer satisfaction and the selection raw material and third party logistics providers. Recent research has highlighted the importance that supply chain managers place on delivery performance (see for example, Rao et al., 2011; Forslund et al., 2009; Lockamy and McCormack, 2004; Vachon and Klassen, 2002). Conceptual frameworks for defining delivery performance in SCM are found in Gunasekaran et al. (2001) and Fawcett et al. (1997). Within these frameworks, delivery performance is classified as a strategic level supply chain performance measure. Our research objective is to provide a modelling framework for integrating the continuous improvement of delivery performance into supply chain operations. Our modelling approach, which develops a financial baseline for justifying the improvement in delivery performance, is in line with the recent findings of Ramaa et al. (2013) who identify the increasing importance being given to financial measures in the evaluation of supply chain performance.
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This paper is organised as follows. In Section 2, we review the literature on delivery performance models in SCM. In Section 3, we present a modelling framework that can be used to guide organisations in modelling improvements in delivery performance and the justification of capital investment needed to support the implementation of a programme to improve delivery performance. In Section 4, we present our research summary and directions for future research.
2
Literature review
In this section, we review two research streams related to performance measurement in chain management. First, we review empirical findings on the importance of delivery performance in SCM. Second, we review models for evaluating supply chain delivery performance. Numerous empirical studies have identified the importance of supply chains meeting customer demand with on-time and reliable delivery performance. A summary of these research studies is presented in Table 2. Examining the set of key findings that are summarised in Table 2, we note that on-time delivery is considered to be a key performance metric across many different industrial sectors. Table 2
Empirical studies on supply chain operations and delivery performance
Research study
Sample description
Key findings with respect to delivery performance
Danese et al. (2013)
266 manufacturing plants in the mechanical, electronics and transportation equipment industries in nine different countries.
The use of an international supplier network moderates the positive relationship between external integration as measured by collaboration, coordination and information exchange with on-time delivery, fast delivery and flexibility to change product mix and volume.
Droge et al. (2012)
57 tier one suppliers as identified by industry experts from the Automotive Industry Action Group.
Product modularity and supplier integration have positive effects on delivery performance; customer integration mediates the relationships between product and process modularity and delivery performance.
Boon-itt and Wong (2011)
151 first tier suppliers in the Thai automotive industry.
Technological and demand uncertainties moderate the relationships between supply chain integration and customer delivery performance.
Golini and Kalchschmidt (2010)
485 companies from 19 countries in metal products, machinery and equipment assembly industries.
A complex interrelationship exists between globalisation of sales, supply chain investment and delivery performance.
da Silveira and Arkader (2007)
243 manufacturers from 13 countries in metal, machinery, electrical and transportation products and equipment industries.
A direct relationship is identified between customer coordination investment and delivery speed and reliability.
A modelling framework for improving supply chain delivery performance Table 2
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Empirical studies on supply chain operations and delivery performance (continued)
Research study Iyer et al. (2004)
Sample description
Key findings with respect to delivery performance
914 US-based manufacturing firms from selected form SIC codes 20-39.
Implementing B2B e-commerce contributes to efficient and effective business processes resulting in enhanced time-based delivery performance in terms of the length of the order cycle and on-time delivery.
128 Swedish manufacturing firms.
Delivery dependability was ranked second to quality when selecting supply chain partners.
Tan et al. (2002)
1500 senior purchasing and materials managers of US firms.
On-time delivery is reported as the most important (out of 24) supply chain performance practices.
Tracey and Tan (2001)
180 manufacturing executives across six different US industries.
On-time delivery performance significantly affected a composite measure of overall firm performance.
Salvador et al. (2001)
164 manufacturing plants from five counties in the electronic, machinery and transportation equipment industrial sectors.
Interactions with suppliers and customers on quality and material flow issues affect time-related performance measures such as punctuality of delivery and operations speed.
