MODELS AND ANALYSES FOR DAMAGE AND YIELD FOR EFFECTIVE SUPPLY
CHAIN NETWORK DESIGN
A Dissertation by
Samir Abdulmoati Alsobhi
Master of Science, Florida Institute of Technology, Melbourne, USA, 2009
Bachelor of Science, King Abdulaziz University, Jeddah, Saudi Arabia, 2004
Submitted to the Department of Industrial and Manufacturing Engineering
and the faculty of the Graduate School of
Wichita State University
in partial fulfillment of
the requirements for the degree of
Doctor of Philosophy
May 2015
© Copyright 2015 by Samir Abdulmoati Alsobhi
All Rights Reserved
MODELS AND ANALYSES FOR DAMAGE AND YIELD FOR EFFECTIVE SUPPLY
CHAIN NETWORK DESIGN
The following faculty members have examined the final copy of this dissertation for form and
content, and recommend that it be accepted in partial fulfillment of the requirement for the degree
of Doctor of Philosophy with a major in Industrial Engineering.
____________________________________
Krishna K. Krishnan, Committee Chair
____________________________________
Deepak Gupta, Committee Co-Chair
____________________________________
Anil Mahapatro, Committee Member
____________________________________
Ramazan Asmatulu, Committee Member
____________________________________
Rajeev Madhavannair, Committee Member
Accepted for the College of Engineering
____________________________________
Royce Bowden, Dean
Accepted for the Graduate School
____________________________________
Abu Masud, Interim Dean
iii
DEDICATION
To my late father
iv
ACKNOWLEDGEMENTS
I would like to express my heartfelt gratitude to my adviser, Dr. Krishna Krishnan, for his
valuable time, guidance, and help. I especially appreciate his patience, encouragement,
perseverance, advising, and expertise, all of which helped me to complete this dissertation. None
of this work would have been possible without his guidance and assistance. Dr. Krishna has been
generous and supportive—academically, professionally, and personally—and the right role model
for me.
I thank my co-advisor Dr. Deepak Gupta for his perseverance and guidance, which allowed
me to make more discoveries in my research area. I also thank my committee members, Dr. Anil
Mahapatro, Dr. Ramazan Asmatulu, and Dr. Rajeev Madhavannair, for their judicious guidance
and valuable comments, all of which were imperative for achieving completion of this dissertation.
I thank all members of the Facilities Planning and Logistics Research Group at Wichita State
University for their stimulus and support.
Most especially, I extend my deepest gratitude to my mother, wife, and family for their
unlimited support and love during my studies.
v
ABSTRACT
In a supply chain (SC) system, products are damaged during shipping due to transportation
hazards and inadequate packaging. The most common hazards in transportation include shocks,
vibrations, accidents, and poor handling. Damage from accidents and handling issues are not
completely within the control of packaging. However, proper packaging can prevent most damages
from shocks and vibrations. In this dissertation, a mathematical model that minimizes total costs
(damage, shipping, and packaging) has been developed to address the issue of damage costs. This
model was implemented in MATLAB and verified by using a total enumeration strategy.
The damages during shipping are stochastic in nature. To minimize the impact of damage,
the selection of routes should consider not only the expected damage but also the variability of
damage. In this research, two models, the first of which aims at minimize total cost in the supply
chain network, which consists of product cost and transportation cost while considering multiple
routes and multiple products under stochastic yield conditions. In the second model, the concept
of robust design has been applied to minimize damage while maximizing yield.
This research also focuses on the recovery of products that are damaged during transit.
Different recovery models based on the type of damage are also developed. Also, a network that
recovers the damaged product at different stages in the supply chain network is considered. A
methodology for determining the best recovery model to ensure maximum profit and meet demand
is developed. Results indicate that the location of inspection and recovery stations influence the
cost models and the subsequent profits.
vi
TABLE OF CONTENTS
Chapter
1.
INTRODUCTION ...............................................................................................................1
1.1.
1.2.
1.3.
2.
Introduction ..............................................................................................................1
Research Objectives .................................................................................................3
References ................................................................................................................5
ANALYSIS OF DAMAGE COSTS IN SUPPLY CHAIN SYSTEMS ..............................7
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
2.7.
3.
Page
Abstract ....................................................................................................................7
Introduction ..............................................................................................................7
Literature Review...................................................................................................10
General Model .......................................................................................................14
2.4.1 Mathematical Formulation for General Model ..........................................15
Case Studies ...........................................................................................................16
2.5.1. Case Study 1 ..............................................................................................16
2.5.1.1
Results and Analysis for Case Study 1 ....................................19
2.5.2. Case Study 2 ..............................................................................................23
2.5.2.1
Results and Analysis for Case Study 2 ....................................25
2.5.3. Case Study 3 ..............................................................................................28
2.5.3.1
Results and Analysis for Case Study 3 ....................................30
2.5.3.2
Sensitivity Analysis for Case Study 3 ......................................34
2.5.4. Case Study 4 ..............................................................................................35
2.5.4.1
Results and Analysis for Case Study 4 ....................................37
2.5.5. Case Study 5 ..............................................................................................40
2.5.5.1
Results and Analysis for Case Study 5 ....................................40
Conclusions and Future Work ...............................................................................43
References ..............................................................................................................43
ROBUST SUPPLY CHAIN SYSTEM UNDER YIELD UNCERTAINTY ....................46
3.1.
3.2.
3.3.
3.4.
Abstract ..................................................................................................................46
Introduction ............................................................................................................46
3.2.1 Supply Chain under Yield Uncertainty ......................................................47
3.2.2 Robust Supply Chain .................................................................................50
3.2.3 Supplier Selection ......................................................................................51
Problem Statement and Formulation .....................................................................53
3.3.1 Model 1: Design to Minimize Total Cost ..................................................55
3.3.2 Model 2: Robust Design under Yield Uncertainty ....................................56
Case Studies ...........................................................................................................57
3.4.1 Case Study 1 ..............................................................................................57
3.4.1.1
Results and Analysis for Case Study 1 ....................................59
vii
TABLE OF CONTENTS (continued)
Chapter
3.5.
3.6.
4.
3.4.1.2
Sensitivity Analysis for Case Study 1 ......................................63
3.4.2 Case Study 2 ..............................................................................................64
3.4.2.1
Results and Analysis for Case Study 2 ....................................67
Conclusions and Future Work ..............................................................................68
References ..............................................................................................................69
DAMAGE RECOVERY MODELS FOR SUPPLY CHAIN SYSTEM ...........................73
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
5.
Page
Abstract ..................................................................................................................73
Introduction ............................................................................................................73
Literature Review...................................................................................................75
Damage Recovery Approach .................................................................................78
4.4.1 Recovery Model 1 ......................................................................................79
4.4.1.1
Mathematical Representation for Recovery Model 1 ..............80
4.4.1.2
Numerical Example 1 for Recovery Model 1 ..........................81
4.4.2 Recovery Model 2 ......................................................................................81
4.4.2.1
Mathematical Representation for Recovery Model 2 ............82
4.4.2.2
Numerical Example 2 for Recovery Model 2 ..........................84
4.4.3 Recovery Model 3 ......................................................................................85
4.4.3.1
Mathematical Representation for Recovery Model 3 ..............86
Numerical Example 3 for Recovery Model 3 ..........................87
4.4.3.2
4.4.4 Recovery Model 4 ......................................................................................89
4.4.4.1
Mathematical Representation for Recovery Model 4 ..............90
4.4.4.2
Numerical Example 4 for Recovery Model 4 ..........................90
4.4.5 Recovery Model 5 ......................................................................................92
4.4.5.1
Mathematical Representation for Recovery Model 5 ..............93
Numerical Example 5 for Recovery Model 5 ..........................94
4.4.5.2
Case Study 1 ..........................................................................................................96
Case Study 2 .........................................................................................................98
4.6.1 Results and Analysis for Case Study 2 ......................................................99
Case Study 3 .......................................................................................................100
4.7.1 Results and Analysis for Case Study 3 ....................................................101
Case Study4 ........................................................................................................101
4.8.1 Results and Analysis for Case Study 4 ....................................................102
Case Study 5 .......................................................................................................102
4.9.1 Results and Analysis for Case Study 5 ....................................................103
Conclusions and Future Work .............................................................................103
References ............................................................................................................104
CONCLUSIONS AND FUTURE WORK ......................................................................106
viii
TABLE OF CONTENTS (continued)
Chapter
5.1
5.2
Page
Conclusions ..........................................................................................................106
Future Work .........................................................................................................108
APPENDIX ..................................................................................................................................109
ix
LIST OF TABLES
Table
Page
2.1
Transportation Network Routes for Case Study 1 ............................................................ 17
2.2
Parameters of Transportation Types and Routes for Case Study 1 .................................. 18
2.3
Damage Probability of Packaging Types for Case Study 1 .............................................. 18
2.4
Damage Cost for Case Study 1 ..........................................................................................19
2.5
Total Costs (Damage + Transportation) for Case Study 1 .................................................19
2.6
Scenarios for Case Study 1 ................................................................................................21
2.7
Optimal Cost for Case Study 1 Using MATLAB ..............................................................22
2.8
Labor and Assembly Costs for Case Study 2.....................................................................23
2.9
Damage Probability Prior to Shipping for Case Study 2 ...................................................24
2.10
Damage Probability at Final Destination for Case Study 2 ...............................................24
2.11
Total Cost of Products if Assembled Prior to Shipping for Case Study 2 .........................25
2.12
Total Cost of Products if Assembled at Final Destination for Case Study 2 .....................25
2.13
Scenarios for Case Study 2 ............................................................................................... 28
2.14
MATLAB Results for Case Study 2 ................................................................................. 28
2.15
Damage Probability when Using Different Packaging for Case Study 3 ......................... 29
2.16
Total Cost when Using Type Z1 Packaging for Case Study 3 ...........................................31
2.17
Total Cost when Using Type Z2 Packaging for Case Study 3 ...........................................31
2.18
Scenarios for Case Study 3 ................................................................................................32
2.19
MATLAB Results for Case Study 3 when All Products Use Different Paths ...................33
2.20
MATLAB Results for Case Study 3 when All Products Use Same Best Path
(1-3-6-10) ...........................................................................................................................33
x
LIST OF TABLES (continued)
Table
Page
2.21
Results of Sensitivity Analysis for Case Study 3...............................................................35
2.22
Product Costs for Case Study 4 .........................................................................................36
2.23
Distance between Nodes for Case Study 4 ........................................................................37
2.24
Optimal Cost and Yield Percentage for Case Study 4 when All Products Use
Different Paths ..................................................................................................................38
2.25
Optimal Cost and Yield Percentage for Case Study 4 when All Products Use Same
Best Path (1-2-6-13-25-34) ............................................................................................... 39
2.26
Product Costs for Case Study 5 .........................................................................................40
2.27
MATLAB Results for Case Study 5 when All Products Use Different Paths ...................41
2.28
MATLAB Results for Case Study 5 when All Products Use Same Best Path
(1-2-4-10-22-34) ................................................................................................................42
3.1
Design Parameters for Case Study 1 ..................................................................................58
3.2
Results of Initial Design for All Products for Case Study 1 ..............................................59
3.3
Results of Robust Design for All Products for Case Study 1 ............................................59
3.4
Results of Initial Design for Product P1 for Case Study 1 .................................................63
3.5
Results of Robust Design for Product P1 for Case Study 1 ...............................................63
3.6
Results of Sensitivity Analysis for All Products for Case Study 1 ....................................64
3.7
Design Parameters of Supplier 1 for Case Study 1 ............................................................66
3.8
Design Parameters of Supplier 2 for Case Study 1 ............................................................66
3.9
Results of Initial Design for Both Suppliers for Case Study 2 ..........................................67
3.10
Results of Robust Design for Both Suppliers for Case Study 2 ........................................68
4.1
Parameters of Numerical Example 1 for Recovery Model 1 .............................................81
xi
LIST OF TABLES (continued)
Table
Page
4.2
Parameters of Numerical Example 2 for Recovery Model 2 .............................................84
4.3
Parameters of Numerical Example 3 for Recovery Model 3 .............................................88
4.4
Parameters of Numerical Example 4 for Recovery Model 4 .............................................91
4.5
Parameters of Numerical Example 5 for Recovery Model 5 .............................................94
4.6
Parameters of Recovery Model 3 for Case Study 1 ...........................................................96
4.7
Parameters of Recovery Model 4 for Case Study 1 ...........................................................97
4.8
Recovery System Parameters for Case Study 2 .................................................................99
4.9
Damage Probability for Each Path in Case Study 2 ..........................................................99
4.10
Design Parameters for Case Study 3 ................................................................................101
4.11
Design Parameters for Case Study 4 ................................................................................102
4.12
Design Parameters for Case Study 5 ................................................................................103
A-1
Damage Percentages for Case Study 4 ............................................................................110
A-2
Damage Percentages for Case Study 5 ............................................................................112
xii
LIST OF FIGURES
Figure
Page
2.1
Transportation network for Case Study 1 ..........................................................................17
2.2
Damage cost when using type Z1 packaging for Case Study 1..........................................20
2.3
Damage cost when using type Z2 packaging for Case Study 1..........................................20
2.4
Optimal cost for Case Study 1 ..........................................................................................21
2.5
Damage cost for products assembled prior to shipping for Case Study 2 .........................26
2.6
Damage cost for products assembled at final destination for Case Study 2 ......................26
2.7
Optimal cost for Case Study 2 ...........................................................................................27
2.8
Transportation network for Case Study 3 ..........................................................................29
2.9
Optimal path for each product for Case Study 3................................................................34
2.10
Transportation network for Case Study 4 ..........................................................................36
2.11
MATLAB output for product P1 for Case Study 4 ............................................................39
2.12
MATLAB output for product P1 for Case Study 5.............................................................42
3.1
Supply chain network for Case Study 1 .............................................................................58
3.2
Results of initial design vs robust design for product P1 for Case Study 1 .......................60
3.3
Results of initial design vs robust design for product P2 for Case Study 1 .......................61
3.4
Results of initial design vs robust design for product P3 for Case Study 1 .......................61
3.5
Results of initial design vs robust design for product P4 for Case Study 1 .......................62
3.6
Supply chain network for Case Study 2 .............................................................................65
4.1
Recovery Model 1 ..............................................................................................................80
4.2
Recovery Model 2 ..............................................................................................................82
xiii
LIST OF FIGURES (continued)
Figure
Page
4.3
Recovery Model 3 ..............................................................................................................86
4.4
Recovery Model 4 ..............................................................................................................89
4.5
Recovery Model 5 ..............................................................................................................92
4.6
Recovery Model Case Study 1 Supply Chain Network .....................................................96
4.7
Recovery Model 3 Integrated into Case Study 1 Supply Chain Network .........................97
4.8
Recovery Model 4 Integrated into Case Study 1 Supply Chain Network ........................97
4.9
Transportation Network for Case Study 2 .........................................................................98
4.10
Transportation Network for Case Study 3 .......................................................................100
4.11
Transportation Network for Case Study 4 .......................................................................102
4.12
Transportation Network for Case Study 5 .......................................................................103
xiv
LIST OF ABBREVIATIONS/NOMENCLATURE
A
Airport
d
Quantity
DEA
Data Envelopment Analysis
G
Retailer
ILP
Integer Linear Programming
k
Route Number
l
Location
LP
Linear Programing
M
Manufacturer
MIP
Mixed Integer Programing
MILP
Mixed-Integer Linear Programming
P
Product
PS
Path Set
R
Route
RC
Recovery Center
S
Supplier
SC
Supply Chain
Sn
Stage
SP
Seaport
SCN
Supply Chain Network
T
Transportation Method
TA
Transportation Alternative
xv
LIST OF ABBREVIATIONS/NOMENCLATURE (continued)
TRC
Total Recovery Cost
U
Shipping Cost
V
Damage type
Z
Packaging Type
φ
Distance
xvi
LIST OF SYMBOLS
Aij
Distance from node i to node j
Bi,j
Cost per mile from node i to node j
Cnd
Damage cost for product n
dn
Quantity for product n
Dn
Disassembly cost for product n
d L nij
Lower limit of damage percentage for product n from node i to node j
d U nij
Upper limit of damage percentage for product n from node i to node j
Fn
Lost cost for product n
Ii+1
Inspection cost at node i+1
Lnl
Labor cost for product n at location l
Mnl
Total Part cost for product n at location l
Oz, n
Repackaging cost for type Z packaging for product n
PZ,n
Packaging cost using type Z packaging for product n
qi+1
Quantity of good units received at node Si+1
Rs
Route from supplier s
Tv,n
Repair cost for type v damages for product n
Uij
Shipping cost from node i to node j
xvii
LIST OF SYMBOLS (continued)
Wn
Cost of product n
Xvnij
Percentage for all types of damages v for product n
Ynsij
Yield for product n for supplier s from node i to node j
φsg
Distance from supplier s to retailer g
γnzsj
Damage percentage for product n using type z packaging from supplier s to retailer
g
σ d nij
Standard deviations of damage distribution functions for product n from node i to
node j
xviii
CHAPTER 1
INTRODUCTION
1.1
Introduction
The expansion and globalization of manufacturing have led to long supply chains (SCs),
which can increase the cost of products because of several factors: fuel costs, waiting times at
ports, high inventory, warehousing costs, and shipping damage. Damage during shipping can
result in increased cost on a per unit basis. Reduction of damages by improved packaging methods
and selection of appropriate methods of transportation can lead to a reduction in cost. Good
packaging helps to ensure that customers receive a product without any damage, whereas poor
packaging may result in it being damaged and hence an increase in SC costs. The practice of good
packaging will also lead to improvement in the efficiency of the SC while maintaining product
quality.
The nature of a product can also influence the type of damage incurred. For example,
fragile products can be easily damaged during shipping, especially if the type of packaging is not
appropriate. Both the nature of the product and the type of packaging influence the percentage of
damaged goods during shipping. Vursavuş and Özgüven (2004) describe numerous types of
packaging that are used to transport goods, such as polystyrene soft cell trays, paper pulp trays,
wood bins, bulk bins, and corrugated fiberboard. Aryanezhad et al. (2012) designed a supply chain
network while considering unreliable suppliers and distribution centers. They pointed out that the
quantity of products delivered may decrease due to unreliable distribution centers. In order to
minimize the total shipping cost, they developed a mathematical model of the problem as a
nonlinear integer program.
1
From another perspective, the type of damage and cost of transportation can depend on
whether the product is shipped in a fully or partially assembled condition or as individual
components. As a result, it is important that the design of the SC system focus on actions that
reduce the negative impacts of packaging type and methods on products damaged in the system,
since these disruptions can be very expensive. The nature of the damage can also vary. In some
cases, damage may be severe and the product may have to be discarded; in other cases, damage
may be minor and the product functionality is retained. Therefore, damage that is caused during
transit may be categorized into three levels: minor, repairable, and severe. Also, because each leg
of the supply chain uses a different method of transportation, the percentage of damage along each
path might vary.
The robustness technique was first presented by Taguchi et al. (1987) and Taguchi and
Phadke (1989). The advantage of this technique is that it finds a robust solution that is less sensitive
to unknown variations. Numerous studies have used the robustness technique to solve engineering
problems. For example, Shi et al. (2013) developed a model to design a robust design configuration
for a cross-docking center in order to minimize variability of the supply chain system. Chauhan
et al. (2006) and Pan and Nagi (2010) developed a robust optimization model for designing an SC
network under demand uncertainty. Their objective function consists of total cost, variability cost,
and penalty cost due to unmet demand. Gutiérrez et al. (1996) developed a robust model for an
incapacitated network design problem by considering uncertainty in the input data. Almaktoom
et al. (2014) presented a novel robust design optimization to assure service level rate requirements
in complex SC networks. Lalmazloumian et al. (2013) developed a robust optimization model for
agile and build-to-order SC planning to minimize the total cost while preserving the customer
service level rate.
2
Thierry et al. (1995) detailed five recovery options: repair, refurbishing, remanufacturing,
cannibalization, and recycling. They focused on remanufacturing, whereby the product is
refurbished to a good-as-new condition by replacing components or by employing used parts. Aras
et al. (2006) focused on profitability of the remanufacturing option and pointed out that it is often
assumed that bringing the product to a good-as-new condition is, on average, less costly than
producing a new one and disposing of the damaged product. According to the Food Marketing
Institute and the Grocery Manufacturers Association (2008), products that cannot be sold represent
about a $15 billion loss to the industry every year. On average, a manufacturer must sell seven
new items to replace the lost profit from one unsold product. The Reverse Logistics Executive
Council pointed out that the cost of handling, transporting, and determining the disposition of
returned products is about $35 billion annually for U.S. firms (Meyer, 1999), and the cost of
remanufacturing is about $50 billion per year (Corbett, 2001).
Based on issues relative to damaged products in the supply chain, as cited above, research
objectives in this dissertation are detailed in the next section.
1.2
Research Objectives
This dissertation is organized into five chapters, each focusing on a particular type of
disruption in the supply chain network (SCN) and consequently a different objective. The overall
objective of this dissertation is to develop mathematical models for products damaged in the SC
network in order to minimize total cost while maximizing profit and meeting customer demand
under yield uncertainty by answering the following questions:
(1) Should the factory ship products in an assembled or disassembled condition?
(2) What packaging methods should be used to minimize damage costs?
(3) What is the impact of different transportation types on the yield at each stage of the SCN?
3
(4) Is there a difference in the probability of damage if different modes of transportation are
used?
(5) If there are multiple suppliers who provide the same product with different types of
packaging and product prices, will it help to maintain the yield in the final destination?
(6) If there is a recovery center in the system to salvage the damaged product, is this better
than ordering more products in order to meet customer demand?
(7) Where can the recovery center be allocated with respect to efficiency?
Chapter 2: Analysis of Damage Costs in Supply Chain Systems. This chapter answers the
first three research questions and also presents a novel model for selecting the appropriate type of
packaging and transportation method for reducing damages. The objective of this model is to
minimize damage and shipping costs while considering multiple routes, products, and packaging
types under deterministic damage probability. It is pointed out that there is a need for addressing
the quality of packaging and its impact on damage to products and parts during shipping. As a
result, this chapter focuses on different models for the selection of best routes and packaging types
to minimize loss that occurs during shipping.
Chapter 3: Robust Supply Chain System under Yield Uncertainty. This chapter answers
questions four and five. It considers the problem when customer demand is known in the SC, and
the yield is stochastic depending on the route and suppliers used to ship products to the customer.
Two models are developed to solve the problem. The first model minimizes total cost, which
consists of both product cost and shipping cost. The second model is a robust design model, which
minimizes the probability of damage to products in order to maximize yield. As a result, this
chapter focuses on different models for the selection of best routes and suppliers to maintain the
4
yield during shipping, and it also focuses on two case studies to illustrate and validate the
procedure.
Chapter 4: Damage Recovery Models for Supply Chain System. This chapter answers the
last two questions. In a supply chain when there are damages that occur in each leg of shipping,
recovery of the damaged products can be attempted. The type of recovery model depends on the
cost of the product, value of the damaged product, cost of shipping, and ease of repair. Most
companies focus on controlling efficiency in their forward SC, paying less attention to the
damaged product, which leads to dissatisfied customers, minimized profits, increased overhead
cost, etc.). Different recovery models that can be applied to varying parameters of costs have been
developed in this chapter. A methodology for determining the cost associated with each recovery
model has been developed to understand the impact. A case study is used to validate the proposed
methodology.
Chapter 5: Conclusions and Future Work.