Tan et al. (1998)
1469 US manufacturing firms across 14 different industries.
Tracking on-time delivery performance was the highest rated supplier evaluation variable.
Fawcett et al. (1997)
131 organisations across multiple countries.
Firms with high levels of delivery performance outperformed firms unable to achieve delivery competence.
Olhager and Selldin (2004)
As a time-based performance measure, on-time delivery is a natural extension of the time-based performance measures advocated in the 1980s by Porter (1980) and Stalk (1988) to modern day SCM. Corbett (1992) demonstrated the use of the delivery window as a means to control delivery reliability (conformance to on-time delivery) in the just-in-time production setting. In using a delivery window, the customer defines benchmarks in time to which delivery times are compared to as a means to classify deliveries as early, on-time or late. Penalty costs per unit time are contractually assigned for early and late deliveries with no penalty cost assigned for on-time deliveries (Schneiderman, 1996). Quantitative models for evaluating supply chain delivery performance have adopted the delivery window concept within their modelling environments to provide a cost-based delivery performance metric. Modelling supply chain delivery performance using a cost-based metric meets the need for linking supply chain performance with cost that has been identified by several researchers (see for example, Whicker et al., 2009; Ellram, 2002; Ballou et al., 2000; Lancioni, 2000). Table 3 presents a summary of cost-based supply chain delivery performance models found in the literature. A major limitation of the models reviewed in Table 2 is their lack of integration of the cost-based delivery performance metric into the continuous improvement of the delivery process. Da Silveira and Arkader (2007), Guiffrida and Nagi (2006a, 2006b), and Wang and Du (2007) identify frameworks for linking supply chain delivery performance
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models and delivery process improvement; however, a more comprehensive treatment of the modelling aspects of this linkage is required. In the next section, we present an integrated cost-based delivery performance modelling framework that can be used to justify the financial investment required to implement improvements to the delivery process of a supply chain. In the development of our model, we build on contemporary management theories advocated by Hopp and Spearman (2008), and Blackhurst et al. (2004) which identify the reduction of variance as a key step in improving the performance of a system. The model presented herein directly incorporates the variability found in the customer delivery time distribution into a cost-based financial delivery performance measure. This financial measure is easy to understand by managers and provides a bound for justifying the capital investment required to reduce delivery uncertainty and improve delivery performance within the supply chain. Table 3
Supply chain delivery performance models
Authors
Key model features
Hsu et al. (2013)
Integrate process capability measures and Six-Sigma concepts to develop a delivery performance chart for measuring supply chain delivery performance.
Roy et al. (2013)
Present a nonlinear optimisation model for determining the cost optimal mix of logistics service providers to meet customer delivery requirements in a make to order supply chain.
Safaei et al. (2013)
Develop a generalised model for evaluating uncertainty in delivery performance for supply chains when chain activity times are dependent.
Bushuev and Guiffrida (2012)
Determine the optimal placement of the customer delivery window for the uniform, logistic, asymmetric Laplace and exponential delivery time distributions.
Shin et al. (2009)
Develop an analytical model for comparing sourcing alternatives based on supplier quality and delivery performance.
Guiffrida and Jaber (2008)
Present an optimisation framework for minimising a convex-concave cost model for delivery performance.
Wang and Du (2007)
Develop a capability index to model continuous improvement when delivery performance is measured subject to a delivery window.
Choudhary et al. (2006)
Present a nonlinear optimisation model for minimising the cost of untimely delivery when delivery performance is measured as a Six-Sigma-based capability index measure.
Garg et al. (2006)
Utilise a Six-Sigma-based delivery capability index to optimally distribute the pool of individual stage activity variance in a multi-stage supply chain to satisfy customer delivery expectations.
Guiffrida and Nagi (2006a)
Develop a framework for financial modelling and evaluation of continuous improvement in delivery performance.
Guiffrida and Nagi (2006b)
Quantify the concept of managerial neglect as the opportunity cost of management neglecting to improve delivery performance.