This chapter ends the dissertation with
conclusions and discusses future work in this area of study.
1.3
References
Almaktoom, A. T., Krishnan, K. K., Wang, P., and Alsobhi, S. (2014). Assurance of system service
level robustness in complex supply chain networks. The International Journal of Advanced
Manufacturing Technology, 74(1-4), 445–460.
Aras, N., Verter, V., and Boyaci, T. (2006). Coordination and priority decisions in hybrid
manufacturing/remanufacturing systems. Production and Operations Management, 15(4):528–43.
Chauhan, S., Proth, J., Sarmiento, A., and Nagi, R. (2006). Opportunistic supply chain formation
from qualified partners for a new market demand. Journal of the Operational Research Society,
57(9), 1089–1099.
Corbett, C. J., and Kleindorfer, P. R. (2001). Environmental management and operations
management: Introduction to part 1 (manufacturing and eco-logistics). Production and Operations
Management, 10(2):107–111.
5
Food Marketing Institute and the Grocery Manufacturers Association. (2008). Retrieved from
GENCO Product Lifecycle Logistics, http://www.genco.com/Damage-Research/damageresearch.php [03-02-2014].
Gutiérrez, G. J., Kouvelis, P., and Kurawarwala, A. A. (1996). A robustness approach to
uncapacitated network design problems. European Journal of Operational Research, 94(2), 362–
376.
Lalmazloumian, M., Wong, K., Y., Govindan, K., and Kannan, D. (2013). A robust optimization
model for agile and build-to-order supply chain planning under uncertainties. Annals of Operations
Research, 1–36.
Aryanezhad, M.-B., Naini, S. G. J., and Jabbarzadeh, A. (2012). An integrated model for designing
supply chain network under demand and supply uncertainty, African Journal of Business
Management, 6(7), 2678–2696.
Meyer, H. (1999). Many happy returns. Journal of Business Strategy, 80(7), 27–31.
Pan, F., and Nagi, R. (2010). Robust supply chain design under uncertain demand in agile
manufacturing. Computers and Operations Research, 37(4), 668–683.
Shi, W., Liu, Z., Shang, J., and Cui, Y. (2013). Multi-criteria robust design of a JIT-based crossdocking distribution center for an auto parts supply chain. European Journal of Operational
Research, 229(3), 695–706.
Taguchi, G., Clausing, D., and Watanabe, L., T. (1987). System of experimental design, in
Engineering Methods to Optimize Quality and Minimize Costs. Vol. 2. UNIPUB/Kraus
International Publications, White Plains, NY.
Taguchi, G., and Phadke, M., S. (1989). Quality Engineering through Design Optimization Quality
Control, Robust Design, and the Taguchi Method (pp. 77–96). Springer.
Thierry, M., Salomon, M., Van Nunen, J., and Van Wassenhove, L. (1995). Strategic issues in
product recovery management. California Management Review, 37(2),114–35.
Vursavuş, K. K., and Özgüven, F., (2004). Determining the effects of vibration parameters and
packaging method on mechanical damage in golden delicious apples. Turkish Journal of
Agriculture and Forestry, 28(5), 311–320.
6
CHAPTER 2
ANALYSIS OF DAMAGE COSTS IN SUPPLY CHAIN SYSTEMS
2.1
Abstract
In a supply chain system, products are damaged during shipping due to transportation
hazards and inadequate packaging. The most common transportation hazards include shocks,
vibrations, accidents, and poor handling. Damage from accidents and handling issues are not
completely within the control of packaging; however, proper packaging can prevent most damage
from shocks and vibrations. The loss due to damage at each stage of the SC network can be reduced
by selecting the appropriate packaging, transportation, and shipping in either an assembled or
unassembled condition. There is a paucity of literature available in the area of goods damaged
during shipping relative to SC systems. In this dissertation, a mathematical model that minimizes
total costs (damage, shipping, and packaging) has been developed to address the issue of damage
costs. This model was implemented in MATLAB and verified by using a total enumeration
strategy.
Case studies to illustrate and validate the procedure were developed and also
implemented in MATLAB.
Keywords: damage cost, supply chain risk, shortest path, shipping cost, packaging cost
2.2
Introduction
The expansion and globalization of manufacturing have led to long supply chains, which
have raised the need for preventing damage during transportation. With poor packaging, products
may get damaged during transportation. This consequently leads to waste (or Muda, as it is called
in Japan). Proper application of lean techniques should be applied to reduce this waste. One
method of reducing waste is by using proper packaging. The type of packaging and method of
transportation influences the amount and type of damage. Proper or appropriate product packaging
7
helps to ensure that customers receive products without any damage. The most common hazards
in transportation include shocks, vibrations, accidents, and poor handling. Damage from accidents
and handling issues are not completely within the control of packaging; however, proper packaging
can prevent most damage from shocks and vibrations.
Shocks occur during handling or
transportation. For example, during the transportation of products in trucks, shocks might occur
when the condition of the road is poor. Good packaging can ensure increased prevention of
damage for products and improve efficiency of the entire SC system (manufacturing processes,
logistics processes, supply chain relationships, and lost sales). Azzi et al. (2012) stated that
“approximately 9% of the cost of any product is likely to be the cost of its packaging.” Some
techniques for monitoring transportation conditions to prevent packaging and products from
damage are available, e.g., shipping containers can be fitted with devices that can monitor
conditions during shipping, such as air temperature, humidity, air pressure, vibration, and shock.
The nature of products can also influence the type of damage. For instance, fragile products
can be easily damaged during transportation, especially if the package type is not appropriate for
the product. Thus, products that are fragile may be shipped as individual components with
appropriate packaging; however, a product that is sturdy may be assembled and shipped with
minimal packaging. On the other hand, in some cases, it might be better to wrap individual
components and package them together to prevent damage.
Each stage of the supply chain may encounter damage. The damage caused during
shipping may be classified into three categories: minor, repairable, and severe. In the case of minor
damage, the product has physical damage but has retained its required functionality. In this case,
the product may have dents and scratches on the surface, but it has not lost any of its functionality.
In the case of repairable damage, the product has physical damage that affects the required
8
functionality; however, the product can be repaired by replacing some parts. In the case of severe
damage, the product has serious physical damage and a repair option is not feasible. However, the
product can be salvaged, because some of its parts could be reused; therefore, some cost can be
recovered while other parts might be scrapped.
Different transportation methods are available in a supply chain.
Often multiple
transportation methods are used within a single SC system. Trucking is the most popular method
of transportation, because it involves shorter times, and is easy and flexible. However, truck
transportation is often subject to higher shocks and vibrations. Transportation involving ships,
which is often the cheapest method, is limited to certain routes and typically cannot be used for
transporting products to the final destination.
The type of packaging can also determine the quantity of damage. Vursavuş and Özgüven
(2004) identified several different types of packaging—polystyrene soft cell trays, paper pulp
trays, wood bins, bulk bins, and corrugated fiberboard—all of which are used to transport goods.
Singh and Xu (1993) stated that up to 80% of apples may be damaged during shipping by truck.
The amount of damage depends on the type of packaging, type of truck, and position of cartons in
which the apples are shipped.
Products can be shipped after final assembly, or they can be shipped as individual
components that will require assembly at some point in the supply chain or at the final destination.
Products can also be shipped in a partially assembled condition. The type of damage and the cost
of transportation will depend on whether the product is shipped in a fully assembled condition, in
a partially assembled condition, or as individual components. Products that are shipped in a fully
assembled condition may be subject to all three types of damages described previously. In the
case of products that are shipped as individual sub-components that require assembly, it is possible
9
that any damage may lead to scrap. However, lean techniques used to maximize performance and
minimize waste caused by product damage during transportation must be evaluated to determine
the best possible options at the lowest cost.
The contents of this chapter are organized as follows: Section 2.3 provides a review of
related literature on supply chain disruptions. In section 2.4, a mathematical model to minimize
the total cost is developed. Section 2.5 illustrates five case studies to demonstrate the effectiveness
of the proposed model, including results and analysis. Finally, conclusions and future work are
discussed in section 2.6.
2.3
Literature Review
Supply chain disruptions have received considerable attention in the last decade. Most SC
disruptions that have been studied are those that have a low probability of occurrence, such as
tsunamis and earthquakes. While these SC disruptions happen with low probability, the results
are typically disastrous and lead to high costs. Azad and Davoudpour (2010) considered a facility
with random disruption risk in order to design a reliable supply chain network. They studied
disruption in distribution centers by location and capacity, formulated the problem as a nonlinear
integer programming model, and then linearized it to obtain the optimal solution. They considered
two different algorithms to solve random disruption risks for large-size cases: tabu search and
simulated annealing algorithms. These authors found the better solution by using the tabu search
algorithm.
Transportation costs from reliable and unreliable distribution centers were also
considered in this model.
Aryanezhad et al. (2012) designed an SCN considering an unreliable suppliers and
distribution centers. They found that the quantity of products delivered may decrease because of
unreliable distribution centers. They formulated the problem as a nonlinear integer program to
10
minimize total cost. They considered the costs of location, transportation, inventory, and lost sales.
Two approaches were developed to solve the problem: Lagrangian relaxation and a genetic
algorithm. In their model, they determined the location of optimal distribution centers and the
subset of customers to be served, assigned customers to distribution centers, and determined the
order quantity. The authors assumed infinite capacity as well with one distribution center to serve
all customers. Darwish et al. (2014) incorporated the quality of items into two vendor-managed
inventory models by considering a single-vendor multi-retailer in a supply chain system. The first
model focused on developing a decentralized SC to maximize the vendor’s profit, and the second
focused on a centralized supply chain to maximize the system profit.
Kristianto and Helo (2010) considered a strategic safety stock allocation to manage the
product development process in order to provide more flexibility to the SC system. Also,
Arshinder (2012) developed contracts for implementing and measuring SC flexibility when
producing newsvendor type products. Hatefi and Razmi (2013) used integer programming with
fuzzy objectives and assigned an optimal order quantity for allocated suppliers as constraints in
their model in order to perform supplier selection and determine order allocation.
Jabbarzadeh et al. (2012) designed a supply chain network based on the risk of disruption
at facilities. Facilities can be disrupted by natural disasters, machine breakdowns, terrorism, and
wars. The authors proposed the problem as a mixed-integer nonlinear model to maximize total
profit for the supply chain system. Two methods were developed to solve the problem: Lagrangian
relaxation and a genetic algorithm. They used Lagrangian relaxation to integrate the entire supply
chain and the genetic algorithm to obtain the optimal solution for the model. Several researchers
(Aryanezhad et al., 2012; Azad and Davoudpour, 2010; Jabbarzadeh et al., 2012) have assumed
that the risk of random disruption can occur at any point in the network. Schmitt and Snyder
11
(2012) examined unreliable suppliers who caused uncertain yield and supply chain disruption, and
they developed cost models to determine the optimal order quantity. Qi et al. (2010) used the
concept of disruptions to develop an integrated SCN that can be used when suppliers and retailers
are unreliable. They formulated the problem as a nonlinear integer programming model to
minimize the total annual cost (including fixed cost, inventory cost, transportation cost, and lost
sales cost). Moreover, they integrated the model to decrease disruptions to retailers by determining
the number of retailers that should be open, location of retailers, and frequency and order size for
each retailer. Therefore, they assumed that suppliers and retailers have a deterministic yield. Jaggi
et al. (2012) developed a model to obtain the retailer’s optimal lot size for the inventory system.
Wang et al. (2010) considered a model to help a firm source from several suppliers in order to
improve supplier reliability. Widodo et al. (2011) proposed three scenarios—lost sales, online
facility return, and conventional store scenario—for managing sales return in a dual-sales channel.
Yu et al. (2009) studied selection methods between a single-source and dual-source
strategy to obtain greater benefits when a supply chain disruption occurs. A more reliable supply
chain is more expensive than an unreliable one because of the additional flexibility that a reliable
supply chain will offer. Tomlin (2006) assumed that the capacity constraint is the supplier, but a
reliable supplier may possess volume flexibility. Tomlin proved that in special situations in which
an unreliable supplier has infinite capacity and the reliable supplier has no flexibility, the dual
source strategy is more efficient to meet multi-objective operations. Yu et al. (2009) explained
that in a dual-source strategy, the two suppliers offer different prices and reliability because they
are in different regions. Also, the authors captured the probability of SC disruption risk and
formulated the expected profit functions with respect to supply chain disruption when the buying
firms used both sources. Gomez-Padilla and Mishina (2013) developed models for option and
12
capacity contracts for one retailer and one vendor SC system, and solved the problem by using
simulation to compare both contracts .
Cui et al. (2010) formulated the reliable facility location problem by using two models.
First, a mixed-integer programing (MIP) model was used to obtain the optimal facility location
and assigned customer, which was solved by using the Lagrangian relaxation algorithm. Second,
a continuum approximation model was developed to minimize setup and transportation costs for
two scenarios: when a facility is reliable and when it is unreliable. This model was used to
calculate the expected total cost of the system and find a close optimal solution. They designed
the supply chain network to be reliable and cost efficient, and formulated the discrete model as a
MIP model to minimize total operating and failure costs. Ramírez et al. (2012) developed an
approach for a two-echelon supply chain (retailer and supplier) in which the retailer faces
stochastic demand, and the supplier is willing to meet their demand. The model minimizes the
cost for inventory and penalty costs. Alenezi and Darwish (2014) integrated a location model with
risk pooling and a transportation problem. They proposed the problem as a large-scale nonlinear
MIP.
Based on the previous literature, there is a need for addressing the quality of packaging and
its impact on damage to products and parts during shipping. This dissertation focuses on different
models of loss that occurs during shipping. A methodology for determining the best routes and
packaging types to ensure minimum total cost has been developed. The methodology is detailed
using example case studies.
13
2.4
General Model
This section details the general formulation developed for modeling the problem. The
objective function of this model minimizes damage and shipping cost, considering multiple routes,
multiple products, and multiple packaging types, as shown in equation (2.1).
Indices:
u
Shipping cost (uU)
n
Type of product (nN)
G
Retailer (g G)
k
Route number (kK)
l
Location (either at the supplier location or right before delivering to retailer)
T
Transportation methods (tT)
r
Route (rR)
s
Supplier (sS)
z
Packaging type (zZ)
Parameters:
Wn
Cost of product “n”
Cnd
Damage cost for product “n”
BT
Cost per mile when using shipping method “T”
dnj
Demand for product “n” at retailer “g”
UnsgT Shipping cost for product “n” from supplier “s” to retailer “g” by using
transportation method “T”
Lnl
Labor cost for product “n” at location “l”
Mnl
Total part cost for product “n” if assembled at location “l”
14
Rksg
Route from supplier “s” to retailer “g” using route number “k”
qn
Quantity of product “n”
φsg
Distance from supplier “s” to retailer “g”
γnzsg
Percentage damage for product “n” when using packaging type “z” from supplier
“s” to retailer “g”
Variables:
Rnk = 1, if route “k” is selected for product “n”, 0 otherwise
Xnl = 1 if product “n” is assembled at location “l”, 0 otherwise
ᴪzn = 1, if Package type “z” is used for product “n”, 0 otherwise
2.4.1 Mathematical Formulation for General Model
The mathematical formulation for the objective function of the general model is
Min Z   (( M nl )  Lnl ) X il  (  nzsg  zi )Wn qn   qnsgT Rnk
r 1
i
subject to
X
nl
n
1
n
X nl  0,1 n
R nk  0,1 k
where
k
(2.1)
T
(2.2)
(2.3)
(2.4)
zn  0,1 z
(2.5)
 nzsg  0 n
(2.6)
qn  d ng n
(2.7)
U nsgT sg * BT * d ng
(2.8)
Cnd   nzsg * d ng * Wn
(2.9)
15
The objective function minimizes the total cost. The first term of the objective function
refers to the assembly cost, the second term refers to the damage cost, and the third term refers to
the shipping cost. Constraint (2.2) ensures that the products are assembled either prior to shipping
or at any designated location during shipping. Constraints (2.3), (2.4), and (2.5) ensure that the
terms can take on only binary values for assembly location, routes, and packaging type,
respectively. Constraint (2.6) represents the non-negativity for the damage percentage. Constraint
(2.7) ensures that the quantity shipped is more than the demand at each retailer. Equation (2.8) is
used to compute the shipping cost. Equation (2.9) is used to compute the damage cost.
2.5
Case Studies
Here, three smaller case studies are first used to demonstrate the effectiveness of the
proposed methodology and then to verify the MATLAB code. Three different scenarios are
compared and analyzed. The first scenario is to ship assembled products, the second scenario is
to ship unassembled product as parts and then assemble them at the destination (before shipping
to customers), and the third scenario is to ship the assembled product by using different routes.
All cases show methods of minimizing the cost of transportation and product damage by using
different types of packaging and routes to ship different products. After the MATLAB code is
verified for the smaller case studies, two larger case studies are developed to show the effectiveness
of this methodology.
2.5.1
Case Study 1
In this case study, a company has three different products—P1, P2, and P3—and uses two
types of packaging—Z1 and Z2. The cost of using packaging type Z1 for products P1, P2, and P3 is
$30, $45, and $60, respectively. The cost of using packaging type Z2 for products P1, P2, and P3
are $32, $47, and $62, respectively. In this study, the transportation cost depends on the type of
16
transportation (truck, train, ship, or airplane). Damage cost depends on the type of packaging.
Figure 2.1 shows the transportation network for Case Study 1, which consists of four routes: R1,
R2, R3, and R4. Table 2.1 describes the sequence of paths taken for each route. Nodes 1, 2, 3, and
4 represent ports, and nodes 5, 6, 7, and 8 represent airports. Nodes S and G represent the supplier
and retailer, respectively. Each route has a different method of transportation, and the associated
distances and shipping costs are shown in Table 2.2. Furthermore, the probability of damage for
Z1 and Z2 packaging types are different for each shipping method and route. These are shown in
Table 2.3.
Figure 2.1. Transportation network for Case Study 1
TABLE 2.1
TRANSPORTATION NETWORK ROUTES FOR CASE STUDY 1
Routes
Nodes
R1
S, 1, 2, G
R2
S, 3, 4, G
R3
S, 5, 6, G
R4
S, 7, 8, G
17
TABLE 2.2
PARAMETERS OF TRANSPORTATION TYPES AND ROUTES FOR CASE STUDY 1
Transportation Type
R1
R2
R3
R4
Distance (miles)
Cost/Mile ($)
Truck 1
1
0
0
0
200
0.02
Truck 2
1
0
0
0
400
0.03
Train 1
0
1
0
0
175
0.01
Train 2
0
1
0
0
800
0.02
Ship
1
1
0
0
11,000
0.01
Airplane
0
0
1
1
10,000
0.07
Truck 3
0
0
1
0
220
0.02
Truck 4
0
0
1
0
430
0.03
Train 3
0
0
0
1
150
0.01
Train 4
0
0
0
1
750
0.02
TABLE 2.3
DAMAGE PROBABILITY OF PACKAGING TYPES FOR CASE STUDY 1
Transportation
Type
Damage Probability for Type Z1
(%)
Damage Probability for Type Z2
(%)
P1
P2
P3
P1
P2
P3
Truck 1
3.40
1.80
1.50
3.90
2.00
1.00
Truck 2
2.50
1.40
1.60
3.95
1.50
0.95
Train 1
1.30
1.70
1.30
0.75
0.75
0.30
Train 2
1.50
1.20
1.00
0.70
0.70
0.20
Ship
2.40
1.50
1.50
0.85
1.70
0.30
Airplane
1.00
0.70
0.40
1.00
0.30
0.10
Truck 3
3.40
1.80
1.50
3.90
2.00
1.00
Truck 4
2.50
1.40
1.60
3.95
1.50
0.95
Train 3
1.30
1.70
1.30
0.75
0.75
0.30
Train 4
1.50
1.20
1.00
0.70
0.70
0.20
18
2.5.1.1 Results and Analysis for Case Study 1
The proposed model for Case Study 1 was formulated and solved using a total enumeration
strategy and MATLAB, and then used to validate the optimization code. Tables 2.4 and 2.5
illustrate the damage cost and minimum total cost of all products obtained using the total
enumeration strategy. Figures 2.2 and 2.3 show the damage cost for type Z1 and Z2 packaging,
respectively. Using the proposed model, the total minimum cost was obtained for all products
using all combinations of transportation methods and packaging types. Figure 2.4 shows the
optimal total cost for all products by applying the four different scenarios shown in Table 2.6.
TABLE 2.4
DAMAGE COST FOR CASE STUDY 1
Type Z1
Type Z2
Route
P1
P2
P3
P1
P2
P3
R1
$2,490
$2,115
$2,760
$2,784
$2,444
$1,395
R2
$1,560
$1,980
$2,280
$736
$1,481
$496
R3
$2,070
$1,755
$2,100
$2,832
$1,786
$1,271
R4
$1,140
$1,620
$1,620
$784
$822.50
$372
TABLE 2.5
TOTAL COSTS (DAMAGE + TRANSPORTATION) FOR CASE STUDY 1
Type Z1
Type Z2
Routes
P1
P2
P3
P1
P2
P3
R1
$128,490
$128,115
$128,760
$128,784
$128,444
$127,395
R2
$129,310
$129,730
$130,030
$128,486
$129,231
$128,246
R3
$719,370
$719,055
$719,400
$720,132
$719,086
$718,571
R4
$717,640
$718,120
$718,120
$717,284
$717,323
$716,872
19
$3,100
Damage cost
$2,600
$2,100
product 1
$1,600
product 2
$1,100
product 3
$600
$100
0
1
2
3
4
5
Routes
Figure 2.2. Damage cost when using type Z1 packaging for Case Study 1
$3,100
Damage cost
$2,600
$2,100
Product 1
$1,600
Product2
$1,100
Product3
$600
$100
0
1
2
3
4
5
Routes
Figure 2.3. Damage cost when using type Z2 packaging for Case Study 1
20
$384,700
$384,623.00
Total cost
$384,500
$384,325.00
$384,300
$384,100
$384,000.00
$383,996.00
$383,900
0
1
2
3
4
5
Scenarios
Figure 2.4. Optimal cost for Case Study 1
TABLE 2.6
SCENARIOS FOR CASE STUDY 1
Z1
Scenario
Description
Routes
for
P1-P2-P3
Total
Cost
($)
√
R2-R1-R1
383,996
Z2
Rank
P1 P2 P3 P1 P2 P3
1
2
3
4
All routes (R) and all
packaging (Z) types
allowed
Only one route (R) and
one type of packaging
(Z) allowed for all
products
Any route (R) selected
and only one type of
packaging (Z) allowed
Only one route (R)
and any type of
packing (Z) allowed
1
√
√
4
√
√
√
R1 R1-R1
384,623
3
√
√
√
R2-R1-R1
384,325
√
R1 R1-R1
384,000
2
√
21
√
First, all combinations of routes (R) and packaging (Z) types were allowed to be used in
determining the lowest cost. Thus, as a result of optimization, product type P1 used packaging
type Z2 and route R2, product type P2 used packaging type Z1 and route R1, and product type P3
used packaging type Z2 and route R1, which resulted in an optimal cost of $383,996. If all products
used the same packaging type and route, then the lowest cost of $384,623 was obtained with route
R1 and packaging type Z2. If the constraints are relaxed to allow only one type of packaging while
using any of the routes, then the optimization model resulted in selecting packaging type Z2 for all
products, and routes R2, R1, and R1 for products P1, P2, and P3, respectively, with an optimal cost
of $384,325. If the constraints are modified to allow only one route while relaxing the constraint
for the type of packaging, the optimization model resulted in selecting route R1 for all products,
while using packaging type Z1 for P1 and P2, and packaging type Z2 for P3, with an optimal cost of
$384,000. By applying the proposed model using MATLAB, the total minimum cost was obtained
for all products, as shown previously in Table 2.6. The minimum total cost for P1 is $128,486
using route R2 and selecting packaging type Z2. The minimum total cost for P2 is $128,115 using
route R1 and selecting packaging type Z1. The minimum total cost for P3 is $127,395 using route
R1 and selecting packaging type Z2. The optimal solution obtained using the total enumeration
strategy and MATLAB provided the same objective function cost, as shown in Table 2.7.
TABLE 2.7
OPTIMAL COST FOR CASE STUDY 1 USING MATLAB
Type Z1
Type Z2
Route
P2
P1
P3
R1
$128,115
–
$127,395
R2
–
$128,486
–
22
2.5.2
Case Study 2
In the first case study, all products are assumed to be assembled at the supplier site, and
hence shipping was limited to the assembled products. Case Study 2 is similar to Case Study 1 in
that it also considers three different products—P1, P2, and P3—but it uses only one type of
packaging—Z1. The transportation cost depends on type of transportation (truck, train, ship, or
airplane). As shown previously in Figure 2.1, the transportation network consists of four routes—
R1, R2, R3, and R4—and eight nodes. Nodes 1, 2, 3, and 4 represent ports, while nodes 5, 6, 7, and
8 represent airports. Nodes S and J represent supplier and retailer, respectively. The transportation
cost depends on type of transportation. Each route has a different method of transportation, as
well as associated distances and shipping costs, also as shown previously in Table 2.2.
In Case Study 2, the products are shipped in either the assembled or unassembled condition.
Table 2.8 illustrates the labor cost and assembly cost for all products assembled prior to shipping
and also at the final destination, and Tables 2.9 and 2.10 illustrate the damage probability prior to
shipping and at the final destination, respectively. Similar to Case Study 1, both total enumeration
and optimization using MATLAB were conducted in Case Study 2 to validate the MATLAB code.
TABLE 2.8
LABOR AND ASSEMBLY COSTS FOR CASE STUDY 2
Assembled Prior to Shipping
Assembled at Final Destination
Product Cost
Labor Cost ($)
Assembly Cost ($)
P1
P2
P3
P1
P2
P3
8
12
15
18
25
28
22
33
45
30
50
80
23
TABLE 2.9
DAMAGE PROBABILITY PRIOR TO SHIPPING FOR CASE STUDY 2
Transportation
Type
Damage Probability Prior to Shipping (%)
P1
P2
P3
Truck 1
3.4
1.8
1.5
Truck 2
2.5
1.4
1.6
Train 1
1.3
1.7
1.3
Train 2
1.5
1.2
1.0
Ship
2.4
1.5
1.5
Airplane
1.0
0.7
0.4
Truck 3
3.4
1.8
1.5
Truck 4
2.5
1.4
1.6
Train 3
1.3
1.7
1.3
Train 4
1.5
1.2
1.0
TABLE 2.10
DAMAGE PROBABILITY AT FINAL DESTINATION FOR CASE STUDY 2
Transportation
Type
Damage Probability at Final Destination (%)
P1
P2
P3
Truck 1
0.17
0.090
0.075
Truck 2
0.125
0.610
0.350
Train 1
0.065
0.085
0.065
Train 2
0.075
0.260
0.050
Ship
0.12
0.075
0.075
Airplane
0.05
0.035
0.020
Truck 3
0.17
0.090
0.075
Truck 4
0.125
0.430
0.190
Train 3
0.065
0.085
0.065
Train 4
0.075
0.001
0.095
24
2.5.2.1 Results and Analysis for Case Study 2
The proposed model for Case Study 2 was formulated and solved using MATLAB. Tables
2.11 and 2.12 illustrate the minimum total cost for all products assembled prior to shipping and
also assembled at the final destination, respectively. Figures 2.5 and 2.6 show the damage cost for
all products assembled prior to shipping and also assembled at the final destination, respectively.
By applying the proposed model, the total minimum cost was obtained for all products for all
combinations of transportation methods.
TABLE 2.11
TOTAL COST OF PRODUCTS ASSEMBLED PRIOR TO SHIPPING FOR CASE STUDY 2
Route
Cost ($)
P1
P2
P3
R1
128,490
128,115
128,760
R2
129,310
129,730
130,030
R3
719,370
719,055
719,400
R4
717,640
718,120
718,120
TABLE 2.12
TOTAL COST OF PRODUCTS ASSEMBLED AT FINAL DESTINATION FOR CASE STUDY 2
Route
Cost ($)
P1
P2
P3
R1
129,725
136,988
140,000
R2
129,853
136,985
139,927
R3
719,834
727,008
729,958
R4
719,707
726,711
729,794
25
$3,000
Damage Cost
$2,500
$2,000
Product 1
$1,500
Product 2
$1,000
Product 3
$500
$0
0
1
2
3
4
5
Routes
Figure 2.5. Damage cost for products assembled prior to shipping for Case Study 2
450
400
Damage Cost
350
300
250
product 1
200
Product 2
150
Product 3
100
50
0
0
1
2
3
4
5
Routes
Figure 2.6. Damage cost for products assembled at final destination for Case Study 2
Figure 2.7 shows the optimal total cost for Case Study 2 for all products assembled prior
to shipping and at the final destination by applying the two different scenarios shown previously
in Tables 2.11 and 2.12.
26
Thousands
Total cost
$680
$620
Assembled product prior
shipping
$560
Assembled at final destination
$500
$440
$380
$320
$406.64
$406.71
$385.37
$385.37
1
2
$260
$200
0
3
Scenarios
Figure 2.7. Optimal cost for Case Study 2
First, all combinations of routes (R) were allowed to be used in determining the lowest
cost. Thus, as a result of optimization, product types P1, P2, and P3 used route R1 and were
assembled prior to shipping, which resulted in an optimal cost of $385,365. When the products
were assembled at the final destination, product type P1 used route R1, while product types P2 and
P3 used route R2, which resulted in an optimal cost of $ 406,637. If the constraints are modified,
to allow only one route to be used for products assembled prior to shipping, the lowest cost of
$385,365 was obtained when using R1 for all products. If the constraints are modified to allow
only one route for all the products assembled at the final destination, the optimization resulted in
selecting route R2 for all products, and then the optimal cost is $406,712. Table 2.13 illustrates the
optimal total cost for all products assembled prior to shipping and at the final destination by using
MATLAB. The optimal solution obtained using the total enumeration strategy and MATLAB
provided the same objective function cost, as shown in Table 2.14.
27
TABLE 2.13
SCENARIOS FOR CASE STUDY 2
Scenario
Assembled
Prior to
Shipping
Description
Assembled
at Final
Destination
P1
P2
P3
√
√
1
Any route used
for all products
√
2
One route used
for all products
√
P1
√
√
P2
√
P3
√
√
√
√
√
Routes
for
P1-P2-P3
Total
Cost ($)
R1-R1-R1
385,365
R1-R2-R2
406,637
R1-R1-R1
385,365
R2-R2-R2
406,712
TABLE 2.14
MATLAB RESULTS FOR CASE STUDY 2
Assembled Prior to Shipping ($)
Assembled at Final Destination ($)
Route
P1
P2
P3
P1
P2
P3
R1
128,490
128,115
128,760
129,725
–
–
R2
–
–
–
–
136,985
139,927
2.5.3 Case Study 3
In Case Study 3, a company has three different products - P1, P2, and P3 and uses two types
of packaging - Z1 and Z2. The cost of using packaging type Z1 for products P1, P2, and P3 is $30,
$45, and $60, respectively. The cost of using packaging type Z2 for products P1, P2, and P3 is $32,
$47, and $62, respectively. In this study, the transportation cost depends on type of transportation
(truck, train, ship, or airplane). Damage cost depends on the type of packaging. In addition, the
products use different paths to minimize the total cost. Figure 2.8 shows the transportation network
for Case Study 3, which consists of 16 paths and 10 nodes. Nodes SP1, SP2, SP3, and SP4 represent
sea ports, and nodes A1, A2, A3, and A4 represent airports. Nodes S and G represent the supplier
28
and retailer, respectively. Each path has a different method of transportation, and the associated
distances and shipping costs are shown previously in Table 2.2. Furthermore, the probabilities of
damage for Z1 and Z2 packaging types are different for each path and are shown in Table 2.15.
Figure 2.8. Transportation network for Case Study 3
TABLE 2.15
DAMAGE PROBABILITY WHEN USING DIFFERENT PACKAGING IN CASE STUDY 3
Path
Number
Damage Probability for Type Z1 (%)
Damage Probability for Type Z2 (%)
P1
P2
P3
P1
P2
P3
1
3.40
0.45
1.50
3.90
2.00
0.75
2
1.30
5.50
1.30
0.30
0.75
4.50
3
3.40
0.45
1.50
3.90
2.00
0.75
4
1.30
5.50
1.30
0.30
0.75
4.50
5
2.40
0.60
1.50
0.20
1.70
0.30
6
2.40
0.60
1.50
0.20
1.70
0.30
7
2.40
0.60
1.50
0.20
1.70
0.30
8
2.40
0.60
1.50
0.20
1.70
0.30
9
1.00
0.70
0.40
1.00
0.30
0.10
29
TABLE 2.15 (continued)
Path
Number
Damage Probability for Type Z1 (%)
Damage Probability for Type Z2 (%)
P1
P2
P3
P1
P2
P3
10
1.00
0.70
0.40
1.00
0.30
0.10
11
1.00
0.70
0.40
1.00
0.30
0.10
12
1.00
0.70
0.40
1.00
0.30
0.10
13
2.50
0.20
1.60
13.00
1.50
0.95
14
1.50
1.30
1.00
0.20
0.70
0.2
15
2.50
0.20
1.60
13.00
1.50
0.95
16
1.50
1.30
1.00
0.20
0.70
0.20
2.5.3.1 Results and Analysis for Case Study 3
The proposed model for Case Study 3 was formulated and solved using the total
enumeration strategy and MATLAB. Tables 2.16 and 2.17 illustrate the damage cost and
minimum total cost for the two types of packaging Z1 and Z2, respectively, for all products. Using
the proposed model, the total minimum cost was obtained for all products for all combinations of
paths and packaging types. Table 2.18 shows the optimal total cost for all products by applying
the four different scenarios.
30
TABLE 2.16
TOTAL COST USING TYPE Z1 PACKAGING FOR CASE STUDY 3
Product P1
Product P2
Shipping
Total Damage
Total
Cost ($) Damage
Cost ($) Cost ($) Cost ($) Cost ($)
Path
Set
Path
PS1
1-5-13
126,000
2,490
128,490
563
PS2
2-8-14
127,750
1,560
129,310
PS3
2-7-13
123,750
1,860
PS4
1-6-14
130,000
PS5
3-9-15
PS6
Product P3
Damage
Cost ($)
Total
Cost ($)
126,563
2,760
128,760
3,330
131,080
2,280
130,030
125,610
2,835
126,585
2,640
126,390
2,190
132,190
1,057
131,058
2,400
132,400
717,300
2,070
719,370
608
717,908
2,100
719,400
4-12-16
716,500
1,140
717,640
3,375
719,875
1,620
718,120
PS7
4-11-15
714,400
1,440
715,840
2,880
717,280
1,980
716,380
PS8
3-10-16
719,400
1,770
721,170
1,103
720,503
1,740
721,140
TABLE 2.17
TOTAL COST USING TYPE Z2 PACKAGING FOR CASE STUDY 3
Product P1
Product P2
Shipping
Total Damage
Total
Cost ($) Damage
Cost ($) Cost ($) Cost ($) Cost ($)
Path
Set
Path
PS1
1-5-13
126,000
5,472
131,472
2,444
PS2
2-8-14
127,750
224
127,974
PS3
2-7-13
123,750
4,320
PS4
1-6-14
130,000
PS5
3-9-15
PS6
Product P3
Damage
Cost ($)
Total
Cost ($)
128,444
1,240
127,240
1,480
129,230
3,100
130,850
128,070
1,856
125,606
3,565
127,315
1,376
131,376
2,068
132,068
775
130,775
717,300
5,728
723,028
1,786
719,086
1,116
718,416
4-12-16
716,500
480
716,980
822
717,322
2,976
719,476
PS7
4-11-15
714,400
4,576
718,976
1,198
715,598
3,441
717,841
PS8
3-10-16
719,400
1,632
721,032
1,410
720,810
651
720,051
31
TABLE 2.18
SCENARIOS FOR CASE STUDY 3
Z1
Scenario
1
2
Description
Any path set and any type
of packaging
One path set and one type
of packaging
Rank
Z2
P1 P2 P3 P1 P2 P3
PS3
377,607
√
4
√
√
√
PS3
378,585
√
√
PS3PS1PS3
378,562
PS3
377,606
Any path set and only one
type of packaging
3
√
4
One path set and any type
of packaging
1
√
√
√
Total
Cost ($)
2
3
√
Path
Set
√
First, all combinations of path sets (PS) and packaging types (Z) were allowed to be used
in determining the lowest cost. Thus, as a result of optimization, product P1 used packaging type
Z1 and path set PS3, product P2 used packaging type Z2 and path set PS3, and product P3 used
packaging type Z1 and path set PS3, which resulted in an optimal cost of $377,606.5. If all products
used the same packaging type and path set, then the lowest cost of $378,585 was obtained when
using path set PS3 and packaging type Z1. If the constraints are relaxed to allow only one type of
packaging while using any of the path sets, then optimization is obtained by selecting packaging
type Z1 for all products, and path sets PS3, PS1, and PS3 for products P1, P2, and P3, respectively,
with an optimal cost of $378,562. If the constraints are modified to allow only one path set while
relaxing the type of packaging constraint, then the optimization model results in selecting path LS3
for all products, while using packaging type Z1 for P1 and P3, and using packaging type Z2 for P2,
with an optimal cost of $377,606. By applying the proposed model using MATLAB, the lowest
total cost for P1 is $125,610 with a yield of 95.40%, packaging type Z1, and selected path (1-3-610). The minimum total cost for P2 was $125,606.5 with a yield of 97.30%, packaging type Z2,
32
and selected path (1-3-6-10). The lowest total cost for P3 was $126,390 with a yield of 96.8%,
packaging type Z1, and selected path (1-3-6-10). Table 2.19 illustrates the optimal total cost for
all products and all paths obtained from MATLAB, and Table 2.20 shows the optimal total cost
when all products use the same path (1-3-6-10). Figure 2.9 shows the optimal path for each product
using MATLAB.
The optimal solution obtained using the total enumeration strategy and
MATLAB gave the same results.
TABLE 2.19
MATLAB RESULTS FOR CASE STUDY 3 WHEN ALL PRODUCTS USE
DIFFERENT PATHS
Type Z1
Type Z2
Product
Path
Optimal Cost ($) Yield (%)
Path
Optimal Cost ($) Yield (%)
P1
1-3-6-10
125,610
95.4
1-3-7-10
127,974
99.3
P2
1-2-6-10
126,563
98.8
1-3-6-10
125,607
97.3
P3
1-3-6-10
126,390
96.8
1-2-6-10
127,240
98.0
TABLE 2.20
MATLAB RESULTS FOR CASE STUDY 3 WHEN ALL PRODUCTS USE SAME
BEST PATH (1-3-6-10)
Type Z1
Type Z2
Product
Optimal Cost ($)
Yield (%)
Optimal Cost ($)
P1
125,610
95.4
128,070
98.5
P2
126,585
93.0
125,607
97.3
P3
126,390
96.8
124,804
98.0
33
Yield (%)
Figure 2.9. Optimal path for each product for Case Study 3
2.5.3.2 Sensitivity Analysis for Case Study 3
The fundamentals of sensitivity analysis are to find the stability of the best solutions with
potential adjustment in parameters (Poh and Ang, 1999). Moreover, efficient solutions can be
measured via sensitivity analysis, which could assist in reducing uncertainty in parameters and
approaches (Triantaphyllou et al., 1998). Therefore, sensitivity analysis is used to perform the
34
ranking of changes in parameters with respect to weights of the primary goals that were proposed
for decision making. Also, it provides guidance for improving the solution, understanding the
model, and reducing the output uncertainties.
From results, the sensitivity analysis provides guidance for changing the total cost for each
route when the damage percentage changes for any path. Table 2.21 illustrates the sensitivity
results for each product with two different types of packaging Z1 and Z2. For example, the original
solution for P1, Z2 was LS2 with a total cost of $127,974. However, if the damage percentage of
path number 8 is changed by 0.3%, then LS3 is found to be optimal, with a total cost of $128,070.
In the same context, changing the damage percentage of path number 1 by 0.06% and 0.13% for
P2-Z1 and P3-Z2, respectively, the respective optimal solutions are found as LS3 with a total cost
of $126,585 and LS3 with a total cost $127,315.
TABLE 2.21
RESULTS OF SENSITIVITY ANALYSIS FOR CASE STUDY 3
Original
Solution
Original
Value ($)
New
Solution
New Value
($)
Difference
(%)
P1, Z1
PS3
125,610
PS1
128,490
9.82
P2, Z1
PS1
126,563
PS3
126,585
0.06
P3, Z1
PS3
126,390
PS1
128,760
3.97
P1, Z2
PS2
127,974
PS3
128,070
0.30
P2, Z2
PS3
125,607
PS1
128,444
6.04
P3, Z2
PS1
127,240
PS3
127,315
0.13
2.5.4 Case Study 4
In this case study, the company has 15 different products that are shipped to the final
destination using five different paths. Table 2.22 lists the cost for each product. Figure 2.10 shows
the transportation network for Case Study 4, which consists of 34 nodes and 46 paths. The
35
associated distances between each node in the network are shown in Table 2.23. The demand for
each product in this case study is assumed to be 1,000 units, and the shipping cost is $0.02 per
mile. The damage percentages for all products on each path of Case Study 4 can be found in the
appendix in Table 2.30.
TABLE 2.22
PRODUCT COSTS FOR CASE STUDY 4
Product
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
Cost ($)
30
33
40
42
32
50
52
45
47
36
60
63
57
70
66
Figure 2.10. Transportation network for Case Study 4
36
TABLE 2.23
DISTANCE BETWEEN NODES FOR CASE STUDY 4
Path
Distance (Miles)
Path
Distance (Miles)
Path
Distance (Miles)
N1,2
50
N8,18
160
N20,33
100
N1,3
70
N9,19
300
N21,34
250
N2,4
90
N9,20
350
N22,34
200
N2,5
100
N10,21
240
N23,34
200
N2,6
75
N10,22
310
N24,34
270
N3,7
95
N11,23
190
N25,34
190
N3,8
100
N12,24
105
N26,34
290
N3,9
87
N13,25
103
N27,34
350
N4,10
140
N14,26
250
N28,34
400
N4,11
150
N14,27
300
N29,34
290
N5,12
140
N15,28
150
N30,34
195
N6,13
135
N16,29
155
N31,34
420
N6,14
170
N17,30
200
N32,34
320
N6,15
180
N18,31
350
N33,34
235
N7,16
210
N19,32
310
N8,17
250
N19,33
150
2.5.4.1 Results and Analysis for Case Study 4
The proposed model for Case Study 4 was formulated and solved using MATLAB, and
using the model, the total minimum cost was obtained for all products. As shown in Table 2.24,
the minimum total cost for P1 is $15,260 with a yield of 86%, and the path is (1-2-6-13-25-34), as
shown in red in Figure 2.11. Table 2.24 also shows the minimum total cost for P2 as $21,080 with
a yield of 85% and the path as (1-2-4-10-22-34). Table 2.24 illustrates the optimal total cost and
yield percentage for all products using different paths; however, if the constraint is modified to
37
allow only one path, the best path (1-2-6-13-25-34), then the optimal cost for P2 is $18,980 and
the yield as 78%, as shown in Table 2.25.
TABLE 2.24
OPTIMAL COST AND YIELD PERCENTAGE FOR CASE STUDY 4 WHEN ALL
PRODUCTS USE DIFFERENT PATHS
Product
Path
Optimal Cost ($)
Yield (%)
P1
1-2-6-13-25-34
15,260
86
P2
1-2-4-10-22-34
21,080
85
P3
1-2-5-12-24-34
30,500
63
P4
1-2-6-13-25-34
29,540
63
P5
1-2-4-10-22-34
28,600
65
P6
1-2-6-14-26-34
39,700
61
P7
1-2-6-13-25-34
43,300
51
P8
1-2-4-10-22-34
29,300
73
P9
1-2-6-14-26-34
35,970
64
P10
1-2-6-13-25-34
30,140
57
P11
1-2-6-14-27-34
49,500
58
P12
1-2-6-14-27-34
43,470
66
P13
1-2-5-12-24-34
40,660
60
P14
1-3-9-19-32-34
46,940
68
P15
1-3-8-17-30-34
54,580
53
38
21
500
10
4
5
2
400
22
11
23
24
25
26
12
13
6
14
300
1
7
200
3
29
8
17
100
0
50
100
28
16
9
0
34
27
15
150
30
18
31
19
32
20
33
200
250
300
350
Figure 2.11. MATLAB output for product P1 for Case Study 4
400
TABLE 2.25
OPTIMAL COST AND YIELD PERCENTAGE FOR CASE STUDY 4 WHEN ALL
PRODUCTS USE SAME BEST PATH (1-2-6-13-25-34)
Product
Optimal Cost ($)
Yield (%)
Product
Optimal Cost ($)
P1
15,260
86.8
P9
36,440
55.4
P2
18,980
78
P10
30,140
57
P3
23,060
73.3
P11
39,260
61
P4
29,540
63
P12
40,670
61
P5
25,140
62.8
P13
44,120
54
P6
38,060
55.9
P14
46,060
58.3
P7
43,300
51.6
P15
50,000
53.2
P8
30,860
62.7
39
Yield (%)
2.5.5 Case Study 5
In this case study there are 30 different products, and the same transportation network as
shown in Figure 2.10 for Case Study 4 is used to further test the consistency of the proposed model.
Table 2.26 shows the product costs for Case Study 5. The associated distances between each node
in the network are also the same as shown in Table 2.23 for Case Study 4. The demand for each
product in this case is assumed to be 1,000 units, and the shipping cost is $0.02 per mile. The
damage percentages for all products at each arc are shown in the appendix as Table 2.31.
TABLE 2.26
PRODUCT COSTS FOR CASE STUDY 5
Product
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
Cost ($)
30
33
40
42
32
50
52
45
47
36
60
63
57
70
66
Product
P16
P17
P18
P19
P20
P21
P22
P23
P24
P25
P26
P27
P28
P29
P30
Cost ($)
35
36
46
49
34
51
57
40
48
35
62
64
59
72
68
2.5.5.1 Results and Analysis for Case Study 5
The proposed model was formulated and solved using MATLAB, and then the total
minimum cost and yield percentage was obtained for all products, as shown in Table 2.27. As can
be seen, the minimum total cost for P1 is $34,100 with a yield of 50.6%, and the path is (1-2-4-1022-34), as shown in red in Figure 2.12. Also shown in Table 27, the minimum total cost for P2 is
$27,680 with a yield of 68.4%, and the path is also (1-2-4-10-22-34). If the constraint is modified
to allow only one path (1-2-4-10-22-34), then the optimal cost for P3 is $47,400 and the yield is
38.3%, as can be seen in Table 2.29.