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Model development
As a starting point in the development of our model, we adopt the penalty cost function found in the delivery performance model of Guiffrida and Nagi (2006a). This penalty cost orientation is in agreement with Zhu et al. (2007) who note that penalty costs are effective in driving supply chain quality improvement. The expected penalty cost per delivery period for untimely (early and late) delivery, Y, is cE
∫
Y = QH ( cE − x ) f ( x)dx + K a
b
∫ (x −c
L
) f ( x)dx
(1)
cL
where Q
constant delivery lot size per cycle
H
supplier’s inventory holding cost per unit per time
K
penalty cost per time unit late (levied by the buyer)
a, b, cE, cL parameters defining the delivery window f(x) density function of delivery time X. According to the delivery window deliveries may be classified as early, X ∈ [a, cE); on-time, X ∈ [cE, cL]; or late, X ∈ (cL, b]. Figure 1 illustrates the supply chain delivery window. The first term in (1) defines the expected cost associated with early delivery; the second term defines the expected cost associated with late delivery. Figure 1
Illustration of delivery window
On-time delivery cE ≤ X ≤ cL
Early delivery a ≤ X < cE a
cE
Late delivery cL < X ≤ b cL
b
X
Continuous improvement of a process requires identifying the sources of waste in the process, evaluating the cost attributed to the waste, and investing in process improvements to eliminate the waste. When an early or late delivery occurs, management can study the delivery process and determine the assignable cause(s) for the untimely delivery. Corrective actions can be initiated with process improvements implemented to remove the cause(s) of untimely delivery. Improvements in delivery performance can occur as a result of: 1
the supplier gaining tighter control over process flow times
2
enhanced coordination of freight transport
3
more efficient material handling of outbound stock by the supplier and inbound stock by the buyer
4
implementation of electronic data interchange (EDI)
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The aforementioned strategies for improving delivery performance require financial investment for their implementation. These investments require justification which is often difficult to document. At a macro level of analysis, we provide a financial bound for justifying investments to improve delivery performance by modelling the improvement in delivery penalty costs that result from reducing the variability of the delivery time distribution. Variation in delivery performance represents uncertainty and uncertainty represents a lack of understanding and control of the delivery system by management. Management must first concentrate on reducing the variance of a process before improvement in the output of the process can be achieved. A reduction in the variance of the delivery distribution can occur due to: 1
a reduction in earliness of delivery (a approaches cE)
2
a reduction in lateness of delivery (b approaches cL)
3
a reduction in both earliness and lateness (a approaches cE, and b approaches cL).
Let τj represent the reduction in the range of the delivery distribution achieved at the jth delivery. Thus, a → a + gτj and b → b – (1 – g)τj where the reduction parameter, g, is defined as a function of the probabilities of early and late delivery g=
∫
cE
a
∫
cE
a
f ( x)
f ( x) +
∫
b
cL
(2) f ( x)
The reduction parameter g reflects the scope of delivery earliness relative to delivery lateness and incorporates this into the improvement of the expected penalty cost accordingly. For an improvement of τj, the expected penalty cost (1) is cE
Y j = QH
∫
a + gτ j
b − (1− g ) τ j
( cE − x )h( x)dx + K
∫
( x − cL )h( x)dx
(3)
cL
In (3), h(x) is the redefinition of the original delivery time density function f(x) in terms of τj. For a fixed mean delivery time, the variance of h(x) is decreasing in τj. Decreasing the variance of h(x) decreases the probabilities of early and late delivery while increasing the probability of on-time delivery. As the probability of on-time delivery increases the expected penalty cost defined by (3) decreases thus achieving improvement in delivery performance. Consider a time horizon of length T years where a demand requirement of D units for a single product will be met with deliveries of size Q. The time between deliveries is t = (QT/D), the number of deliveries is n = D/Q. When there is no improvement in the variance of delivery time the expected penalty cost to be incurred on a given delivery is constant and the net present value (NPV) of the expected penalty cost stream over time horizon T, YTnpv , compounded continuously at period interest rate i is defined to be
YTnpv = Y {e−it } + Y {e−2it } + … + Y {e−( n −1)it } + Y {e− nit }
(4)
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⎧⎪ n −1 k⎫ ⎪ = {Ye −it } ⎨ ⎡⎣ e −it ⎤⎦ ⎬ ⎩⎪ k = 0 ⎭⎪
(5)
⎧1 − e−iT = Y ⎨ it ⎩ e −1
(6)
∑
⎫ ⎬. ⎭
When improvement in the variance of the delivery distribution occurs the expected penalty cost decreases and the present worth of the expected penalty cost stream at the jth delivery is n
YTInpv =
∑ Y ( e ). j
− jit
(7)
j =1
The NPV estimates defined by (6) and (7) estimate in current dollars the costs expected to be incurred as a result of early and late deliveries over time horizon T. The difference in the two cost streams Γ = YTnpv − YTInpv
(8)
represents a bound for justifying the capital investment required to improve delivery performance. We argue that management should be willing to invest an amount equivalent to Γ to integrate continuous improvement within the supply chain by improving delivery performance. In the next two subsections, we illustrate the framework for variance-based improvement in the supply chain delivery performance and demonstrate the investment bound for justifying capital improvement in delivery performance. As identified in Bushuev and Guiffrida (2012), several candidate forms of probability density functions exist for defining supply chain delivery time distributions. We illustrate our continuous improvement framework for the following two densities: 1
the uniform
2
the Gaussian.
Our motivation for the selection of these two particular densities is to demonstrate the generalisability of the continuous improvement framework for delivery time densities that exist in closed form (uniform) and non-closed form (Gaussian).
3.1 Closed form illustration: uniformly distributed delivery times When delivery times are uniformly distributed, delivery times are defined such that X ~ Uniform(a, b) with f(x) = 1/(b – a). Entering f(x) = 1/(b – a) into (1) and simplifying yields the expected penalty cost per delivery (Guiffrida, 1999) Y=
1 ⎡QH ( cE − a )2 + K ( cL − b )2 ⎤ ⎦ 2(b − a) ⎣
and the probability density function of X is
(9)
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1 for a + gτ j ≤ x ≤ b − (1 − g ) τ j , b−a−τj
(10)
with variance Vj (X ) =
( b − a − τ j )2 12
(11)
.
To make sure that X is properly defined in (10), we assume that 0 < τj ≤ (b – a). Given (10), the expected penalty cost (9) is redefined as QH ( cE − a − gτ j ) + K (cL − b + τ j {1 − g}}2 2
Yj =
2 (b − a − τ j )
.
(12)
Equation (12) now provides a means to evaluate the expected penalty cost of early and late delivery as a function of the reduction in the variance of the delivery time distribution achieved at delivery j.
3.1.1 Numerical example Consider a supply chain where delivery time to the final customer is uniformly distributed with a = 5 days and b = 13 days. The on-time portion of the delivery window is defined by cE = 7 and cL = 10. The ratio of the costs associated with late and early K , was varied from 5 to 15 in increments of 5 with K ranging from delivery, QH $1,000 per day late to $4,000 per day late for a delivery lot size of Q = 200 units and an inventory holding cost per unit per year of H = $20. An annual demand of D = 4,800 units was used and the delivery time horizon was set to T = 2 years. For the 24 deliveries to be made over horizon T no order crossing was assumed and the delivery lot size was assumed to be defect free. The range reduction factor, τ, was varied from 0.02 to 0.10 in increments of 0.02 to reflect the improvement in delivery performance that is achievable by investing to reduce the variability of the delivery time distribution. An annual interest rate of i = 0.36 was used. The results of the numerical analyses conducted are presented in Table 4. For the parameters selected, Figure 2 illustrates the reduction in the expected penalty cost per delivery as a function of the reduction in the variance of the delivery time distribution for τ = 0.02, 0.06 and 0.10. Figure 3 illustrates the change in the difference of the NPV [equation (8)] of the penalty costs incurred over time horizon T under conditions of improvement through variance reduction and no improvement for varying ratios of the cost associated with late and early delivery. As defined in equation (8) and illustrated numerically in Figure 3, the difference between the new present values of the delivery penalty cost streams (improvement versus no improvement) over time horizon T provides a bound on the financial investment needed to support continuous improvement activities to reduce the variability of delivery times.