40
TABLE 2.27
MATLAB RESULTS FOR CASE STUDY 5 WHEN ALL PRODUCTS
USE DIFFERENT PATHS
Product
Path
Optimal Cost ($)
P1
1-2-4-10-22-34
34,100
50.6
P2
1-2-4-10-22-34
27,680
68.4
P3
1-3-9-19-32-34
39,340
61.2
P4
1-3-8-17-30-34
35,620
61.5
P5
1-2-4-10-22-34
42,360
37.8
P6
1-3-8-17-30-34
47,800
48.6
P7
1-2-4-11-23-34
48,960
44.8
P8
1-2-4-11-23-34
36,100
58.1
P9
1-3-9-20-33-34
37,990
60.3
P10
1-2-4-10-21-34
28,360
68.2
P11
1-2-6-14-26-34
56,900
47.7
P12
1-2-4-10-21-34
56,980
46.2
P13
1-3-8-17-30-34
48,220
54.9
P14
1-2-6-14-27-34
55,300
56.9
P15
1-3-7-16-29-34
57,980
49.6
P16
1-2-6-15-28-34
36,350
55.2
P17
1-2-5-12-24-34
46,060
34.7
P18
1-2-5-12-24-34
42,280
48
P19
1-3-9-20-33-34
42,810
57
P20
1-3-8-17-30-34
37,380
49.2
P21
1-2-4-11-23-34
36,040
63
P22
1-2-6-13-25-34
57,230
39.6
P23
1-3-8-18-31-34
44,800
52
P24
1-2-4-10-21-34
42,760
54.4
P25
1-2-6-13-25-34
29,960
55.5
P26
1-2-4-11-23-34
29,100
77
P27
1-2-4-11-23-34
57,120
47.7
P28
1-2-5-12-24-34
55,780
45.1
P29
1-2-4-10-21-34
59,320
51.6
P30
1-3-7-16-29-34
66,040
43.5
41
Yield (%)
21
500
10
4
2
400
22
11
5
23
24
25
26
12
13
6
14
300
1
7
200
3
29
8
17
100
0
50
100
28
16
9
0
34
27
15
150
30
18
31
19
32
20
33
200
250
300
350
400
Figure 2.12. MATLAB output for product P1 for Case Study 5
TABLE 2.28
MATLAB RESULTS FOR CASE STUDY 5 WHEN ALL PRODUCTS
USE SAME BEST PATH (1-2-4-10-22-34)
Product
Optimal Cost ($)
Yield (%)
Product
Optimal Cost ($)
Yield (%)
P1
34,100
50.6
P16
30,500
63
P2
27,680
68.4
P17
33,440
60
P3
47,400
38.3
P18
64,100
29
P4
34,700
62
P19
45,200
50.3
P5
42,360
37.8
P20
25,660
74
P6
38,300
62
P21
40,280
60.2
P7
49,080
48
P22
52,280
50
P8
35,150
63
P23
47,000
42
P9
38,360
60
P24
35,480
65
P10
30,200
65
P25
43,100
42
P11
48,200
56
P26
38,740
68
P12
49,190
56
P27
52,280
54
P13
50,000
51
P28
53,560
50
P14
59,900
50
P29
53,240
57
P15
57,380
49.4
P30
72,240
38
42
2.6
Conclusions and Future Work
Products can become damaged during shipping due to improper package selection or
transportation hazards. Appropriate product packaging helps to ensure that customers receive
products without any damage. During each stage of the supply chain network, the amount of
damage can be different. Multiple transportation methods and packaging types exist for shipping
products to the final destination. In this dissertation, a multi-objective model is proposed for
minimizing the total cost, which includes the cost of damage, shipping, and packaging, by
considering different transportation methods and their respective probability of damage and
different packaging types. Three case studies were presented with different conditions, including
whether the products were shipped in assembled condition or as unassembled products. These
case studies were used to validate the MATLAB code. Two larger case studies were also tested
to validate the procedure. For the five case studies shown in this chapter, analyses showed that
shipping unassembled products was more cost effective than shipping assembled products, since
unassembled products have a low probability of damage. However, this could change with a
changes in parameters of the study. In future work, the model could be expanded to consider
damage-recovery approaches when using multiple stages of transportation in various scenarios.
2.7
References
Alenezi, A., R., and Darwish, M., A. (2014). Integrated location model with risk pooling and
transportation mode selection. International Journal of Industrial and Systems Engineering, 16(2),
200–222.
Arshinder. (2012). Flexibility in supply chain using coordination: Issues and learnings.
International Journal of Industrial and Systems Engineering, 11(3), 308–330.
Aryanezhad, M.-B., Naini, S. G. J., and Jabbarzadeh, A. (2012). An integrated model for designing
supply chain network under demand and supply uncertainty. African Journal of Business
Management, 6(7), 2678–2696.
Azad, N., and Davoudpour, H. (2010). Designing a reliable supply chain network model under
disruption risks. Journal of American Science, 6(12), 1091–1097.
43
Azzi, A., Battini, D., Persona, A., and Sgarbossa, F. (2012). Packaging design: General framework
and research agenda. Packaging Technology and Science, 25(8), 435–456.
Cui, T., Ouyang, Y., and Shen, Z.,J., M. (2010). Reliable facility location design under the risk of
disruptions. Operations Research, 58(4, part 1), 998–1011.
Darwish, M., Odah, O., and Goyal, S. (2014). Vendor managed inventory models for single-vendor
multi-retailer supply chains with quality consideration. International Journal of Industrial and
Systems Engineering, 18(4), 499–528.
Gomez-Padilla, A., and Mishina, T. (2013). Option and capacity reservation contracts in a supply
chain. International Journal of Industrial and Systems Engineering, 13(3), 266–279.
Hatefi, S., M., and Razmi, J. (2013). An integrated methodology for supplier selection and order
allocation in the presence of imprecise data. International Journal of Industrial and Systems
Engineering, 15(1), 51–68.
Jabbarzadeh, A., Jalali Naini, S., G., Davoudpour, H., and Azad, N. (2012). Designing a supply
chain network under the risk of disruptions. Mathematical Problems in Engineering, 2012, 1–23.
Jaggi, C., K., Aggarwal, K., and Verma, M. (2012). Optimal retailer’s ordering policies under twostage partial trade credit financing in a supply chain. International Journal of Industrial and
Systems Engineering, 10(3), 277–299.
Kristianto, Y., and Helo, P. (2010). Designing supply chain by coordinating manufacturing process
and product development process. International Journal of Industrial and Systems Engineering,
6(3), 360–380.
Poh, K., and Ang, B. (1999). Transportation fuels and policy for Singapore: An AHP planning
approach. Computers and Industrial Engineering, 37(3), 507–525.
Qi, L., Shen, Z. J. M., and Snyder, L. V. (2010). The effect of supply disruptions on supply chain
design decisions. Transportation Science, 44(2), 274–289.
Ramírez, D. G., Daza-Escorcia, J. M., Martinez, J. V., Paternina-Arboleda, C. D., and Garcia, A.
(2012). The dynamic demand game: A Markov state fictitious play approach to a two-echelon
supply chain problem under demand uncertainty. International Journal of Industrial and Systems
Engineering, 10(3), 319–335.
Schmitt, A. J., and Snyder, L. V. (2012). Infinite-horizon models for inventory control under yield
uncertainty and disruptions. Computers and Operations Research, 39(4), 850–862.
Singh, S., and Xu, M. (1993). Bruising in apples as a function of truck vibration and packaging.
Applied engineering in agriculture, 9(5), 455–460.
Tomlin, B. (2006). On the value of mitigation and contingency strategies for managing supply
chain disruption risks. Management Science, 52(5), 639–657.
44
Triantaphyllou, E., Shu, B., Sanchez, S., N., and Ray, T. (1998). Multi-criteria decision making:
An operations research approach. Encyclopedia of electrical and electronics engineering, 15, 175–
186.
Vursavuş, K. K., and Özgüven, F., (2004). Determining the effects of vibration parameters and
packaging method on mechanical damage in golden delicious apples. Turkish Journal of
Agriculture and Forestry, 28(5), 311–320.
Wang, Y., Gilland, W., and Tomlin, B. (2010). Mitigating supply risk: Dual sourcing or process
improvement? Manufacturing and Service Operations Management, 12(3), 489–510.
Widodo, E., Takahashi, K., Morikawa, K., Pujawan, I., N., and Santosa, B. (2011). Managing sales
return in dual sales channel: Its product substitution and return channel analysis. International
Journal of Industrial and Systems Engineering, 9(2), 121–149.
Yu, H., Zeng, A. Z., and Zhao, L. (2009). Single or dual sourcing: Decision-making in the presence
of supply chain disruption risks. Omega, 37(4), 788–800.
45
CHAPTER 3
ROBUST SUPPLY CHAIN SYSTEM UNDER YIELD UNCERTAINTY
3.1
Abstract
Products are often damaged during shipping. These damages are stochastic in nature. To
minimize the impact of damage, the selection of routes should consider not only the expected
damage but also the variability of damage. In this research, the first model is of the supply chain
network in order to minimize total cost, which consists of product cost and transportation cost
while considering multiple routes and multiple products under stochastic yield conditions. In the
second model, the concept of robust design has been applied to minimize damage while
maximizing yield. The research uses two case studies to demonstrate the procedures and the
models.
Keywords: Yield uncertainty, transportation disruption, supply chain risk
3.2
Introduction
Supply chains have raised the need for preventing and minimize damage during
transportation. Many factors are involved when a product is damaged in transit, such as shocks,
vibrations, accidents, poor handling, etc. This may lead to a stochastic yield at the retailer due to
damage during shipping. Therefore, the type of packaging and modes of transportation affect the
amount and type of damage. Appropriate product packaging helps to ensure that customers receive
the product without any damage. The type of packaging can also cause changes in the quantity of
damage.
Vursavuş and Özgüven (2004) identified several different types of packaging that are used
to transport goods, such as the polystyrene soft cell trays, paper pulp trays, wood bins, bulk bins,
and corrugated fiberboard. Singh and Xu (1993) stated that up to 80% of apples may be damaged
46
during shipping by truck. The amount of damage depends on the type of packaging, type of truck,
and position of cartons in which the apples are shipped. Damage from accidents and handling
issues are not completely within the control of packaging; however, proper packaging can prevent
most damage from shocks and vibrations. Shocks can occur during handling or transportation.
For example, during the transportation of products in trucks, shocks might occur when the
condition of the road is poor.
This chapter focuses on the development of models for situations in which the customer
demand is known; however, the number of products supplied is stochastic, and the stochastic
quantity delivered depends on the routes and suppliers that are used to ship the product to the
customer. Different studies have been related to the randomness or uncertainty of transporting
products throughout the supply chain, but none of them consider the uncertainty associated with
random damages that occur during shipping.
The remainder of this section is organized as follows: Subsection 3.2.1 reviews the related
literature about the SC under yield uncertainty. Subsection 3.2.2 reviews some studies about the
robust SC network. Finally, subsection 3.2.3 provides a review of related literature on supplier
selection.
3.2.1
Supply Chain under Yield Uncertainty
Supply chain disruptions have received much attention in the last decade. Most disruptions
that have been studied are those that have a low probability of occurrence, such as tsunamis and
earthquakes, which typically lead to disastrous results. According to Shear et al. (2002), the value
of products returned every year is around $100 billion. Guide Jr. et al. (2006) and Shear et al.
(2002) have pointed out different reasons for returning products (customer satisfaction, product
evaluation, shipping damage, defective merchandise, end of lease, and end of life). Azad and
47
Davoudpour (2010) considered a facility with random disruption risk in order to design a reliable
supply chain network. They studied disruption in distribution centers by location and capacity,
formulated the problem as a nonlinear integer programming model, and then linearized it to obtain
the optimal solution. Moreover, numerous algorithms have solved random disruption risks by
using two different algorithms for large-size cases: tabu search and simulated annealing
algorithms. Azad and Davoudpour found a better solution by using the tabu search algorithm.
Additionally, transportation costs from reliable and unreliable distribution centers were considered
in the model.
Schmitt and Snyder (2012) examined unreliable suppliers who caused uncertain yield and
supply chain disruption, and developed cost models to determine the optimal order quantity. Qi
et al. (2010) used the concept of disruptions to develop an integrated SCN, which can be used
when suppliers and retailers are unreliable. They formulated the problem as a nonlinear integer
programming model to minimize the total annual cost (fixed cost, inventory cost, transportation
cost, and lost sales cost). Moreover, they integrated the model to decrease disruptions to retailers
by determining the number of retailers that should be opened, their locations, and frequency and
order size for each retailer. They assumed that suppliers and retailers have a deterministic yield.
Yu et al. (2009) studied single-source and dual-source strategy selection methods to obtain greater
benefits when an SC disruption occurs. A more reliable SC is more expensive than an unreliable
one because it has more flexibility. Tomlin (2006) assumed that the capacity constraint is the
supplier, but a reliable supplier may possess volume flexibility. He proved that in special situations
in which an unreliable supplier has infinite capacity and the reliable supplier has no flexibility, the
dual-source strategy is more efficient to meet multi-objective operations. Yu et al. (2009)
explained that in a dual-source strategy, two suppliers offer different prices and reliability because
48
they are in different regions. Also, the authors captured the probability of SC disruption risk and
formulated the expected profit functions with respect to SC disruption when the buying firms used
both sources. Cui et al. (2010) formulated the reliable facility location problem by using two
models. First, a mixed-integer programing model was used to obtain the optimal facility location
and assigned customer, which was solved by using the Lagrangian relaxation algorithm. Second,
a continuum approximation model was developed to minimize setup and transportation costs for
two scenarios: when a facility is reliable and when a facility is unreliable. This model was used to
calculate the expected total cost of the system and can be used to find close optimal solutions.
They designed the SC network to be reliable and cost efficient. They also formulated the discrete
model using MIP to minimize the total operating and failure cost.
Transportation disruption is different from other forms of SC disruptions Wilson (2007).
Transportation disruption stops only the flow of products. But other disruptions may also stop the
production of products. Giunipero and Eltantawy (2004) illustrated that a potential transportation
disruption is a cause of risk and that it could cripple the entire supply chain system. Guiffrida and
Jaber (2008) noted that transportation disruption can cause late deliveries, which may lead to
production stoppages costs, lost sales, and loss of customer goodwill. Guo and Jiang (2006)
developed a decision model to get rid of electronics by considering three levels of recycling
(product reuse, part reuse, and material reuse). Tan and Kumar (2008) used a linear programing
(LP) model to evaluate three end-of-life options for each part (namely, repair, repackage, or scrap).
Schmitt and Snyder (2012) stated that in 2011, the floods in Bangkok caused damage to product
inventories, which led to an increase in cost of the raw materials. They developed cost models to
determine the optimal order quantity by considering unreliable suppliers who caused uncertain
yield and supply chain disruptions.
49
3.2.2
Robust Supply Chain
The robustness technique was first presented by Taguchi et al. (1987) and Taguchi and
Phadke (1989). The advantage of this technique is to find a robust solution that is less sensitive to
unknown variations. Numerous studies have used the robustness technique to solve engineering
problems.
For example, Shi et al. (2013) developed a model to design a robust design
configuration for a cross-docking center to minimize the variability of the supply chain system.
Chauhan et al. (2006) and Pan and Nagi (2010) developed a robust optimization model for
designing an SCN under demand uncertainty. Their objective function consisted of total cost,
variability cost, and penalty cost due to unmet demand. Gutiérrez et al. (1996) developed a robust
model for an incapacitated network design problem by considering uncertainty in the input data.
Almaktoom et al. (2014) presented a novel robust design optimization to assure service level rate
requirement in complex supply chain networks. Lalmazloumian et al. (2013) developed a robust
optimization model for agile and build-to-order SC planning to minimize the total cost while
preserving the customer service level rate.
Pishvaee et al. (2012) proposed robust possibilistic programming to solve the problem of
a socially responsible SCN design under uncertainty. Simchi-Levi et al. (2013) used process
flexibility and inventory to increase SC robustness. Pishvaee et al. (2011) proposed a robust
optimization model for a closed-loop SCN design problem under uncertainty. Hasani et al. (2012)
also developed a model for a closed-loop SCN design under interval data uncertainty for perishable
goods in agile manufacturing by considering multiple periods, products, and SC levels. Amin and
Zhang (2013) presented a multi-objective mixed-integer linear programming (MILP) for a threestage closed-loop SC configuration under uncertainty.
50
3.2.3
Supplier Selection
Supplier selection has been a critical issue for decision-makers in order to choose the best
suppliers for a long period of time. Sawik (2011) considered two scenarios about disruption—
when there is an independent local disruption for each supplier, and when the disruption is local
and global for all suppliers. Moreover, a value-at-risk analysis and conditional value-at-risk
analysis were used to measure risk in a supply disruption. Sawik solved the problem by applying
the portfolio approach and static MIP formulations. In the scenario analysis, the low-probability
and high-impact supply disruptions are combined with high-probability and low-impact supply
delays. The optimal solution was found after applying this approach. However, the author did not
consider this heuristic approach to solve large-size problems. Wang et al. (2010) considered a
model to help a firm to source from several suppliers in order to improve supplier reliability.
Sawik (2013) also considered the supply chain disruption risk based on optimal selection
and protection of parts suppliers and order quantity to decide on the right supplier to be selected.
The emphasis of this approach was to decide on the suppliers to be selected for parts delivery,
allocation of order quantities among the selected suppliers, identification of selected suppliers that
must be protected against disruptions, and allocation of emergency inventory among the protected
suppliers. Sawik formulated the problem as an MIP to minimize the entire SC cost. The objective
function reduced the risk due to supplier’s protection, safety stock, parts ordering, purchasing,
transportation, and shortage by minimizing the worst-case cost.
Burke et al. (2009) used
newsvendor model to decide on a supplier selection source in order to maximize the profit using
either a single- or multiple-supplier sourcing strategy. Sawik also defined the supplier’s resilience
as “a capability of supplying parts in the face of disruption events.”
51
Burke et al. (2009) described the total-order quantity based on how many suppliers are
selected and allocation of the order quantity between the selected suppliers when there is
uncertainty in demand and supply. They examined the sourcing optimality among uncertainties
in product demand and supply reliability, and found that the single-supplier source is optimal for
environments characterized by high levels of demand uncertainty or high salvage values.
Songhori et al. (2011) solved a supplier-selection and order-allocation problem by
proposing a structured framework for a two-phase process of selection and allocation. In the
selection phase, the quantitative and qualitative criteria values for each supplier and transportation
alternative (TA) are identified to determine the efficiency of both. In the allocation phase, a multiobjective MIP model is used in which the objective is to minimize the total cost and maximize the
efficiencies.
Agarwal et al. (2011) reviewed different approaches to develop the best solution for
supplier evaluation and selection problem. Most approaches use data envelopment analysis to
emphasize supplier performance and to maximize efficiency. Talluri and Narasimhan (2003) are
the first researchers who proposed performance variability to evaluate multiple suppliers. They
developed two LP models to maximize delivery performance, reliability, etc. Hong et al. (2005)
proposed an MILP model to solve the supplier selection problem to maximize the profit, while
meeting customer demands. Also, the objective of this research was to identify the optimal number
of suppliers and order quantity so as to maximize the profit when there is variability on customer
demands and supplier performances. Rajan et al. (2010) used an integer linear programming (ILP)
model to solve the supplier selection problem for a multi-product, multi-vendor environment.
Ghodsypour and O’Brien (2001) proposed a mixed-integer non-linear programming model to
52
solve the supplier selection problem in order to determine the best suppliers and allocate the order
to each supplier so that the purchasing cost can be minimized.
Karpak et al. (2001) are the first researchers to use goal programming models to evaluate
suppliers. The objective function identifies the optimal order quantities, subject to demand and
supply constraints. Narasimhan et al. (2006) and Wadhwa and Ravindran (2007) proposed a multiobjective programming model to solve the supplier-selection problem.
Based on previous literature, not much attention has been given to the issue of uncertainty
in supply chains when there are product losses as the result of damage during shipping. This can
lead to two issues: identifying the best routes to reduce product damage and increase yield at the
final destination, and designing supply chains to provide the best yield. This chapter focuses on a
yield uncertainty model due to loss that occurs during shipping. A methodology for determining
the best routes and suppliers to ensure minimum total cost and maximum yield received at the final
destination has been developed. Also presented are details of a robust design model designed to
minimize each route’s variability. The methodology is detailed using two case studies.
The contents of this chapter are organized as follows: In section 3.3, a mathematical model
to minimize the total cost is developed. In subsection 3.3.2, a novel method for a robust design
model is developed. Section 3.4 illustrates two case studies to demonstrate the effectiveness of
the proposed models including results and analysis. Finally, conclusions and future work are
provided in section 3.5.
3.3
Problem Statement and Formulation
This section presents the model formulation that is used for describing the problem. The
network for this problem consists of different nodes, (i) and (j), which represent different
53
transportation company and distribution centers. The first node represents supplier n (Sn) and the
last node represents the retailer, where (φij) is the distance between nodes (i) and (j).
This section also details the general formulation developed for modeling the problem. The
objective function of this model attempts to minimize total cost, which consists of product cost,
and transportation cost, while considering multiple routes and multiple products as shown in
equation (3.1).
Indices:
U
Shipping cost (uU)
N
Type of product (nN)
R
Route (rR)
S
Supplier (sS)
Y
Yield (yY)
Parameters:
dn
quantity of product “n”
Bnij
Cost per mile for shipping product “n” from node “i” to node “j”
Wn
Cost of product “n”
Unij
Shipping cost for product “n” from node “i” to node “j”
Rijkn
Route from node “i” to node “j” using route number “k” for product “n”
Ynsij Yield for product “n” when selecting supplier “s” from node “i” to node “j”
γnsij
Percentage damage for product “n” when selecting supplier “s” from node “i” to
node “j”
54
Two models were designed to solve the problem. The first is a design that minimizes total
cost, and the second is a robust design model that minimizes the probability of damaged products
in order to maximize yield.
3.3.1 Model 1: Design to Minimize Total Cost
The first model is designed to minimize total cost, which consists of the product cost and
shipping cost, and is expressed as
N
N
j
n
n
i
Min Z  Wn d n  U nij Bnij d n Rijk n
subject to
(3.1)
R ijkn  0,1 k , n
(3.2)
Ynsij  (Yni 1, j 1 ) n
(3.3)
 nsij  0 n, s, i, j
(3.4)
Wn  { p1 , p2 ,........, pn }
(3.5)
Bn {B1, B2 ,........, Bn }
(3.6)
Equation (3.1) is the objective function that minimizes the total cost. The first term of the
objective function refers to the product cost, which is the product of the number of units shipped.
The second term refers to the shipping cost, which is the number of products shipped by using a
different route. The first constraint, equation (3.2), ensures that the term can only take on binary
values for selecting the route. Equation (3.3) ensures that the yield for product “n” for supplier
“s” at stage “i” is less than the yield at the next stage “j” due to damage percentage. Equation 3.4
ensures the non-negativity constraint for the damage percentage. Equation (3.5) represents the
cost of product “n.” And finally, equation (3.6) represents the shipping cost per mile for each
product “n.”
55
3.3.2
Model 2: Robust Design to Minimize Product Damage under Yield Uncertainty
The second model is a robust design that minimizes the probability of damaged products
in order to maximize yield. In general, the objective of a robust design is to decrease the impact of
uncertainty by minimizing variation around the mean without excluding the cause of uncertainty.
The objective of developing a robust model is to guarantee the robustness of the yield in the supply
chain and to minimize the impact of uncertainty in the system. The objective function of the model
minimizes the cost associated with damaged products. The robust model is expressed as
Min
N
j
n
i

d nij
(3.7)
subject to
d L nij  d nij  d U nij
(3.8)
Ynsij  Ynsi, j+1 n , s
(3.9)
0   nsij  1n, s
(3.10)
0  Ynsij  1n, s
(3.11)
 nsij , Ynsij  0 n, s, i, j
(3.12)
Equation (3.7) is the objective function that minimizes standard deviations for the damaged
product. In Equation (3.8), d L nij and d U nij are the lower and upper bounds, respectively, for the
design variable. Equation (3.9) ensures that the yield for product “n” for supplier “s” at stage “i”
is less than the yield at the next stage “j” due to damage percentage. Equations (3.10) and (3.11)
are used to ensure that the damage percentage  nsij and yield value Ynsij are between 0 and 1.
Equation (3.12) denotes the non-negativity constraint for damage percentage, and yield.
56
3.4
Case Studies
In this section, two case studies are used to demonstrate the effectiveness of the proposed
methodology. These case studies are also used to verify the MATLAB code. Two different
scenarios are compared and analyzed. Case study 1 represents the first scenario in which the
objective is to ship all products from one supplier to a single retailer. The second case study
represents the second scenario, in which two different suppliers are compared, and the best supplier
that ensures a maximum yield robustness is selected. All cases show methods for minimizing the
cost of transportation and maximizing the yield by using different routes.
3.4.1 Case Study 1
In this case study, a company has four different products—P1, P2, P3, and P4—with a cost
of $41, $60, $55, and $70, respectively. The demand for each product is assumed to be 1,000 units.
The company wants to ship all products from the supplier to the final destination, which is the
retailer, in such a way as to minimize the overall cost and maximize the yield at retailer. Figure
3.1 shows a transportation network that consists of 12 nodes. Node 1 and node 12 represent the
supplier and retailer, respectively. This network consists of five different routes to ship a product
from the supplier to the retailer, and each path of the network has a different distance. Table 3.1
shows the design parameters of path distances and shipping cost per mile for each path.
57
Figure 3.1. Supply chain network for Case Study 1
TABLE 3.1
DESIGN PARAMETERS FOR CASE STUDY 1
Path
Distance (miles)
Cost/Mile ($)
1-2
50
0.02
1-3
70
0.02
2-4
90
0.02
2-5
100
0.02
2-6
75
0.02
3-7
295
0.02
3-8
100
0.02
4-9
87
0.02
5-9
140
0.02
6-10
150
0.02
7-12
170
0.02
8-11
135
0.02
58
TABLE 3.1 (continued)
Path
Distance (miles)
Cost/Mile ($)
9-12
170
0.02
10-12
180
0.02
11-12
210
0.02
3.4.1.1 Results and Analysis for Case Study 1
This model was formulated and solved using MATLAB, and the case study was used to
validate the model. Table 3.2 illustrates the overall yield for each route and total cost for the initial
design, and Table 3.3 illustrates the robust design optimization for all products.
TABLE 3.2
RESULTS OF INITIAL DESIGN FOR ALL PRODUCTS FOR CASE STUDY 1
Product
Route
Probability of Damage (%)
Yield (%)
Total Cost ($)
P1
1-2-4-9-12
34
65
48,940
P2
1-2-4-9-12
44
55
67,940
P3
1-2-4-9-12
30
69
62,940
P4
1-2-4-9-12
38
61
77,940
TABLE 3.3
RESULTS OF ROBUST DESIGN FOR ALL PRODUCTS FOR CASE STUDY 1
Product
Route
Probability of Damage (%)
Yield (%)
Total Cost ($)
P1
1-3-7-12
22
77
51,300
P2
1-3-7-12
19
80
70,300
P3
1-3-7-12
19
80
65,300
P4
1-3-7-12
20
80
80,300
As shown in the initial design, the minimum optimal total cost obtained for product P2
when selecting route (1-2-4-9-12) with a maximum yield of 55% is a total cost of $67,940. By
59
comparison, when applying the robust design model, the optimal solution for product P2 is
obtained as route (1-3-7-12) with an overall yield of 80% and a total cost of $70,300. It is clear
that the robust design provided a higher yield than the initial design. In the initial design, the
maximum yield for product P3 is around 69% when selecting route (1-2-4-9-12) with a minimum
total cost of $62,940. In contrast, when applying the robust design model, the optimal solution is
obtained when selecting route (1-3-7-12) with an overall yield of 80% and a total cost of $65,300.
Finally, the highest yield for product P4 obtained when applying the initial design is to select route
(1-2-4-9-12) with a yield of 61% and a total cost of $77,940. In contrast, the overall yield is
increased to 80% when applying the robust design model and the new route selected is (1-3-7-12)
with a total cost of $80,300. Figures 3.2 to 3.5 illustrate the damage probability distributions for
both the initial design and the robust design for products P1 to P4, respectively. It can be seen that
the robust design model reduced the variation more than the initial design.
Figure 3.2. Results of initial design vs robust design for product P1 for Case Study 1
60
Figure 3.3. Results of initial design vs robust design for product P2 for Case Study 1
Figure 3.4. Results of initial design vs robust design for product P3 for Case Study 1
61
Figure 3.5. Results of initial design vs robust design for product P4 for Case Study 1
Table 3.4 illustrates the overall yield at each path and total cost for the initial design for
product P1. For example, the overall yield for path 1-2 is 85%, and the overall yield for path 2-4
is 76% because of the uncertainty of the damage probability in the previous path. For path 4-9 the
yield is around 89% and the overall yield is 68%. Finally, the yield at path 9-12 is 95% and the
overall yield is 65%, with a minimum total cost of $48,940.
Table 3.5 shows the result for product P1 when applying robust design optimization. The
robust route for product P1 is (1-3-7-12). As can be seen, the yield for path 1-3 is 92%, which is
higher than the yield of 85% on path 1-2 in the initial design. Also, the overall yield for path 3-7
is 84% and for path 7-12 is 77%. Figure 3.1, previously shown, illustrates the damage probability
distribution for the initial design and robust design for product P1, and it can be seen that the robust
design model reduced the variation more than in the initial design. In brief, when comparing the
62
robust design system with the initial design system, it is obvious that the former provides a
maximum yield to the overall system with a little higher total cost.
TABLE 3.4
RESULTS OF INITIAL DESIGN FOR PRODUCT P1 FOR CASE STUDY 1
Path
Yield (%)
Overall Yield (%)
Number of Products
Total Cost ($)
1-2
85
85
852
42,000
2-4
90
76
765
43,800
4-9
89
68
685
45,540
9-12
95
65
650
48,940
TABLE 3.5
RESULTS OF ROBUST DESIGN FOR PRODUCT P1 FOR CASE STUDY 1
Path
Yield (%)
Overall Yield (%)
Number of Products
Total Cost ($)
1-3
92
92
924
42,400
3-7
91
84
842
48,300
7-12
92
77
777
51,300
3.4.1.2 Sensitivity Analysis for Case Study 1
The fundamentals of sensitivity analysis are to find the stability of the best solutions with
potential adjustment in the parameters (Poh and Ang, 1999). Moreover, efficient solutions can be
measured via sensitivity analysis, which could assist in reducing uncertainty in parameters and
approaches (Triantaphyllou et al., 1998). Therefore, sensitivity analysis is used to perform the
ranking of changes in parameters with respect to weights of the primary goals that were proposed
for decision making. Also, it provides guidance for improving the solution, understanding the
model, and reducing the output uncertainties.
63
From the results, the sensitivity analysis provides guidance for changing the total cost for
each route when the shipping cost changes for any path. Table 3.6 illustrates sensitivity analysis
results for each product with the original solution and the new solution. For example, the original
solution for P1 was to select route (1-2-4-9-12) with a total cost of $48,940. However, if the
shipping cost for path (2-4) is changed to $0.04, then route (1-2-6-10-12) is found to be optimal,
with a total cost of $50,100. The same is true for P2, whereby the optimal solution obtained when
selecting route (1-2-4-9-12) is a minimum total cost of $67,940, but by changing the shipping cost
to $0.04 for path (9-12), the new optimal solution by selecting route (1-2-6-10-12) is $69,100. The
original solution for P3 was to select route (1-2-4-9-12) with a total cost of $62,940. However, if
the shipping cost for path (2-5) is changed to $0.07, then route (1-3-8-11-12) is found to be optimal
with a total cost of $65,800. Moreover, the original minimum cost for P4 by selecting route (1-24-9-12) is a total cost of $77,940, but by changing the shipping cost for path (1-2) to $0.08, route
(1-2-6-10-12) is the new optimal solution with a total cost of $80,300.
TABLE 3.6
RESULTS OF SENSITIVITY ANALYSIS FOR ALL PRODUCTS FOR CASE STUDY 1
Product
Original
Solution
Original Value
($)
New
Solution
New Value
($)
Difference
(%)
P1
1-2-4-9-12
48,940
1-2-6-10-12
50,100
2.3
P2
1-2-4-9-12
67,940
1-2-6-10-12
69,100
1.7
P3
1-2-4-9-12
62,940
1-3-8-11-12
65,800
7
P4
1-2-4-9-12
77,940
1-3-8-11-12
80,300
6
3.4.2 Case Study 2
This case study considers two different suppliers and one retailer. Each supplier provides
a different quality of products. Also, each supplier has a different packaging type, thus leading to
a variability of damage during shipping due to uncertainty. Here, it is assumed that one supplier
64
provides a better product quality, and the decision-maker wants to decide which supplier can
supply more product with less damage during shipping without ordering more product, in order to
minimize total cost and maximize profit. Figure 3.6 shows the supply chain network for Case
Study 2, which consists of 11 nodes and four different routes to ship the product from each supplier
to the retailer. Nodes 1, 2, and 11 represent supplier 1, supplier 2, and retailer, respectively. Nodes
3 to 10 represent sea ports and airports. Each path of the network has a different distance. Tables
3.7 and 3.8 show the associated distances and shipping cost per product per mile for each path for
suppliers 1 and 2, respectively.
Figure 3.6. Supply chain network for Case Study 2
65
TABLE 3.7
DESIGN PARAMETERS OF SUPPLIER 1 FOR CASE STUDY 2
Path
Distance (miles)
Cost/Mile ($)
1-3
60
0.02
1-4
120
0.02
1-5
100
0.02
1-6
150
0.02
3-7
90
0.02
4-8
200
0.02
5-9
120
0.02
6-10
125
0.02
7-11
110
0.02
8-11
170
0.02
9-11
160
0.02
10-11
140
0.02
TABLE 3.8
DESIGN PARAMETERS OF SUPPLIER 2 FOR CASE STUDY 2
Path
Distance (miles)
Cost/Mile ($)
2-3
150
0.03
2-4
140
0.03
2-5
90
0.03
2-6
50
0.03
3-7
90
0.03
4-8
200
0.03
5-9
120
0.03
6-10
125
0.03
7-11
110
0.03
8-11
170
0.03
9-11
160
0.03
10-11
140
0.03
66
3.4.2.1 Results and Analysis for Case Study 2
Tables 3.9 and 3.10 illustrate the yield for each supplier and total cost for the initial design
and robust design optimization for all products. For instance, the optimal total cost obtained for
product P1 when selecting route (1-3-7-11) with maximum yield of 738 and total cost of $46,200
in initial design for supplier 1. In comparison, by applying robust design model the optimal
solution obtained when selecting route (1-4-8-11) with yield of 826 and total cost of $50,800. The
maximum yield for product P1 when receiving the product from supplier 2 is 855 by selecting route
(2-6-10-11) with total cost of $55,400 for initial design. In contrast, when applying robust design
model the optimal solution obtained when selecting route (2-4-8-11) with yield of 894 product and
total cost of $61,300. It is clear that the robust design provided higher yield than the initial design
due to minimize the variation for the damage distribution. For product P2 the maximum yield is
875 for supplier 2 when using route (2-6-10-11) with total cost of $74,450 for initial design. In
comparison, by applying robust design model the optimal solution obtained when selecting route
(2-3-7-11) with yield of 880 and total cost of $75,500. Finally, when comparing a robust design
system with initial design system for both suppliers it is obvious that robust design provide
maximum yield to overall system and with a little higher total cost.
TABLE 3.9
RESULTS OF INITIAL DESIGN FOR BOTH SUPPLIERS FOR CASE STUDY 2
Supplier
Product
Route
1
1
1-3-7-11
2
1
1
2
Probability of Damage (%)
Yield (%)
Total Cost ($)
26
738
46,200
2-6-10-11
12.7
873
55,450
2
1-3-7-11
24
753
65,200
2
2-6-10-11
12.5
875
74,450
67
TABLE 3.10
RESULTS OF ROBUST DESIGN FOR BOTH SUPPLIERS FOR CASE STUDY 2
Supplier Product
Route
Probability of Damage (%)
Yield (%)
Total Cost ($)
1
1
1-4-8-11
17
826
50,800
2
1
2-4-8-11
10
894
61,300
1
2
1-4-8-11
17
831
69,800
2
2
2-3-7-11
12
880
75,500
From the results and analysis of both designs shown in Tables 3.9 and 3.10, it is clear that
damage uncertainty is one of the main causes of yield reduction. Furthermore, having more than
one supplier who can provide a good packaging type to prevent damage to products during
shipping can help manage yield at the final destination.
3.5
Conclusions and Future Work
Products can become damaged during shipping due to imperfect packaging or
transportations hazards. Suitable product packaging helps to ensure that retailers or customers
receive the product without damage. During each echelon of the supply chain network, the
quantity of the product can be different. In this chapter, two models were developed to solve the
problem. The first initial design model minimized total cost, which consists of both the cost of the
product and cost of shipping. The second robust design model minimized the probability of
damage to products in order to maximize yield. Two case studies were used to validate the
procedure and MATLAB code. From these case studies, the analysis showed that when comparing
a robust design system with an initial design system for both cases, it is obvious that the robust
design provides maximum yield to the overall system at a little higher total cost. Therefore, a
sensitivity analysis was used in Case Study 1 to rank changes in parameters with respect to weights
of the primary goals that were proposed for decision-making. In future work, the robust design
68
model could be extended by considering vehicle capacity constraints and the lead time for each
transportation mode.
3.6
References
Agarwal, P., Sahai, M., Mishra, V., Bag, M., and Singh, V. (2011). A review of multi-criteria
decision making techniques for supplier evaluation and selection. International Journal Of
Industrial Engineering Computations, 2(4), 801–810.
Almaktoom, A. T., Krishnan, K. K., Wang, P., and Alsobhi, S. (2014). Assurance of system service
level robustness in complex supply chain networks. The International Journal of Advanced
Manufacturing Technology, 74(1-4), 445–460.
Amin, S., H., and Zhang, G. (2013). A three-stage model for closed-loop supply chain
configuration under uncertainty. International Journal of Production Research, 51(5), 1405–1425.
Azad, N., and Davoudpour, H. (2010). Designing a reliable supply chain network model under
disruption risks. Journal of American Science, 6(12), 1091–1097.
Burke, G., J., Carrillo, J., E., and Vakharia, A., J. (2009). Sourcing decisions with stochastic
supplier reliability and stochastic demand. Production and Operations Management, 18(4), 475–
484.
Chauhan, S., Proth, J., Sarmiento, A., and Nagi, R. (2006). Opportunistic supply chain formation
from qualified partners for a new market demand. Journal of the Operational Research Society,
57(9), 1089–1099.
Cui, T., Ouyang, Y., and Shen, Z.,J. M. (2010). Reliable facility location design under the risk of
disruptions. Operations Research, 58(4-part-1), 998–1011.
Ghodsypour, S. H., and O'Brien, C. (2001). The total cost of logistics in supplier selection, under
conditions of multiple sourcing, multiple criteria and capacity constraint. International Journal of
Production Economics, 73(1), 15–27.
Giunipero, L. C., and Eltantawy, R. A. (2004). Securing the upstream supply chain: A risk
management approach. International Journal of Physical Distribution & Logistics Management,
34(9), 698–713.
Guide Jr., V. D. R., Souza, G. C., Van Wassenhove, L. N., and Blackburn, J. D. (2006). Time value
of commercial product returns. Management Science, 52(8), 1200–1214.
Guiffrida, A. L., and Jaber, M. Y. (2008). Managerial and economic impacts of reducing delivery
variance in the supply chain. Applied Mathematical Modelling, 32(10), 2149–2161.
Guo, W., and Jiang, J. (2006). Green remanufacturing—Sustainable development strategy. 2nd
International Eco-electronics Conference, December 7–9.
69
Gutiérrez, G., J., Kouvelis, P., and Kurawarwala, A., A. (1996). A robustness approach to
uncapacitated network design problems. European Journal of Operational Research, 94(2), 362–
376.
Hasani, A., Zegordi, S., H., and Nikbakhsh, E. (2012). Robust closed-loop supply chain network
design for perishable goods in agile manufacturing under uncertainty. International Journal of
Production Research, 50(16), 4649–4669.
Hong, G., H., Park, S., C., Jang, D., S., and Rho, H., M. (2005). An effective supplier selection
method for constructing a competitive supply-relationship. Expert Systems with Applications,
28(4), 629–639.
Karpak, B., Kumcu, E., and Kasuganti, R., R. (2001). Purchasing materials in the supply chain:
managing a multi-objective task. European Journal of Purchasing and Supply Management, 7(3),
209–216.
Lalmazloumian, M., Wong, K., Y., Govindan, K., and Kannan, D. (2013). A robust optimization
model for agile and build-to-order supply chain planning under uncertainties. Annals of Operations
Research, 1–36.
Narasimhan, R., Talluri, S., and Mahapatra, S., K. (2006). Multiproduct, multicriteria model for
supplier selection with product life‐cycle considerations. Decision Sciences, 37(4), 577–603.
Pan, F., and Nagi, R. (2010). Robust supply chain design under uncertain demand in agile
manufacturing. Computers and Operations Research, 37(4), 668–683.
Pishvaee, M., Razmi, J., and Torabi, S., A. (2012). Robust possibilistic programming for socially
responsible supply chain network design: A new approach. Fuzzy Sets and Systems, 206, 1–20.
Pishvaee, M., S., Rabbani, M., and Torabi, S., A. (2011). A robust optimization approach to closedloop supply chain network design under uncertainty. Applied Mathematical Modelling, 35(2),
637–649.
Poh, K., and Ang, B. (1999). Transportation fuels and policy for Singapore: An AHP planning
approach. Computers and Industrial Engineering, 37(3), 507–525.
Qi, L., Shen, Z. J. M., and Snyder, L., V. (2010). The effect of supply disruptions on supply chain
design decisions. Transportation Science, 44(2), 274–289.
Rajan, A., J., Ganesh, K., and Narayanan, K. (2010). Application of integer linear programming
model for vendor selection in a two stage supply chain. Paper presented at the International
Conference on Industrial Engineering and Operations Management, Dhaka, Bangladesh.
Sawik, T. (2011). Selection of a dynamic supply portfolio in make-to-order environment with
risks. Computers and Operations Research, 38(4), 782–796.
Sawik, T. (2013). Selection of resilient supply portfolio under disruption risks. Omega, 41(2), 259–
269.
70
Schmitt, A. J., and Snyder, L. V. (2012). Infinite-horizon models for inventory control under yield
uncertainty and disruptions. Computers and Operations Research, 39(4), 850–862.
Shear, H., Speh, T., and Stock, J. (2002). Many happy (product) returns. Harvard Business Review,
80(7), 16–17.
Shi, W., Liu, Z., Shang, J., and Cui, Y. (2013). Multi-criteria robust design of a JIT-based crossdocking distribution center for an auto parts supply chain. European Journal of Operational
Research, 229(3), 695–706.
Simchi-Levi, D., Wang, H., and Wei, Y. (2013). Increasing supply chain robustness through
process flexibility and strategic inventory. Working paper, MIT, Cambridge, MA.
Singh, S., and Xu, M. (1993). Bruising in apples as a function of truck vibration and packaging.
Applied Engineering in Agriculture, 9(5), 455–460.
Songhori, M., J., Tavana, M., Azadeh, A., and Khakbaz, M., H. (2011). A supplier selection and
order allocation model with multiple transportation alternatives. The International Journal of
Advanced Manufacturing Technology, 52(1-4), 365–376.
Taguchi, G., Clausing, D., and Watanabe, L., T. (1987). System of Experimental Design:
Engineering Methods to Optimize Quality and Minimize Costs (Vol. 2). UNIPUB/Kraus
International Publications, White Plains, NY.
Taguchi, G., and Phadke, M., S. (1989). Quality engineering through design optimization. In
Quality Control, Robust Design, and the Taguchi Method (pp. 77–96). Springer.
Talluri, S., and Narasimhan, R. (2003). Vendor evaluation with performance variability: A max–
min approach. European Journal of Operational Research, 146(3), 543–552.
Tan, A., and Kumar, A. (2008). A decision making model to maximise the value of reverse
logistics in the computer industry. International Journal of Logistics Systems and Management,
4(3), 297–312.
Tomlin, B. (2006). On the value of mitigation and contingency strategies for managing supply
chain disruption risks. Management Science, 52(5), 639–657.
Triantaphyllou, E., Shu, B., Sanchez, S., N., and Ray, T. (1998). Multi-criteria decision making:
An operations research approach. Encyclopedia of Electrical and Electronics Engineering, 15,
175–186.
Vursavuş, K. K., and Özgüven, F., (2004). Determining the effects of vibration parameters and
packaging method on mechanical damage in golden delicious apples. Turkish Journal of
Agriculture and Forestry, 28(5), 311–320.
Wadhwa, V., and Ravindran, A., R. (2007). Vendor selection in outsourcing. Computers and
Operations Research, 34(12), 3725–3737.
71
Wang, Y., Gilland, W., and Tomlin, B. (2010). Mitigating supply risk: Dual sourcing or process
improvement? Manufacturing and Service Operations Management, 12(3), 489–510.
Wilson, M. C. (2007). The impact of transportation disruptions on supply chain performance.
Transportation Research Part E: Logistics and Transportation Review, 43(4), 295–320.
Yu, H., Zeng, A. Z., and Zhao, L. (2009). Single or dual sourcing: Decision-making in the presence
of supply chain disruption risks. Omega, 37(4), 788–800.
72
CHAPTER 4
DAMAGE RECOVERY MODELS FOR SUPPLY CHAIN SYSTEM
4.1
Abstract
The growth and development in the field of manufacturing has led to long supply chains,
which have raised the need for preventing damage during transportation. During each stage of the
SC, the yield is different because of disruptions during transit, which may lead to a random yield
at the retailer. Decision-makers strive to implement strategies that enable the SC to quickly return
to the steady state, while minimizing the significant costs associated with recovery of the
disruption. This chapter focuses on the recovery of products that are damaged in transit. Recovery
models have been developed by considering different types of damage. Also detailed here is the
application of these models in a network that recovers the damaged product in different stages in
the SC network. A methodology for determining a cost-effective recovery model has been
developed to ensure maximum profit and meet customer demand. Five case studies are used to
validate the proposed methodology. Results specify that specific options for recovering the
damaged product can lead to significantly different expected profits.
Keywords: Damage recovery, transportation disruption, damage cost, supply chain risk
4.2
Introduction
Supply chains have raised the need for preventing damage during shipping and
transportation. With poor packaging, products may get damaged during transit. The types of
packaging and method of shipping influence the amount and type of damage. Good product
packaging helps to ensure that customers receive the product without any damage. The most
common hazards of transportation include shocks, vibrations, accidents, and handling. Damage
from accidents and handling issues are not completely within the control of packaging; however,
73
shocks occur during handling and transportation. For example, during the transportation of
products in trucks, shocks might occur when the condition of the road is poor.