98,373 88,179 78,258 68,696
0.08
0.10
108,775
YTINPV
0.06
142,499
YTNPV
0.04
0.02
τ
73,803
64,241
54,320
44,126
33,724
Γ 273,365
YTNPV
133,495
151,749
170,528
189,710
209,202
YTINPV
K = 10 QH
139,870
121,616
102,837
83,654
64,163
Γ 404,231
YTNPV
198,293
225,240
252,877
281,048
309,630
YTINPV
K = 15 QH
205,938
178,991
151,354
123,183
94,601
Γ
Table 4
K =5 QH
A modelling framework for improving supply chain delivery performance Results for uniform numerical example under varying model parameters
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Figure 2
Reduction in delivery penalty cost under improvement
Figure 3
NPV of delivery penalty cost reduced by improvement
3.2 Non-closed form illustration: Gaussian distributed delivery times When delivery times are Gaussian distributed, delivery times are defined such that 1 ⎡ ( x − μ)2 ⎤ exp ⎢ X ~ Gaussian(μ, σ2) with f ( x) = ⎥ for –∞ < x < ∞, σ > 0, and 2 σ 2π ⎣ 2σ ⎦ –∞ < μ < ∞. To insure a valid probability density function over the finite endpoints a and b of the delivery window, we define the bounded Gaussian density as h( x ) =
f ( x) b
∫ f ( x)dx a
(13)
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For ease of notation let φ(∙) and Φ(∙)as the standard normal density and cumulative distribution functions respectively. Using (9), the expected penalty cost per delivery (Guiffrida, 1999) is QH ⎡ ⎤ Yj = ⎢ ⎛ {b − (1 − g ) τ j } − μ ⎞ ⎛ {a + gτ j } − μ ⎞ ⎥ ⎢Φ⎜ ⎟−Φ⎜ ⎟⎥ σ σ ⎠ ⎝ ⎠ ⎦⎥ ⎣⎢ ⎝ ⎡ ⎛ ⎛c −μ⎞ ⎛ {a + gτ j } − μ ⎞ ⎞ × ⎢σ ⎜ φ ⎜ E − φ⎜ ⎟ ⎟⎟ ⎟ σ ⎝ ⎠⎠ ⎢⎣ ⎜⎝ ⎝ σ ⎠ ⎛ ⎛ {a + gτ j } − μ ⎞ ⎛ cE − μ ⎞ ⎞ ⎤ + ( cE − μ ) ⎜ Φ ⎜ ⎟−Φ⎜ ⎟ ⎟⎥ σ ⎠ ⎝ σ ⎠ ⎠⎦ ⎝ ⎝ K ⎡ ⎤ +⎢ ⎥ ⎛ {b − (1 − g ) τ j } − μ ⎞ ⎛ {a + gτ j } − μ ⎞ ⎥ ⎢Φ⎜ ⎟ − Φ⎜ ⎟⎥ ⎢⎣ ⎝ σ σ ⎠ ⎝ ⎠⎦
(14)
⎡ ⎛ ⎛c −μ⎞ ⎛ {b − (1 − g ) τ j } − μ ⎞ ⎞ × ⎢σ ⎜ φ ⎜ L ⎟⎟ ⎟ − φ⎜ σ ⎝ ⎠⎠ ⎣⎢ ⎝ ⎝ σ ⎠ ⎛ ⎛ {b − (1 − g ) τ j } − μ ⎞ ⎛ cL − μ ⎞ ⎟⎞ ⎥⎤ − ( cL − μ ) ⎜ Φ ⎜ ⎟−Φ⎜ ⎟ σ ⎠ ⎝ σ ⎠ ⎠ ⎦⎥ ⎝ ⎝
3.2.1 Numerical example In this example, a comparative analysis will be drawn between a Gaussian and a uniformly distributed delivery time distribution using a set of common parameters. Consider a supply chain with a delivery window defined by the following set of parameters (all in days): a = 3, cE = 6, cL = 7 and b = 10. The Gaussian delivery time distribution is defined by X ~ Gaussian(6.5, 2.25); the uniform delivery time distribution is defined by X ~ Uniform(3, 10). For Q = 500 units, K = $10,000 and H = $5 the expected penalty costs per delivery under no improvement (τj = 0) are Yj = $4,423 for the Gaussian distributed delivery time distribution and Yj = $8,036 for the uniformly distributed delivery time distribution. Both delivery time distributions had mean and modal delivery times of 6.5 days with the probabilities of early, on-time and late deliveries equal to 0.367 (0.429), 0.266 (0.142), and 0.367 (0.429) for the bounded Gaussian (uniform). Figure 4 illustrates the change in the difference of the NPV of the penalty costs incurred under varying levels of improvement across 24 deliveries for a time horizon of T = 2 years with i = 0.36%. When no improvement is initiated, the expected penalty cost of the uniform ($8,036) exceeds that of the Gaussian ($4,423) do to the thickener tails found in the uniform. Over the common delivery window, the thicker tails result in the uniform having more probability mass positioned in the early and late portions of the delivery window and less probability mass within the on-time portion of the window when compared to the Gaussian. As illustrated in Figure 4, the improvement in the expected penalty cost for each density is possible when improvements are initiated to reduce the variability of