Supply chain disruptions are expensive, and appropriate actions to decrease negative
effects to the SC system must be taken into account to ensure smooth performance of the system.
When disruptions occur, strategies that will enable the SC to quickly return to the steady state
while minimizing significant costs associated with recovery of the disruption must be developed.
Disruptions and damages can occur at all stages of the supply chain. For instance,
disruption may occur during production at the factory or during shipping when there are unusually
long delays at ports. The type of disruption and damage is different at each stage. As a result, the
percentage of damaged goods during each stage is also different. Thus, the yield at the final stage
is dependent on the damage that occurs at each stage. The damage caused during transit may be
categorized into three levels: minor damage, repairable, and severe. At the minor damage level,
the product has physical damage but there is no loss to its functionality, i.e., the product may have
scratches and dents but is still functional. In the case of repairable damage, the product has
physical damage that affects its required functionality; however, the product can be repaired by
replacing some of its parts. In the case of severe damage, the product has physical damage and
the damage is severe enough that a repair option is not feasible; however, the product can be
salvaged as parts from the damaged product, the parts could be reused, and some cost can be
recovered.
Depending on the level of the damage to the product, a decision must be made to determine
the appropriate level of recovery for the damaged products. In a forward supply chain, damaged
products can be recovered in different ways based on the type of damage. When the product is
damaged but not enough to affect its functionality, the product can be sold with scratches and dents
74
or shipped back to the recovery center for repair. If the damaged product has functional damage
and the repair option is not possible, then two options are available: first, the products are rejected
and send to a recovery center to be disassembled and sold as parts; or the products are rejected and
shipped back to the home factory. This chapter proposes a comprehensive approach that considers
all types of damage that may occur during transit and recommends models and methods to
maximize profit and meet customer demand.
4.3
Literature Review
Supply chain disruptions have received much attention in the last decade. Most of these
disruptions that have been studied have a low probability of occurrence, such as tsunamis and
earthquakes. These low-probability SC disruptions are typically disastrous and lead to high costs.
However, the loss as the result of damage during shipping is also high. According to Shear et al.
(2002), the value of products that are returned every year is around $100 billion. Guide Jr. et al.
(2006) and Shear et al. (2002) discuss different reasons for returning products (customer
satisfaction, product evaluation, shipping damage, defective merchandise, end of lease, and end of
life). Thierry et al. (1995) indicate five recovery options: repair, refurbishing, remanufacturing,
cannibalization, and recycling. They focused on remanufacturing and refurbishing the product to
restore it to an “as-good-as-new” condition by changing components or reusing used parts.
Additionally, the remanufacturing supply is controlled by the number of available returned
products. The quantity, quality, and time of product return are usually difficult to forecast and will
increase uncertainty along with demand risks in recovery systems.
The management of product returns has been studied in different models. Fleischmann et
al. (1997) reviewed quantitative models for reverse logistics in three operational areas (distribution
planning, inventory control, and production planning). Francas and Minner (2009) studied the
75
network design problem when a company manufactures new products and remanufactures returned
products in the same facilities, and they examined the performance and capacity for the
manufacture of these products when demand and returned products are uncertain. Fleischmann et
al. (2000) classified the general characteristics of recovery networks into three categories (product
characteristics, supply chain characteristics, and resource characteristics).
With respect to
profitability of the remanufacturing option, it is assumed that refurbishing used products costs less
than producing new products (Aras et al., 2006). Hishamuddin et al. (2013) developed a recovery
model for a two-stage production and inventory system under transportation disruption and
developed a heuristic model to obtain the result.
Transportation disruption is different from other forms of SC disruptions (Wilson, 2007).
Transportation disruption stops only the flow of products. But other disruptions may also stop the
production of products. Giunipero and Eltantawy (2004) illustrated that a potential transportation
disruption is a cause of risk and could cripple an entire SC system. Guiffrida and Jaber (2008)
noted that transportation disruption can cause late deliveries, which may lead to production
stoppage costs, lost sales, and loss of customer goodwill. Guo and Jiang (2006) developed a
decision model to eliminate electronics by considering three levels of recycling (product reuse,
part reuse, and material reuse). Jorjani et al. (2004) developed a piecewise linear concave program
to decide the optimal allocation of disassembled parts to five disposal options (refurbish, resell,
reuse, recycle, and landfill) in order to maximize the overall return. Tan and Kumar (2008) used
a linear programing model to evaluate three end-of-life options for each part (repair, repackage, or
scrap). Schmitt and Snyder (2012) reported that in 2011, the floods in Bangkok caused damage to
product inventories, which led to an increase in the cost of raw materials. They developed cost
76
models to determine the optimal order quantity by considering unreliable suppliers who caused
uncertain yield and SC disruption.
Azad and Davoudpour (2010) considered facilities with random disruption risk to design a
reliable supply chain network. They considered disputes in distribution centers by location and
capacity, formulated the problem as a nonlinear integer programming model, and then linearized
it to obtain the optimal solution. Numerous studies have solved random disruption risks by using
two different algorithms for large-size cases: tabu search and simulated annealing algorithms. The
tabu search algorithm provided better solutions in these models in which transportation disruption
costs were not considered. Similarly, Aryanezhad et al. (2012) designed an SCN considering
unreliable supplier and distribution centers. They found that the quantity of products delivered
may decrease due to unreliable distribution centers. Also, they formulated the problem as a
nonlinear integer programing to minimize total cost. The costs they considered included location,
transportation, inventory, and lost sales. Two approaches were developed to solve the problem:
Lagrangian relaxation and the genetic algorithm. They determined optimal distributions centers
and the subset of customers to be served, assigned customers to distribution center, and determined
the order quantity.
Qi et al. (2010) used the concept of disruptions to develop an integrated supply chain
network that can be used when suppliers and retailers are unreliable. They formulated the problem
as a nonlinear integer programming model to minimize the total annual cost, including fixed cost,
inventory cost, transportation cost, and lost sales cost). Moreover, they integrated the model to
decrease disruptions to retailers by determining the number of retailers that should be opened, their
locations, and frequency and order size for each retailer. They did not consider dynamic sourcing
to serve customers when the retailer is disrupted, which led to a random yield at the suppliers and
77
retailers. Wang et al. (2010) considered a model for helping a firm to source from several suppliers
with a goal to improving supplier reliability. Yu et al. (2009) studied selection methods involving
a single-source strategy and a dual-source strategy to obtain greater benefits when a SC disruption
occurs. A more reliable SC is more expensive than an unreliable one because it has more
flexibility.
Based on the previous literature, there is a need for addressing the recovery of the product
in a forward supply chain, and it is important to maximize profit while meeting demand. This
chapter focuses on the recovery of products that are damaged during shipping. Several models that
can be used to develop an understanding of the forward SC recovery process are developed to
show generic methodologies to calculate cost and percentage of met demand. A methodology for
using the best recovery model to ensure minimum total costs and maximize profit has been
developed. This methodology is detailed using an example case study.
4.4
Damage Recovery Approach
Most companies are focused on controlling efficiency in their forward SC and often pay
less attention to the damaged products, which leads to customer dissatisfaction, loss in profits,
increased overhead cost, etc. Depending on the product price, damage level, and lead time,
companies may adopt strategies to recover products damaged during shipping. The purpose of this
chapter is to develop models for analyzing the damaged-product recovery process. Five different
models are considered here. The first model deals with an SC network where no recovery is being
done. The second model is used when the recovery of products with type 1 and type 2 damages
are performed, while products with type 3 damage are disassembled and sold as parts. In the third
model, products with all types of damages are collected and shipped back to the manufacturer for
recovery. In the fourth model, the recovery of products with type 1 and type 2 damages are
78
performed, while products with type 3 damage are rejected. And finally in the fifth model,
products with type 1 damage are sent to a local recovery center, products with type 2 damage are
shipped back to the manufacturer, and products with type 3 damage are rejected after inspection.
Notations:
Aij
Distance from node “i” to node “j”
Bij
Cost per mile from node “i” to node “j”
dn
Quantity shipped for product “n”
D n, i
Disassembly cost for product “n” at node “i”
Fn
Lost cost for product “n”
Ii+1
Inspection cost at stage “Si+1”
Ozn
Repackaging cost using type “z” packaging for product “n”
PZ, n
Packaging cost using type “z” packaging for product “n” at the first stage
qi+1
Quantity of good units received at node “Si+1”
Tv, n
Repair cost for type “v” damages for product “n”
Uij
Shipping cost from node “i” to node “j”
Wn
Cost of product “n”
Xvnij
Percentage of type “v” damages for product “n” when shipping from node “i” to
node “j”
Ω3,n
4.4.1
Sales price for type 3 damages for product “n”
Recovery Model 1
Recovery Model 1 considers a system consisting of two nodes (Si, and Si+1) and one route.
Products that are shipped from node Si reach node Si+1 and can be inspected for damage at that
node. The damaged products are identified and separated. There is no recovery of damaged
79
products. The objective of this model is to obtain the cost per unit and quantity of good products
received at the final destination. Figure 4.1 shows the system for Recovery Model 1. This type of
model is used when the product costs are low and the repair and recovery costs are relatively high.
This type of model may also be used when the product costs are high, but the recovery and repair
costs are difficult and expensive.
Figure 4.1. Recovery Model 1
4.4.1.1 Mathematical Representation for Recovery Model 1
This subsection shows the calculations needed for Recovery Model 1 in order to determine
the total cost and quantity of good units delivered. The total shipping cost (U) (Equation 4.1) from
node S(i) to node S(i+1) is the product of the following: quantity of products shipped, distance from
node S(i) to node S(i+1), cost per mile, and packaging cost. In this the packaging cost is added only
for node S0.
Uij  d n (Aij Bij  PZ , n  Ii 1 )
(4.1)
The total quantity that arrives in good condition at node S(i+1) is given by Equation 4.2.
3


qi 1  d n  1   X vnii 1 
 v 1

(4.2)
After obtaining the total shipping cost, and total quantity arriving at node S(i+1), the cost per unit
can be obtained by
Unit cost  (Uij  d nWn ) / qi 1
80
(4.3)
4.4.1.2 Numerical Example 1 for Recovery Model 1
This numerical example illustrates the steps necessary to calculate the unit cost and
quantity received at the S(i+1) node of this model. The parameters for numerical example 1 are
shown in Table 4.1.
TABLE 4.1
PARAMETERS OF NUMERICAL EXAMPLE 1 FOR RECOVER MODEL 1
Parameter
Value
Parameter
Value
Aij (mile)
500
Wn ($)
30
Bij ($)
0.03
Pz,n ($)
2
Xvnij (%)
11
dn (unit)
100
Ii+1($)
2
Step 1: Calculate the shipping cost from node S(i) to node S(i+1)
Uij  100(500 * 0.03  2  2)  $1, 900
Step 2: Calculate the good quantity received at node S(i+1)
q i1  100 1  0.11  89
Step 3: After obtaining the total shipping cost, and total quantity arriving at node S(i+1), the cost
per unit is
Unit cost  (1, 900  3, 000) / 89  $55
In summary, the total shipping cost is $1,900, cost per unit is $55, and number of units of goods
received at node S(i+1) is 89.
4.4.2
Recovery Model 2
In this model, consider the system shown in Figure 4.2, which consists of two nodes and
one recovery center. Here, there is damage during shipping between two nodes, and products are
inspected at node S(i+1). The damaged products are separated and shipped to the recovery center
81
for repair. At the recovery center, types 1 and 2 damaged products are recovered, and type 3
damaged products are disassembled and sold as parts. The recovered products are shipped back to
node S(i+1). This model is used when product costs are high. In addition, the parts have significant
value when recovered. The repair or recovery costs are relatively low when compared to the cost
of the parts or the product.
Figure 4.2. Recovery Model 2
4.4.2.1 Mathematical Representation for Recovery Model 2
The shipping cost (U) from node S(i) to node S(i+1) can be obtained by using equation (4.1).
The shipping cost for damaged products from node S(i+1) to the recovery center is
3
U i 1, r  Bij  Aij  d n  X vnij
(4.4)
v 1
The shipping cost for the repaired products from the recovery center to S(i+1) is
2
Ur
, i 1
 Bij  Aij  d n (  X vnij )
v 1
82
(4.5)
The total shipping cost is the sum of the shipping costs shown in equations (4.1), (4.4), and (4.5):
Total shipping cost  Uij  U i 1 , r  U r
, i 1
(4.6)
The damaged products at S(i+1) are identified and separated, and the damaged products are shipped
to the recovery center for repair. After sorting all types of damage at the recovery center, all costs
associated with damaged products can be calculated to determine the total recovery cost (TRC).
The repair cost associated with the recovery of type 1 damaged products is
Repair cost for type1 damage  d n ( X 1nij T1, n )
(4.7)
The repair cost associated with the recovery of type 2 damaged products is
Repair cost for type 2 damage  d n ( X 2 nij T2, n )
(4.8)
The cost associated with the disassembly cost is
Disassembly cost  d n ( X 3nij Dn )
(4.9)
The repackaging cost at the recovery center is
2
Repackaging cost  d n (  X vnij )  OZ , n
(4.10)
v 1
The total recovery cost is the sum of costs in equations (4.7) to (4.10):
2
2
v 1
v 1
TRC  d n [[(  X vnijTv n )  ((  X vnij )  OZ , n )]  (X 3nij Dn )]
(4.11)
The total quantity that arrived in good condition at node S(i+1), including recovered products is
qi 1  d n 1  X 3nij 
(4.12)
sale price for type 3 damages  d n X 3nij * 3, n
(4.13)
The sale price for type 3 damages is
83
The cost per unit at node (i+1), which is the ratio of the total shipping cost, total recovery cost,
product cost, minus the sales price for type 3 damages to the number of good units arriving at node
(i+1), is
Unit cost  [(Uij  U i 1 , r  U r )  TRC  d nWn  (d n X 3nij * 3, n )] / qi 1
(4.14)
, i 1
4.4.2.2 Numerical Example 2 for Recovery Model 2
This example illustrates steps for calculating the unit cost and quantity received at the last node of
Recovery Model 2. The parameters for numerical example 2 are shown in Table 4.2.
TABLE 4.2
PARAMETERS OF NUMERICAL EXAMPLE 2 FOR RECOVERY MODEL 2
Parameter
Value
Parameter
Value
Parameter
Value
Aij (mile)
500
Ai+1,r (mile)
70
Wn ($)
30
Bij ($)
0.03
Ar,i+1(mile)
70
dn (unit)
100
Xvnij (%)
13
Pz,n ($)
2
Br,i+1 ($)
0.03
X1nij (%)
4
T1,n ($)
3.5
Bi+1, r ($)
0.03
X2nij (%)
5
T2,n ($)
5.5
Dn ($)
X3nij (%)
4
OZ,n ($)
2
Ω3,n ($)
15
Ii+1 ($)
2
2
Step 1: Calculate the total shipping cost from the first node to the next node, and shipping cost
from to the recovery center can be calculate the following:
U ij  d n (Aij Bij  PZ, n  Ii1 )  100 (500 * 0.03  2  2)  $1, 900
U n 1 , r  Bij  Ai1, r  d n X vnij  0.03* 70 * (100 * 0.13)  $27
Ur
, n 1
 Bij  A r, i 1  d n (X1nij  X2nij )  0.03* 70 *100 (0.04  0.05)  $18
Total shipping cost  Uij  U i1 , r  U r
, i 1
 1, 900  27  18  $1, 946
84
Step 2: Calculate the total recovery cost for all types of damages can be obtained by calculate the
following:
Repair cost for type1 damage  d n (X1nijT1, n )  100 (0.04 * 3.5)  $14
Repair cost for type 2 damage  d n (X2nijT2, n )  100 (0.05*5.5)  $27
Disassembly cost  d n (X3nij Dn )  100 (0.04 * 2)  $8
Repackaging cost  d n ( X1nij  X2nij )  OZ, n  100(0.04  0.05) * 2  $18
TRC  100 [(0.04 * 3.5)  (0.05 *5.5)  ((.04  .05) * 2)]  8  $67
Step 3: Calculate the good quantity received at node (i+1)
qi1  100 1  0.04   96
Step 4: Calculate the sale price for type 3 damages
Sale price for type 3 damages  di X3nij * 3, n  100 * 0.04 *15  $60
Step 5: Calculate the unit cost for a good quantity received at node (i+1)
Unit cost  [1, 946  67  3, 000  60] / 96  $52
In summary, the total shipping cost is $1,946, the cost per unit is $52, the number of units of goods
received at node S (i+1) is 96, and the sale price for type 3 damages is $60.
4.4.3
Recovery Model 3
In this model, consider the system shown in Figure 4.3, which consists of two nodes. Here
there is damage during shipping between the two nodes, and the products are inspected at node
S(i+1). The damaged products are separated and shipped to the recovery center at the main stage
(S0) for repairs. At the recovery center, products with all types of damages are recovered and then
shipped back to node S(i+1). This model is used when the products are expensive. This type of
recovery model is used when either the expertise may not exist at the recovery center or it is too
85
expensive to duplicate recovery centers. This may also be applied to systems wherein the
manufacturer does not want to disclose product details and would want to protect technical knowhow.
Figure 4.3. Recovery Model 3
4.4.3.1 Mathematical Representation for Recovery Model 3
This subsection illustrates all calculations needed for Recovery Model 3 to obtain the total
cost and quantity received in good condition. Shipping cost (U) from node Si to node S(i+1) is
j
Uij  d n  ( Aij Bij  PZ , n  I i 1 )
(4.15)
i 0
Shipping cost for damaged products from node S(i+1) to the first stage (S0) is
i
U i 1,0  d n X vnij  Bij  Aij
i 0
86
(4.16)
The total shipping cost is the sum of shipping costs from equations (4.15) and (4.16):
Total shipping cost  Uij  Ui 1 , 0
(4.17)
After identifying the damaged products at S(i+1), they are shipped back to stage S0 for repair.
After sorting all damaged products at stage (S0), all costs associated with damaged products are
calculated to determine the total recovery cost. The repair cost associated with the recovery of type
1 and type 2 damaged products can be obtained by using Equations (4.7), and (4.8)
The repair cost associated with the recovery of type 3 damaged products is
Repair cost for type 3 damage  d n ( X 3nijT3, n )
(4.18)
The repackaging cost at the recovery center can be obtained by using Equations (4.10)
The total recovery cost is
3
3
v 1
v 1
TRC  d n [(  X vnij Tv, n )  ((  X vnij )  OZ , n )]
(4.19)
The total quantity that arrives in good condition at node S(n+1), is
3


qi 1  d n (1   X vnij ) 
v 1


(4.20)
The cost per unit at node S(n+1) can be calculated as the sum of the total shipping cost, total recovery
cost, and product cost divided by the number of good units that arrive at node S(n+1):
Unit cost  [Uij  Ui 1 ,0 TRC  d nWn ] / qi 1
(4.21)
4.4.3.2 Numerical Example 3 for Recovery Model 3
This example illustrates the steps to calculate the unit cost and quantity received at the last
node of this model. The parameters for numerical example 3 are shown in Table 4.3. In summary,
the total shipping cost is $3,520, the cost per unit is $68, the number of units of goods received at
node s (i+1) is 98, and the lost product cost is $60.
87
TABLE 4.3
PARAMETERS OF NUMERICAL EXAMPLE 3 FOR RECOVERY MODEL 3
Parameter
Value
Parameter
Value
Parameter
Value
V0,1(mile)
500
X3nij (%)
8
Wn ($)
30
Vi,i+1(mile)
300
Ai+1,i (mile)
300
dn (unit)
100
Bij ($)
0.03
Ai,0 (mile)
500
T1,n($)
3.5
Xvnij (%)
16
Oz,n ($)
2
T2,n($)
5.5
X1nij (%)
3
PZ,n ($)
2
T3,n($)
7
X2nij (%)
5
Ii+1 ($)
2
Step 1: Calculate the total shipping cost from the first node to the next node, and shipping cost
from the last stage to the main source can be calculate the following:
Uij  100 *[((500 * 0.03)  2)((300 * 0.03)  2)]  $2,800
i
U n 1,0  d n X vnij  Bij  Aij  100 * 0.16[(300 * 0.03)  (500 * 0.03)]  $384
i 0
Total shipping cost  Uij  Ui1,0  2,800  284  $3,184
Step 2: Calculate the total recovery cost for all types of damages can be obtained by calculate the
following:
Repair cost for type1 damage  d n (X1nijT1, n )  100(0.03* 3.5)  $10.5
Repair cost for type 2 damage  d n (X2nijT2, n )  100(0.05*5.5)  $27.5
Repair cost for type 3 damage  d n (X3nijT3, n )  100(0.08* 7)  $56
Repackaging cost  d n (X1nij  X2nij  X3nij )  OZ, n  100(.03  .05  .08) * 2  $32
TRC  100 [(0.03* 3.5)  (0.05*5.5)  (0.08* 7)  (.03  .05  .08) * 2]  $126
Step 3: Calculate the good quantity received at the last destination
88
q i 1  d n (1  X vnij )   100 [(1  0.16)]  84
Step 4: Calculate the unit cost for a good quantity received at node (i+1)
Unit cost  [2,800  384  126  3, 000] / 84  $75
4.4.4
Recovery Model 4
This model consists of two nodes and one recovery center, as shown in Figure 4.4. In this
system, there is damage during shipping between the two nodes, and inspection occurs at node
S(i+1). The damaged products are separated and shipped to the recovery center for repair. During
inspection at the recovery center, only products with types 1 and 2 damage are recovered and those
with type 3 damage are rejected. The recovered products are shipped back to node S(i+1) . This
model is used when the parts cannot be salvaged economically and/or may not have significant
value. However, the repair and recovery of products with damages of Type 1 and Type 2 can be
easily done and the product will have significant value as refurbished or can be sold as new.
Figure 4.4. Recovery Model 4
89
4.4.4.1 Mathematical Representation for Recovery Model 4
The shipping cost (U) from node S(i) to node S(i+1) can be obtained by using equation (4.1).
The shipping cost for types 1 and 2 damaged products from node S(i+1) to the recovery center can
be calculated as
2
U i 1, r  Bn 1, r  An 1, r  d n (  X vnij )
(4.22)
v 1
The shipping cost for repaired products from the recovery center back to node S(i+1) can be obtained
by using equation (4.5).
The total shipping cost is the sum of shipping costs from equations (4.1), (4.5), and (4.22):
Total shipping cost  Uij  Ui 1, r  U r, i 1
(4.23)
Equations (4.7) and (4.8) can be used to obtain repair costs for types 1 and 2 damage. The
cost associated with type 3 damage at node S(i+1) can be obtained by
Fn  d n ( X 3nij Wn )
(4.24)
Repackaging costs at the recovery center can be obtained by solving equation (4.10). The total
recovery cost is the sum of the costs in Equations (4.7), (4.8), (4.10), and (4.24):
2
2
v 1
v 1
TRC  d n [(  X vnijTv ,n )  ((  X vnij )  OZ ,n )]  Fn
(4.25)
The total quantity that arrives in good condition at stage S(i+1), including recovered products, can
be obtained by Equation 4.12. The cost per unit at stage S(i+1) is the sum of the total shipping cost,
total recovery cost, and product cost divided by the number of good units arriving at node S(i+1):
Unit cost  [(Uij  Ui1, r  Ur, i1 )  TRC  d n Wn ] / q i1
4.4.4.2 Numerical Example 4 for Recovery Model 4
90
(4.26)
This numerical example illustrates steps to calculate the unit cost and quantity received at
the last stage of Recovery Model 4. The parameters for this numerical example are shown in Table
4.4.
TABLE 4.4
PARAMETERS OF NUMERICAL EXAMPLE 4 FOR RECOVERY MODEL 4
Parameter
Value
Parameter
Value
Parameter
Value
Ai,i+1(mile)
500
dn (unit)
100
Oz,n($)
2
Bij ($)
0.03
Ar,i+1(mile)
70
Ii+1 ($)
2
Xvnij (%)
13
Ai+1,r (mile)
70
Wn ($)
30
X1nij (%)
4
Pz,n ($)
2
Br,i+1 ($)
0.03
X2nij (%)
5
T1,n ($)
3.5
Bi+1, r ($)
0.03
X3nij (%)
4
T2,n ($)
5.5
Step 1: Calculate the total shipping cost from the first node to the next node, and shipping cost
from the last stage to the recovery center can be calculate the following:
Uij  d n (Aij Bij  PZ, n  Ii1 )  100 ((500 * 0.03)  2  2)  $1, 900
Ui1, r  Bi1, r  Ai1, r  d n (X1nij  X2 nij )  0.03* 70 *100 (0.04  0.05)  $19
Ur
, i 1
 Br, i 1  A r, i 1  d n (X1nij  X2 nij )  0.03* 70 *100 (0.04  0.05)  $19
Total shipping cost  Uij  Ui1, r  Ur, i1  1, 900  19  19  $1, 938
Step 2: Calculate the total recovery cost for types 1 and 2 damage can be obtained by calculate
the following:
Fn  d n (X3nij  Wn )  100 (0.04 * 30)  $120
Repackaging cost  d n ( X1nij  X2nij )  OZ, n  100(0.04  0.05) * 2  $18
TRC  100 [(0.04 * 3.5)  (0.05* 5.5)  ((0.04  0.05) * 2)]  120  $180
91
Step 3: Calculate the good quantity received at the last destination
q i 1  d n (1  X3nij )  100 [1  0.04]  96
Step 4: Calculate the cost per unit at node S(i+1) is the sum of the total shipping cost, total
recovery cost, and product cost divided by the number of good units arrived:
Unit cost  [1, 900  19  19  180  3, 000] / 96  $53
In summary, the total shipping cost for this example is $1,938, the cost per unit is $53, the number
of units of goods received at node S(i+1) is 96, and the lost product cost is $120.
4.4.5
Recovery Model 5
This model reflects the system shown in Figure 4.5, which considers shipping from node
Si to node S(i+1) and one recovery center. Here, damage occurs during shipping between nodes S(i)
and S(i+1), and the shipment is inspected at node S(i+1). The damaged products are separated and
shipped to the recovery center for repair. At the recovery center, products with type 1 damage are
recovered, products with type 2 damage are sent back to the first stage (S0), and products with type
3 damage are rejected at the inspection stage. The recovered products are shipped back to node
S(n+1) .
Figure 4.5. Recovery Model 5
92
4.4.5.1 Mathematical Representation for Recovery Model 5
This subsection illustrates the calculations needed to obtain the total cost and quantity
received for Recovery Model 5. The shipping cost (U) from node Si to node S(i+1) can be obtained
by using Equation 4.15. The shipping cost for type 2 damaged products from node S(i+1) to the
first source S0 can be calculated as
j
Ui 1, i  d n X 2 nij  Bij  Aij
(4.27)
i 0
The shipping cost for type 1 damage from S(i+1) to the recovery center is
Ui 1, r  Bi 1, r  Ai 1, r  d n X 1nij
(4.28)
The shipping cost for repaired products from recovery center r to S(i+1) is
U r , i 1  Br , i 1  Er , i 1  d n X1nij
(4.29)
The total shipping cost is the sum of shipping costs from Equations (4.15), (4.27), (4.28),
and (4.29):
Total shipping cost  Uij  Ui 1, r  U r , i 1  Ui 1, i
(4.30)
After separating the damaged products at S(i+1), products with type 1 damage are shipped
to the recovery center, products with type 2 damage are shipped to stage S0 for repair, and products
with type 3 damage at node S(i+1) are rejected. After sorting all damaged products, the costs
associated with them can be calculated to determine the total recovery cost. Repair costs for
products with type 1 damage can be obtained using Equation (4.7), and repair costs for products
with type 2 damage can be obtained using Equation (4.8). Repackaging costs at the recovery center
and at the main source are given by Equation 4.31.
s0
Repackaging cost   d n ( X 1nij  X 2 nij )  OZ , n
r
93
(4.31)
The cost associated with type 3 damaged products at node S(i+1) can be obtained by solving
equation (4.24).
The total recovery cost is the sum of costs from Equations (4.7), (4.8), (4.24), and (4.31).
2
s0
v 1
r
TRC  d n [(  X vnij T1, n )  (  d n ( X 1nij  X 2 nij )  OZ , n )]  Fn
(4.32)
The total quantity that arrives in good condition at node S(i+1), including recovered products, is
given by Equation 4.33.
3


qi 1  d n 1  (  X vnij ) 
v 2


(4.33)
The cost per unit at stage S(i+1) is the sum of the total shipping cost, total recovery cost, and product
cost divided by the number of good units arriving at node S(i+1).
(4.37)
Unit cost  [Uij  Ui 1, r  U r , i 1  Ui 1, i  TRC  d nWn ] / qi 1
4.4.5.2 Numerical Example 5 for Recovery Model 5
This example illustrates steps to calculate the unit cost and quantity received at the last
stage of this model. The parameters for this numerical example are shown in Table 4.5.
TABLE 4.5
PARAMETERS OF NUMERICAL EXAMPLE 5 FOR RECOVERY MODEL 5
Parameter
Value
Parameter
Value
Parameter
Value
A0,i(mile)
500
Ai+1,r (mile)
70
OZ,n ($)
2
Ai,i+1(mile)
300
Ar,i+1(mile)
70
T1,n ($)
3.5
Bij ($)
0.03
Pz,n ($)
2
T2,n ($)
5.5
Xvnij (%)
13
dn (unit)
100
X1nij (%)
5
Ii+1 ($)
X2nij (%)
7
Bi+1, r ($)
X3nij (%)
1
Wn ($)
2
0.03
30
94
Br,i+1 ($)
0.03
Step 1: Calculate the total shipping cost from the first node to the next node, and shipping cost
from the last node to the recovery center can be calculate the following:
j
Uij  d n  ( Aij Bij  PZ, n  I i 1 )  100 * ((500 * 0.03  2  0)  (300 * 0.03  0  2))  $2,800
i 0
Ui1, r  Bi1, r Ai1, r  d n X1nij  0.03* 70 *100 * 0.05  $11
U r,i 1  B
r, i 1
A r, i 1  d n X1nij  0.03* 70 *100 * 0.05  $11
i
Ui1, 0  d n X 2nij  Bij  Aij  100 * 0.07((0.03* 300 ) (0.03* 500))  $168
0
Total shipping cost  1, 700  1,100  11  11  168  $2, 989
Step 2: Calculate the total recovery cost for the damage product can be obtained by calculate the
following:
Fn  d n (X3nij  Wn )  100 (0.01* 30)  $30
s0
Repackaging cost   d n ( X1nij  X 2nij )  OZ, n  100 (0.05  0.07) * 2  $24
r
TRC  100 [(0.05* 3.5)  ((0.07 * 5.5)  (0.05  0.07) * 2)]  30  $110
Step 3: Calculate the good quantity received at the last destination
q i1  d n 1  (X2nij  X3nij )  100 [1  (0.07  0.01)]  92
Step 4: Calculate the cost per unit at node S(i+1) is the sum of the total shipping cost, total
recovery cost, and product cost divided by the number of good units arrived:
Unit cost  [2, 989  110  3, 000] / 92  $66
In summary, the total shipping cost is $2,989, cost per unit is $66, number of units of good received
at node S (i+1) is 92, and lost product cost is $30.
95
4.5
Case Study 1
In this case study, consider the supply chain network shown in Figure 4.6, which consists
of five stages. The distances between stages are shown in miles. The demand here is 100 units of
product. Inspection does not occur until stage S2, which has a recovery system similar to Recovery
Model 3 for the recovery of damaged products. Stage S3 has a recovery system similar to Recovery
Model 4. The parameters of Recovery Models 3 and 4 in this Case Study 1 are shown in Tables
4.6 and Table 4.7, respectively. In summary, the total shipping cost for stage S3 is $4,132, cost per
unit is $75, and number of units of goods received at stage S3 is 96.
Figure 4.6. Recovery Model Case Study 1 Supply Chain Network
TABLE 4.6
PARAMETERS OF RECOVERY MODEL 3 IN CASE STUDY 1
Parameter
Value
Parameter
Value
0.03
Pz,n ($)
2
Xvnij (%)
15
Ii+1 ($)
2
X1nij (%)
3
T2,n ($)
3.5
X2nij (%)
7
T3,n ($)
5.5
X3nij (%)
3
OZ,n ($)
2
Wn ($)
30
Bij($)
96
TABLE 4.7
PARAMETERS OF RECOVERY MODEL 4 IN CASE STUDY 1
Parameter
Value
Bij ($)
Parameter
Value
0.03
Pz,n ($)
2
Xvnij (%)
5
T1,n ($)
3.5
X1nij (%)
2
T2,n ($)
5.5
X2nij (%)
2
OZ,n ($)
2
X3nij (%)
1
Ii+1 ($)
2
Wn ($)
30
Now Recovery Model 3 is integrated with stage S3 to form stage S5, as shown in Figure
4.7. In summary, the total shipping cost until stage 4 is $5,252, cost per unit is $86, number of
units of goods received at stage S4 is 95.
Figure 4.7. Recovery Model 3 Integrated into Case Study 1 Supply Chain Network
Now Recovery Model 4 is integrated with stage S3 to form stage S6 and the final solution,
as shown in Figure 4.8.
Figure 4.8. Recovery Model 4 Integrated into Case Study 1 Supply Chain Network
97
After integrating both Recovery Models 3 and 4 into the supply chain network, the units
of goods arriving at stage S5 and the cost per unit can be determined. For this case study, the unit
cost is $98, and the number of units of goods received at S5 is 94.
4.6
Case Study 2
Here, a case study is used to demonstrate the effectiveness of the proposed methodology.
In this study, a company has one product P1 and uses two routes for shipping the product. The
product cost is $30, and the customer demand is 1,000 units. The shipping cost depends on the
type of transportation (truck, train, or ship). Figure 4.7 shows a transportation network that
consists of two routes (R1 and R2), a manufacturer (M), four facilities (F1, F2, F3, and F4), two
recovery centers (RC1, and RC2), and one retailer (G). Each route has different methods of
transportation, distances, and shipping costs. The associated distances and shipping costs per mile
are shown in Figure 4.9. Tables 4.8 and 4.9 shows the recovery system parameters and damage
probability for each path, respectively, for this case study.
Figure 4.9. Transportation network for Case Study 2
98
TABLE 4.8
PARAMETERS OF RECOVERY SYSTEM IN CASE STUDY 2
Value
($)
Parameter
Value
($)
Parameter
Repair cost for X1nij damage at RC1
3.5 Repair cost for X3nij damage at RC2
Repair cost for X2nij damage at RC1
5.5 Repackaging cost at RC1
12
1.5
Repair cost for X3nij damage at RC1
10
Repackaging cost at RC2
3
Repair cost for X1nij damage at RC2
4
Disassembly cost/ product
2
Repair cost for X2nij damage at RC2
7
Inspection cost
0.25
TABLE 4.9
DAMAGE PROBABILITY FOR EACH PATH IN CASE STUDY 2
Damage Probability for Type Z Packaging
Path
4.6.1
X1nij (%)
X2nij (%)
X3nij (%)
M-F1
3.40
1.45
0.70
M-F3
2.30
5.50
1.30
F1-F2
3
0.45
1.20
F3-F4
1.30
5.50
1.40
F2-J
2.40
1.60
0.55
F4-J
2.00
2.60
1.50
Results and Analysis for Case Study 2
The models were developed and solved using the total enumeration strategy and
AMTLAB. This case study was used to validate the proposed models. By applying the proposed
models, the units of goods and cost per unit received at the final destination for all recovery models
were determined. When applying Recovery Model 1, the optimal quantity received at the final
destination was 860 units at a cost per unit of $72 by selecting route R1. When Recovery Model 2
was applied, the maximum quantity arriving at the final destination was 904 units at a cost per unit
99
of $68 when using route R1. However, the maximum quantity obtained for Recovery Model 3 was
996 units at $69 per unit, and all damaged products were returned to the main stage for recovering
by using route R1. The optimal solution obtained for Recovery Model 4 was 910 units at a cost
per unit of $68 when selecting route R1. Finally, the maximum quantity for Recovery Model 5
was 904 units at a cost of $68 per unit.
4.7
Case Study 3
In this study, the supply chain network shown in Figure 4.10 is consider, which consists of
five stages. There is high damage that occurs during stage S0 to S1, and shipping cost from stage
S1 to S2 is very expensive and the distance between S1 and S2 is also long. Inspection at stage S1
is less expensive than the inspection at the retailer stage as shown in Figure 4.10. In this case study
there are two options to shipping product from stage S0 to retailer stage. In the first option,
inspection does not occur until the product reaches the retailer. The second option is to perform
inspection at stage S1 and return the damaged product to the first stage S0 for recovering. The
repair cost per product at stage S0 is $5.50. The parameters for this study are shown in Table 4.10.
Figure 4.10. Transportation network for Case Study 3
100
TABLE 4.10
DESIGN PARAMETERS FOR CASE STUDY 3
Stage
S1
S2
S3
Retailer
Distance (mile)
200
500
100
150
Cost per mile ($)
0.15
1
0.25
0.10
30
5
3
6
Damage (%)
4.7.1. Results and Analysis for Case Study 3
When applying option one, in which inspection occurs at the last stage, the total shipping
cost is $57,300. The unit cost is $1077 with 56 good units received at the last stage for every 100
units that are shipped. When applying the second option, the total shipping cost is $57,963. The
unit cost is $1093 with 56 good units received at the last stage for every 100 units that are shipped.
After obtaining the results for both options, it is clear that performing inspection at stage S1 and
return the damaged products to stage S0 for recovering is more cost efficient.
4.8
Case Study 4
In this case study, the supply chain network shown in Figure 4.11 is considered. This
supply chain network consists of five stages. Inspection is performed at stage S2. The repair cost
at stage S0 is three dollars while stage S2 repair cost is $22. In this study, there are two options to
shipping product from stage S0 to retailer stage. The first option is perform inspection at Stage S2
and recover the damaged products. The second option is to return the damaged product to the first
stage ‘S0’ for recovering. The parameters for this study are shown in Table 4.11.
101
Figure 4.11. Transportation network for Case Study 4
TABLE 4.11
DESIGN PARAMETERS FOR CASE STUDY 4
Stage
S1
S2
S3
Retailer
Distance (mile)
200
400
100
150
Cost per mile ($)
Damage (%)
0.15
7
0.45
5
0.25
3
0.10
6
4.8.1
Results and Analysis for Case Study 4
When applying option one, the total shipping cost is $26,425. The unit cost is $327 with
91 good units received at the last stage for every 100 units that are shipped. By applying the
second option the total shipping cost is $27,740. The unit cost is $390 with 79 good units
received at the last stage for every 100 units that are shipped. After obtained the result for both
options, it is clear that repairing the damaged products at stage S2 it is more cost efficient.
4.9
Case Study 5
For case study 5, consider the supply chain network shown in Figure 4.12. The supply
chain network consists of five stages and two inspection stages. Inspection is performed at stages
S1and S3. The repair cost at stage S0 and stage S3 are the same. In this study, two recovery centers
102
exist - one at stage S0, and the other at stage S3. Thus, the damaged products at stage S1 can be
returned to stage S0 due to the short distance, and the damaged products that occur between stage
S2 and stage S3 can be recovered at stage S3. The parameters for this study are shown in Table
4.12.
Figure 4.12. Transportation network for Case Study 5
TABLE 4.12
DESIGN PARAMETERS FOR CASE STUDY 5
Stage
S1
S2
S3
Retailer
Distance (mile)
200
500
100
150
Cost per mile ($)
0.15
0.35
0.25
0.10
8
4
35
3
Damage (%)
4.9.1
Results and Analysis for Case Study 5
For this study, the total shipping cost is $25,087. The unit cost at retailer stage is $321
with 89 good units received for every 100 units that are shipped.
4.10
Conclusions and Future Work
In this chapter, different recovery models were developed to maximize quantity of products
at the final destination in order to meet demand by considering different recovery scenarios. This
chapter proposed a new approach for recovering all types of damage that occur during transit.
103
Since the amount and type of damage is different at each stage of the supply chain network,
different models for recovering the products were developed, depending on the type of damage
that occurred. Five case studies were applied to validate the proposed models with different
recovery scenarios depending on the type of damage. The analysis shows that Recovery Model 3
provided the maximum quantity, whereby all damaged products returned to the main source to be
recovered. In future work, the models could be expanded to consider multi-suppliers in order to
reduce damage during shipping and transportation cost.
4.11
References
Aras, N., Verter, V., and Boyaci, T. (2006). Coordination and priority decisions in hybrid
manufacturing/remanufacturing systems. Production and Operations Management, 15, 528–543.
Aryanezhad, M. B., Naini, S. G. J., and Jabbarzadeh, A. (2012). An integrated model for designing
supply chain network under demand and supply uncertainty. African Journal of Business
Management, 6, 2678–2696.
Azad, N., and Davoudpour, H. (2010). Designing a reliable supply chain network model under
disruption risks. Journal of American Science, 6, 1091–1097.
Fleischmann, M., Bloemhof-Ruwaard, J. M., Dekker, R., Van der Laan, E., Van Nunen, J. A., and
Van Wassenhove, L.N. (1997). Quantitative models for reverse logistics: A review. European
Journal of Operational Research, 103, 1–17.
Fleischmann, M., Krikke, H. R., Dekker, R., and Flapper, S. D. P. (2000). A characterisation of
logistics networks for product recovery. Omega, 28, 653–666.
Francas, D., and Minner, S. (2009). Manufacturing network configuration in supply chains with
product recovery. Omega, 37, 757–769.
Giunipero, L. C., and Eltantawy, R. A. (2004). Securing the upstream supply chain: A risk
management approach. International Journal of Physical Distribution and Logistics Management,
34, 698–713.
Guide Jr., V. D. R., Souza, G. C., Van Wassenhove, L. N., and Blackburn, J. D. (2006). Time value
of commercial product returns. Management Science , 52, 1200–1214.
Guiffrida, A. L., and Jaber, M. Y. (2008). Managerial and economic impacts of reducing delivery
variance in the supply chain. Applied Mathematical Modelling, 32, 2149–2161.
Guo, W., and Jiang, J. (2006). Green remanufacturing—Sustainable development strategy. 2nd
International Eco-electronics Conference, December 7–9.
104
Hishamuddin, H., Sarker, R. A., and Essam, D. (2013). A recovery model for a two-echelon serial
supply chain with consideration of transportation disruption. Computers and Industrial
Engineering, 64, 552–561.
Jorjani, S., Leu, J., and Scott, C. (2004). Model for the allocation of electronics components to
reuse options. International Journal of Production Research, 42, 1131–1145.
Qi, L., Shen, Z. J. M., and Snyder, L. V. (2010). The effect of supply disruptions on supply chain
design decisions. Transportation Science, 44, 274–289.
Schmitt, A. J., and Snyder, L. V., (2012). Infinite-horizon models for inventory control under yield
uncertainty and disruptions. Computers and Operations Research, 39, 850–862.
Shear, H., Speh, T., and Stock, J. (2002). Many happy (product) returns. Harvard Business Review,
80, 16–17.
Tan, A., Kumar, A. (2008). A decision making model to maximise the value of reverse logistics
in the computer industry. International Journal of Logistics Systems and Management, 4, 297–
312.
Thierry, M., Salomon, M., Van Nunen, J., and Van Wassenhove, L. (1995). Strategic issues in
product recovery management. California Management Review, 37(2),114–35.
Wang, Y., Gilland, W., and Tomlin, B. (2010). Mitigating supply risk: Dual sourcing or process
improvement? Manufacturing and Service Operations Management, 12, 489–510.
Wilson, M. C. (2007). The impact of transportation disruptions on supply chain performance.
Transportation Research Part E: Logistics and Transportation Review, 43, 295–320.
Yu, H., Zeng, A. Z., and Zhao, L. (2009). Single or dual sourcing: Decision-making in the presence
of supply chain disruption risks. Omega, 37(4), 788–800.
105
CHAPTER 5
CONCLUSIONS AND FUTURE WORK
5.1
Conclusions
The overall objective of this dissertation was to develop mathematical models that are
required to minimize the total cost while maximizing profit and meeting customer demands when
yield is uncertain due to damages that occur during shipping in a supply chain network. To achieve
this objective, four research objectives were defined. This research first developed models for
selecting the appropriate types of packaging and transportation method for reducing damage
during shipping
. Then a methodology for selecting the best supplier and robust route to maintain yield during
shipping was developed. In connection with this, two models were developed. The first model
minimizes the total cost, which consists of product cost and shipping cost, while the second model
performs a robust design to minimize the probability of damaged products and maximizes the
yield. In the final research objective, models for identifying the effectiveness of recovery centers
when there are damaged products in the SCN were developed. These models calculate shipping
costs and network yield when damage occurs during shipping.
Appropriate product packaging helps to ensure that customers receive products without
any damage. During each stage of the SCN, the amount of damage can be different. There are
multiple transportation methods and packaging types for shipping products to the final destination.
In the models developed for identifying best routes and suppliers, a multi-objective model was
proposed for minimizing the total cost, which includes the cost of damage, shipping, and
packaging, by considering different transportation modes with their respective probability of
damage and different types of packaging. Three case studies with different conditions including
106
whether the products were shipped assembled or unassembled were presented. These case studies
were used to validate the MATLAB code. Two larger case studies were also tested to validate the
procedure from the case studies shown in this chapter. Analyses showed that shipping the
unassembled products was more cost effective than shipping assembled products, since
unassembled products have a low probability of damage.
Previous researchers have not paid much attention to the issue of uncertainty in supply
chains when there are product losses as the result of damage from shipping. Identification of the
best routes to reduce product damage and increase the yield at the final destination must be
developed. In addition to identifying the best routes, this research focused on the design/selection
of supply chains to provide the best yield. The emphasis here was on a yield uncertainty model
due to loss that occurs during shipping. A methodology for determining the best routes and
suppliers to ensure minimum total cost and maximum yield received at the final destination was
developed. This research also provided details of a robust design model, which could be used to
minimize the variability for each route. The detailed methodology used two case studies with two
different scenarios relative to whether the products were shipped using one or two suppliers. These
case studies were used to validate the procedure and MATLAB code. Analyses showed that when
comparing a robust design system with the initial design system, in both cases, it is obvious that
the robust design provides maximum yield to the overall system, but with a little higher total cost.
Therefore, sensitivity analysis was used to perform the ranking of the changes in parameters with
respect to the weights of the primary goals that were proposed for decision-making.
This research also developed five recovery models to determine the cost of the supply chain
network and to identify the network yield. This dissertation proposes a new approach for various
types of product damage that occur during transit. Since the amount and type of damage are
107
different at each stage of the supply chain network, different models were developed for recovering
the damaged products, depending on the type of damage that occurred. Two case studies were
applied to demonstrate these models.