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delivery times. Over a set time horizon, a greater opportunity to achieve a reduction in the expected penalty cost exists when the delivery time density has thicker tails. Figure 4
4
Comparison in improvement of net present value of delivery penalty costs for Gaussian and uniform delivery time distribution
Summary and conclusions
In this paper, we have presented a model for evaluating the financial investment required to improve supply chain delivery performance. The model determines the expected penalty cost for untimely (early or late) delivery subject to conditions for early, on-time and late delivery as specified by a stated delivery window. The model herein can be used to financially quantify the benefit of reducing the variance of the delivery time distribution. By analysing the present worth of the expected cost streams resulting from no improvement and from improvement based on reducing the variance of delivery time, an investment bound for improvement of the delivery process can be established. The modelling framework presented is capable of guiding supply chain managers to assign investment to delivery performance and in order achieve to continuous improvement in supply chain operations. The generalisability of the continuous improvement framework was demonstrated for delivery time distributions which exist in closed form (uniform) and non-closed form (Gaussian). The results of the numerical illustrations are clearly dependent on the values assigned to the model parameters. Given this assertion, we maintain that the model presented herein does provide a useful framework for financially evaluating the impact of improving delivery performance by reducing the variability of the delivery time distribution. An attractive feature of this model is that the investment needed to support the improvement in delivery performance is reported in the metric of cost which is regarded as a key measure for supply chains (see for example, Melnyk et al., 2010). There are several aspects of this research that can be extended. First, other forms of the probability density function used to define the delivery time distribution could be used. An interesting class of density functions would be those that exhibit varying degrees of both skewness and excess kurtosis such as the Gamma. Second, the optimal positioning of the delivery window could be introduced into the model formulation.
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Third, a non-constant delivery lot size could be introduced into the model thus allowing for weight and quantity-based transportation costs to be integrated into the model. Lastly, a budget constraint governing the investment available for process improvement could be introduced and the level of variance reduction in the delivery process could be investigated.
Acknowledgements The authors thank the reviewers for their useful comments and suggestions which lead to improvements in the manuscript.
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