5.2
Future Work
In the future, this dissertation could be expanded to extend the current models discussed in
each chapter. The proposed multi-objective model considered in Chapter 2 could be further
enhanced by considering multiple suppliers and retailers for minimizing the total cost and
maximizing the yield downstream of the supply chain network. The robust design model discussed
in Chapter 3 could be extended by considering vehicle capacity constraints and the lead time for
each transportation mode.
The models in Chapter 4 could be expanded to consider multi-suppliers, and multi-recovery
centers in order to reduce damage during shipping and transportation cost. Also, developing an
optimization model for stochastic damage probability in order to obtain the required recovery
center capacity could be studied. The recovery models could be expanded to consider multiproduct and optimal location for the recovery center based on the cost of transportation and ability
to recover the products, thus reducing the lead time to recover products. Finally, using the
economic order quantity (EOQ) model to determine the optimal order quantity during
transportation when there is a shortage could be an interesting area of research.
108
APPENDIX
109
TABLE A-1
DAMAGE PERCENTAGES FOR CASE STUDY 4
Products
Arc
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
N1,2
0.02
0.02
0.07
0.07
0.10
0.12
0.14
0.04
0.07
0.08
0.06
0.10
0.13
0.10
0.18
N1,3
0.03
0.05
0.08
0.09
0.12
0.13
0.15
0.07
0.08
0.10
0.08
0.14
0.13
0.06
0.03
N2,4
0.04
0.03
0.08
0.11
0.09
0.12
0.12
0.04
0.13
0.15
0.16
0.08
0.09
0.05
0.17
N2,5
0.06
0.06
0.09
0.12
0.13
0.11
0.13
0.06
0.13
0.16
0.15
0.10
0.08
0.06
0.19
N2,6
0.03
0.25
0.05
0.10
0.30
0.11
0.12
0.19
0.12
0.15
0.14
0.07
0.08
0.04
0.16
N3,7
0.05
0.03
0.06
0.13
0.15
0.12
0.14
0.11
0.14
0.18
0.16
0.12
0.11
0.09
0.18
N3,8
0.04
0.05
0.06
0.11
0.14
0.14
0.16
0.10
0.15
0.16
0.17
0.20
0.15
0.10
0.08
N3,9
0.06
0.06
0.07
0.15
0.15
0.16
0.15
0.12
0.13
0.20
0.18
0.17
0.14
0.15
0.17
N4,10
0.03
0.03
0.05
0.09
0.02
0.10
0.15
0.08
0.17
0.15
0.13
0.16
0.14
0.18
0.14
N4,11
0.10
0.11
0.06
0.07
0.12
0.15
0.14
0.12
0.14
0.17
0.14
0.18
0.15
0.19
0.12
N5,12
0.09
0.10
0.06
0.11
0.11
0.14
0.12
0.11
0.08
0.15
0.16
0.16
0.07
0.18
0.13
N6,13
0.02
0.06
0.25
0.06
0.07
0.22
0.11
0.10
0.16
0.14
0.12
0.15
0.13
0.17
0.10
N6,14
0.04
0.05
0.02
0.12
0.14
0.01
0.16
0.15
0.05
0.16
0.14
0.14
0.16
0.18
0.15
N6,15
0.04
0.07
0.08
0.10
0.08
0.12
0.13
0.14
0.16
0.15
0.19
0.16
0.15
0.20
0.16
N7,16
0.05
0.04
0.05
0.15
0.14
0.14
0.13
0.12
0.12
0.15
0.20
0.17
0.21
0.19
0.24
N8,17
0.07
0.07
0.07
0.16
0.13
0.12
0.15
0.18
0.14
0.19
0.22
0.20
0.19
0.22
0.09
N8,18
0.08
0.08
0.05
0.07
0.12
0.11
0.12
0.12
0.08
0.17
0.18
0.19
0.21
0.24
0.17
N9,19
0.07
0.07
0.08
0.09
0.11
0.10
0.14
0.13
0.09
0.16
0.19
0.17
0.22
0.08
0.15
N9,20
0.06
0.06
0.07
0.10
0.08
0.10
0.15
0.14
0.10
0.18
0.17
0.22
0.14
0.19
0.16
N10,21
0.06
0.05
0.08
0.12
0.17
0.11
0.14
0.17
0.12
0.12
0.15
0.11
0.05
0.16
0.10
N10,22
0.07
0.04
0.08
0.11
0.10
0.10
0.16
0.11
0.13
0.14
0.16
0.17
0.04
0.16
0.12
N11,23
0.07
0.07
0.10
0.13
0.16
0.15
0.17
0.19
0.14
0.15
0.14
0.18
0.03
0.18
0.10
N12,24
0.06
0.06
0.11
0.12
0.16
0.15
0.18
0.17
0.16
0.12
0.18
0.19
0.06
0.16
0.17
N13,25
0.04
0.07
0.07
0.10
0.14
0.09
0.13
0.16
0.12
0.11
0.29
0.24
0.12
0.15
0.09
110
TABLE A-1 (continued)
Products
Arc
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
N14,26
0.05
0.06
0.08
0.14
0.19
0.17
0.14
0.15
0.14
0.19
0.14
0.12
0.11
0.17
0.20
N14,27
0.06
0.07
0.08
0.13
0.20
0.21
0.17
0.16
0.13
0.20
0.03
0.04
0.06
0.16
0.21
N15,28
0.12
0.13
0.10
0.11
0.15
0.11
0.15
0.17
0.18
0.18
0.15
0.18
0.08
0.19
0.18
N16,29
0.11
0.10
0.12
0.17
0.18
0.17
0.14
0.18
0.17
0.16
0.15
0.19
0.10
0.17
0.16
N17,30
0.10
0.09
0.07
0.11
0.15
0.13
0.16
0.19
0.18
0.12
0.07
0.20
0.18
0.21
0.20
N18,31
0.08
0.08
0.12
0.13
0.20
0.10
0.17
0.17
0.20
0.14
0.11
0.24
0.21
0.26
0.23
N19,32
0.08
0.10
0.13
0.15
0.18
0.17
0.15
0.18
0.21
0.19
0.12
0.12
0.15
0.02
0.19
N19,33
0.09
0.09
0.08
0.11
0.15
0.15
0.16
0.20
0.19
0.17
0.17
0.14
0.16
0.19
0.10
N20,33
0.16
0.14
0.17
0.16
0.17
0.14
0.18
0.17
0.13
0.21
0.16
0.21
0.17
0.20
0.13
N21,34
0.14
0.15
0.10
0.12
0.14
0.12
0.18
0.10
0.08
0.12
0.07
0.11
0.21
0.10
0.17
N22,34
0.17
0.04
0.13
0.15
0.09
0.14
0.15
0.03
0.09
0.13
0.09
0.15
0.20
0.12
0.15
N23,34
0.16
0.14
0.17
0.16
0.17
0.14
0.18
0.17
0.13
0.21
0.16
0.21
0.17
0.20
0.13
N24,34
0.06
0.07
0.10
0.12
0.14
0.07
0.11
0.14
0.10
0.06
0.11
0.14
0.16
0.14
0.16
N25,34
0.03
0.06
0.09
0.11
0.10
0.20
0.12
0.05
0.25
0.05
0.06
0.10
0.15
0.13
0.14
N26,34
0.20
0.18
0.15
0.19
0.13
0.05
0.13
0.12
0.03
0.08
0.09
0.11
0.19
0.04
0.19
N27,34
0.18
0.16
0.19
0.12
0.15
0.11
0.14
0.09
0.12
0.06
0.14
0.04
0.20
0.22
0.19
N28,34
0.16
0.17
0.15
0.14
0.21
0.14
0.16
0.14
0.14
0.10
0.10
0.15
0.17
0.12
0.18
N29,34
0.06
0.07
0.10
0.12
0.11
0.15
0.19
0.10
0.12
0.14
0.10
0.20
0.24
0.16
0.15
N30,34
0.09
0.10
0.12
0.13
0.16
0.16
0.14
0.08
0.15
0.13
0.17
0.24
0.22
0.17
0.18
N31,34
0.10
0.12
0.13
0.14
0.12
0.11
0.22
0.15
0.16
0.13
0.20
0.23
0.25
0.18
0.20
N32,34
0.21
0.23
0.18
0.20
0.16
0.10
0.21
0.22
0.25
0.18
0.14
0.22
0.20
0.05
0.19
111
TABLE A-2
DAMAGE PERCENTAGES FOR CASE STUDY 5
Products
Arc
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
N1,2
0.11
0.08
0.15
0.32
0.16
0.23
0.07
0.22
0.07
0.07
0.19
0.34
0.30
0.08
0.22
N1,3
0.27
0.30
0.07
0.03
0.22
0.34
0.16
0.24
0.28
0.35
0.28
0.00
0.11
0.17
0.19
N2,4
0.25
0.01
0.04
0.19
0.40
0.37
0.00
0.08
0.20
0.07
0.14
0.24
0.24
0.40
0.17
N2,5
0.22
0.21
0.34
0.04
0.13
0.25
0.19
0.02
0.05
0.10
0.22
0.33
0.05
0.06
0.15
N2,6
0.21
0.18
0.31
0.05
0.33
0.28
0.31
0.33
0.30
0.17
0.22
0.18
0.16
0.11
0.10
N3,7
0.21
0.16
0.14
0.35
0.19
0.28
0.29
0.20
0.08
0.12
0.34
0.26
0.01
0.24
0.27
N3,8
0.10
0.08
0.18
0.07
0.26
0.10
0.34
0.03
0.17
0.18
0.02
0.17
0.11
0.13
0.19
N3,9
0.01
0.28
0.08
0.31
0.33
0.16
0.22
0.08
0.05
0.11
0.12
0.36
0.12
0.40
0.17
N4,10
0.03
0.06
0.37
0.12
0.07
0.33
0.29
0.14
0.08
0.00
0.21
0.01
0.09
0.29
0.31
N4,11
0.36
0.00
0.39
0.04
0.32
0.09
0.02
0.12
0.21
0.39
0.16
0.37
0.29
0.18
0.33
N5,12
0.26
0.16
0.24
0.37
0.29
0.34
0.10
0.15
0.21
0.34
0.19
0.36
0.33
0.24
0.00
N6,13
0.38
0.27
0.37
0.23
0.19
0.30
0.11
0.34
0.38
0.06
0.32
0.33
0.30
0.21
0.32
N6,14
0.09
0.11
0.03
0.14
0.19
0.11
0.10
0.02
0.19
0.19
0.09
0.38
0.05
0.23
0.32
N6,15
0.31
0.14
0.35
0.35
0.13
0.14
0.30
0.02
0.33
0.39
0.33
0.32
0.31
0.35
0.04
N7,16
0.04
0.01
0.33
0.04
0.40
0.04
0.40
0.31
0.22
0.10
0.10
0.12
0.25
0.01
0.04
N8,17
0.03
0.15
0.28
0.13
0.34
0.02
0.35
0.11
0.35
0.13
0.12
0.38
0.13
0.12
0.38
N8,18
0.23
0.27
0.18
0.30
0.03
0.09
0.18
0.23
0.20
0.40
0.17
0.27
0.03
0.17
0.23
N9,19
0.14
0.27
0.27
0.37
0.20
0.21
0.39
0.30
0.00
0.30
0.11
0.00
0.11
0.19
0.24
N9,20
0.34
0.36
0.19
0.22
0.31
0.24
0.03
0.09
0.00
0.27
0.12
0.29
0.16
0.16
0.23
N10,21
0.27
0.15
0.19
0.16
0.11
0.32
0.37
0.03
0.08
0.05
0.26
0.01
0.04
0.36
0.20
N10,22
0.21
0.15
0.26
0.15
0.16
0.36
0.35
0.13
0.11
0.21
0.32
0.38
0.02
0.25
0.06
N11,23
0.07
0.16
0.28
0.14
0.02
0.32
0.22
0.00
0.08
0.03
0.10
0.36
0.10
0.09
0.06
N12,24
0.33
0.00
0.16
0.39
0.19
0.02
0.29
0.40
0.34
0.06
0.38
0.00
0.27
0.26
0.31
N13,25
0.25
0.37
0.26
0.09
0.28
0.13
0.38
0.03
0.34
0.40
0.22
0.10
0.13
0.40
0.24
N14,26
0.21
0.03
0.10
0.08
0.11
0.02
0.30
0.13
0.08
0.23
0.00
0.31
0.13
0.03
0.36
112
TABLE A-2 (continued)
Products
Arc
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
P12
P13
P14
P15
N14,27
0.38
0.20
0.22
0.01
0.07
0.27
0.35
0.37
0.40
0.23
0.03
0.25
0.36
0.03
0.40
N15,28
0.16
0.37
0.17
0.02
0.30
0.00
0.06
0.24
0.27
0.32
0.22
0.25
0.38
0.31
0.31
N16,29
0.11
0.20
0.14
0.28
0.38
0.34
0.08
0.07
0.03
0.24
0.37
0.34
0.37
0.07
0.04
N17,30
0.07
0.29
0.28
0.12
0.34
0.14
0.40
0.06
0.14
0.18
0.36
0.31
0.17
0.35
0.22
N18,31
0.01
0.21
0.12
0.16
0.35
0.35
0.23
0.10
0.12
0.24
0.39
0.04
0.03
0.16
0.09
N19,32
0.39
0.23
0.02
0.31
0.09
0.13
0.30
0.36
0.28
0.17
0.06
0.26
0.30
0.06
0.20
N19,33
0.36
0.34
0.38
0.17
0.07
0.26
0.26
0.15
0.38
0.22
0.20
0.04
0.16
0.16
0.27
N20,33
0.35
0.26
0.29
0.15
0.30
0.12
0.07
0.30
0.10
0.19
0.31
0.24
0.06
0.27
0.23
N21,34
0.20
0.38
0.06
0.16
0.25
0.07
0.30
0.39
0.08
0.17
0.05
0.06
0.35
0.01
0.20
N22,34
0.01
0.06
0.34
0.07
0.04
0.05
0.13
0.31
0.02
0.15
0.29
0.06
0.32
0.30
0.12
N23,34
0.17
0.24
0.23
0.33
0.30
0.13
0.37
0.08
0.26
0.39
0.24
0.32
0.39
0.35
0.07
N24,34
0.19
0.11
0.39
0.17
0.32
0.08
0.12
0.04
0.08
0.04
0.23
0.35
0.20
0.20
0.11
N25,34
0.06
0.07
0.36
0.13
0.33
0.16
0.05
0.29
0.26
0.20
0.04
0.37
0.19
0.23
0.09
N26,34
0.38
0.26
0.30
0.28
0.30
0.11
0.11
0.23
0.12
0.35
0.17
0.29
0.19
0.36
0.40
N27,34
0.19
0.27
0.36
0.08
0.03
0.21
0.18
0.21
0.26
0.05
0.16
0.03
0.30
0.07
0.18
N28,34
0.04
0.16
0.09
0.12
0.33
0.22
0.09
0.11
0.11
0.18
0.36
0.28
0.09
0.29
0.25
N29,34
0.31
0.11
0.21
0.29
0.38
0.01
0.40
0.17
0.07
0.32
0.40
0.00
0.05
0.37
0.09
N30,34
0.15
0.05
0.16
0.11
0.21
0.03
0.16
0.27
0.30
0.26
0.01
0.14
0.04
0.12
0.12
N31,34
0.29
0.11
0.13
0.11
0.33
0.19
0.11
0.18
0.37
0.26
0.39
0.18
0.40
0.23
0.03
N32,34
0.00
0.07
0.00
0.35
0.07
0.37
0.09
0.39
0.35
0.15
0.37
0.29
0.03
0.38
0.30
N33,34
0.25
0.14
0.05
0.07
0.16
0.35
0.26
0.02
0.02
0.11
0.33
0.22
0.31
0.19
0.25
113
TABLE A-2 (continued)
Products
Arc
P16
P17
P18
P19
P20
P21
P22
P23
P24
P25
P26
P27
P28
P29
P30
N1,2
0.01
0.15
0.35
0.38
0.00
0.08
0.19
0.23
0.08
0.19
0.02
0.18
0.18
0.16
0.24
N1,3
0.02
0.37
0.23
0.09
0.34
0.03
0.16
0.33
0.38
0.24
0.03
0.28
0.25
0.30
0.27
N2,4
0.26
0.30
0.09
0.11
0.39
0.11
0.31
0.20
0.17
0.40
0.01
0.21
0.13
0.14
0.04
N2,5
0.20
0.05
0.18
0.25
0.39
0.14
0.22
0.01
0.20
0.00
0.40
0.07
0.12
0.03
0.30
N2,6
0.19
0.15
0.26
0.20
0.13
0.37
0.32
0.32
0.28
0.01
0.21
0.18
0.25
0.03
0.23
N3,7
0.12
0.03
0.24
0.39
0.28
0.11
0.17
0.24
0.01
0.40
0.22
0.16
0.20
0.18
0.06
N3,8
0.23
0.03
0.40
0.11
0.14
0.17
0.11
0.02
0.36
0.00
0.28
0.36
0.10
0.06
0.16
N3,9
0.07
0.17
0.20
0.12
0.20
0.14
0.14
0.07
0.38
0.17
0.13
0.31
0.39
0.11
0.18
N4,10
0.24
0.39
0.39
0.25
0.23
0.29
0.07
0.24
0.12
0.26
0.27
0.06
0.21
0.00
0.36
N4,11
0.35
0.28
0.33
0.02
0.17
0.05
0.37
0.19
0.19
0.35
0.09
0.12
0.12
0.29
0.36
N5,12
0.10
0.37
0.04
0.33
0.35
0.03
0.13
0.15
0.36
0.33
0.34
0.39
0.08
0.18
0.10
N6,13
0.22
0.11
0.22
0.07
0.15
0.39
0.02
0.05
0.35
0.20
0.30
0.38
0.03
0.27
0.38
N6,14
0.32
0.30
0.16
0.11
0.11
0.22
0.13
0.32
0.02
0.08
0.02
0.21
0.25
0.19
0.18
N6,15
0.12
0.23
0.19
0.32
0.27
0.07
0.29
0.16
0.12
0.06
0.20
0.23
0.16
0.34
0.05
N7,16
0.33
0.28
0.30
0.12
0.13
0.17
0.00
0.00
0.37
0.40
0.31
0.06
0.31
0.20
0.29
N8,17
0.40
0.08
0.20
0.27
0.07
0.11
0.19
0.29
0.27
0.01
0.25
0.22
0.16
0.11
0.37
N8,18
0.08
0.14
0.36
0.14
0.39
0.25
0.38
0.08
0.00
0.06
0.37
0.28
0.25
0.40
0.01
N9,19
0.36
0.34
0.09
0.40
0.17
0.17
0.21
0.06
0.18
0.16
0.40
0.15
0.07
0.01
0.10
N9,20
0.38
0.11
0.30
0.09
0.34
0.30
0.21
0.08
0.39
0.25
0.31
0.24
0.28
0.29
0.39
N10,21
0.03
0.09
0.06
0.39
0.15
0.11
0.31
0.35
0.10
0.17
0.19
0.24
0.27
0.15
0.19
N10,22
0.00
0.22
0.11
0.34
0.26
0.35
0.17
0.27
0.39
0.15
0.13
0.08
0.15
0.09
0.24
N11,23
0.15
0.15
0.14
0.07
0.05
0.12
0.34
0.33
0.35
0.14
0.09
0.09
0.21
0.27
0.37
N12,24
0.32
0.25
0.02
0.40
0.13
0.08
0.08
0.38
0.32
0.07
0.08
0.35
0.07
0.19
0.38
N13,25
0.37
0.38
0.13
0.09
0.17
0.22
0.22
0.22
0.34
0.08
0.04
0.01
0.03
0.17
0.28
N14,26
0.26
0.05
0.33
0.24
0.38
0.34
0.25
0.14
0.23
0.09
0.07
0.18
0.16
0.08
0.30
114
TABLE A-2 (continued)
Products
Arc
P16
P17
P18
P19
P20
P21
P22
P23
P24
P25
P26
P27
P28
P29
P30
N14,27
0.07
0.14
0.09
0.13
0.28
0.03
0.16
0.12
0.18
0.00
0.04
0.03
0.18
0.16
0.30
N15,28
0.15
0.32
0.21
0.19
0.37
0.02
0.17
0.18
0.26
0.10
0.08
0.02
0.29
0.19
0.21
N16,29
0.03
0.13
0.25
0.00
0.11
0.33
0.28
0.11
0.35
0.40
0.13
0.11
0.37
0.06
0.05
N17,30
0.20
0.37
0.30
0.10
0.04
0.28
0.20
0.38
0.13
0.31
0.14
0.01
0.24
0.19
0.34
N18,31
0.39
0.02
0.23
0.12
0.37
0.07
0.17
0.13
0.33
0.15
0.16
0.40
0.06
0.26
0.35
N19,32
0.08
0.28
0.35
0.01
0.23
0.34
0.34
0.29
0.10
0.02
0.19
0.32
0.17
0.25
0.05
N19,33
0.03
0.26
0.22
0.25
0.32
0.04
0.27
0.12
0.40
0.27
0.20
0.01
0.28
0.40
0.02
N20,33
0.11
0.06
0.24
0.15
0.24
0.31
0.00
0.14
0.20
0.01
0.22
0.22
0.02
0.27
0.16
N21,34
0.20
0.10
0.28
0.19
0.25
0.00
0.11
0.22
0.10
0.03
0.30
0.33
0.14
0.16
0.04
N22,34
0.11
0.07
0.37
0.02
0.01
0.05
0.10
0.05
0.25
0.15
0.14
0.37
0.15
0.33
0.08
N23,34
0.00
0.09
0.27
0.08
0.23
0.08
0.28
0.28
0.10
0.02
0.04
0.08
0.16
0.28
0.11
N24,34
0.26
0.09
0.04
0.04
0.00
0.37
0.24
0.09
0.20
0.13
0.00
0.10
0.27
0.11
0.17
N25,34
0.11
0.28
0.15
0.34
0.39
0.03
0.06
0.21
0.02
0.06
0.36
0.23
0.28
0.19
0.28
N26,34
0.36
0.17
0.27
0.13
0.20
0.03
0.28
0.14
0.08
0.39
0.16
0.01
0.03
0.33
0.23
N27,34
0.07
0.30
0.25
0.22
0.13
0.26
0.37
0.13
0.20
0.08
0.10
0.06
0.30
0.33
0.34
N28,34
0.08
0.10
0.17
0.16
0.12
0.20
0.03
0.28
0.28
0.09
0.25
0.27
0.04
0.07
0.19
N29,34
0.30
0.15
0.36
0.35
0.07
0.23
0.27
0.39
0.33
0.19
0.02
0.33
0.06
0.13
0.06
N30,34
0.12
0.37
0.28
0.24
0.03
0.21
0.26
0.02
0.35
0.02
0.01
0.11
0.23
0.09
0.27
N31,34
0.13
0.17
0.31
0.23
0.35
0.12
0.30
0.01
0.04
0.32
0.14
0.18
0.18
0.00
0.01
N32,34
0.39
0.01
0.23
0.36
0.40
0.26
0.27
0.07
0.23
0.27
0.34
0.00
0.03
0.15
0.38
N33,34
0.40
0.16
0.09
0.08
0.32
0.21
0.38
0.12
0.17
0.26
0.36
0.22
0.26
0.12
0.02
115