a multi-product, multi-supplier, multi-period and multi
TRANSCRIPT
The Pennsylvania State University
The Graduate School
Harold and Inge Marcus Department of Industrial and Manufacturing Engineering
A MULTI-PRODUCT, MULTI-SUPPLIER, MULTI-PERIOD AND MULTI-
PRICE DISCOUNT LEVELS SUPPLIER SELECTION AND ORDER
ALLOCATION MODEL UNDER DEMAND UNCERTAINTY
A Thesis in
Industrial Engineering and Operations Research
` by
Swapnil C. Phansalkar
Β© 2016 Swapnil C. Phansalkar
Submitted in Partial Fulfillment of the Requirements
for the Degree of
Master of Science
May 2016
The thesis of Swapnil C. Phansalkar will be reviewed and approved* by the following:
M. Jeya Chandra
Professor of Industrial and Manufacturing Engineering
Thesis Advisor
Vittal Prabhu
Professor of Industrial and Manufacturing Engineering
Janis Terpenny
Professor of Industrial and Manufacturing Engineering
Department Head, Industrial and Manufacturing Engineering
*Signatures are on file in the Graduate School
ii
Abstract
In this thesis, a multi criteria decision making model for supplier selection and order
allocation is developed under demand uncertainty. The primary goal of this model is to
assist the buyer in making the decision regarding the selection of suppliers and allocation
of order quantities to the selected suppliers to satisfy demand for each product in every
time period.
The multi criteria decision making model has the following three objectives: to
minimize the total cost over planning horizon, to minimize the total number of quality
defects and to minimize the weighted average lead time. The demand is assumed to follow
normal distribution and shortages are allowed in this model. Further, each supplier is
assumed to provide all unit type of price discount for each part. A mathematical
formulation is done for each objective and then a general model for multiple products,
multiple suppliers, multiple time periods and multiple price discount levels is developed.
The multi criteria model includes the buyerβs demand constraints, the supplierβs
capacity constraints and the supplierβs price break constraints. The following decisions are
to be made in this model: to select the appropriate suppliers considering the objectives of
the model and to determine the optimal order quantity from the selected suppliers to satisfy
the buyerβs demand for each time period. This model uses weighted objective method to
determine the optimal solution to the multi criteria decision making problem.
iii
TABLE OF CONTENTS
List of Tables....................................................................................................................vi
Acknowledgements........................................................................................................viii
Chapter 1: Introduction..................................................................................................... 1
1.1 Multi criteria decision making in Supplier selection and order allocation................. 1
1.2 Multi criteria decision making in Supplier selection and order allocation with
deterministic demand .............................................................................................. 2
1.3 Multi criteria decision making in Supplier selection and order allocation with
stochastic demand ....................................................................................................3
1.4 Multi criteria decision making in Supplier selection and order allocation with price
discounts....................................................................................................................4
Chapter 2: Model Formulation...........................................................................................8
2.1 Model Assumptions .................................................................................................8
2.2 Objective functions ................................................................................................10
2.2.1 To minimize the total cost over the planning horizon......................................10
2.2.2 To minimize the number of defective parts ....................................................15
2.2.3 To minimize the weighted average lead time..................................................15
2.3 Constraints..............................................................................................................16
2.4 Weighted objective method....................................................................................17
Chapter 3: Numerical Example and Sensitivity analysis...................................................19
3.1 Problem Description...............................................................................................19
3.2 Mathematical Model...............................................................................................23
3.3 Sensitivity Analysis................................................................................................30
iv
Chapter 4: Summary, Conclusions and scope for future research.....................................39
4.1 Summary and Conclusions.....................................................................................39
4.2 Scope for future research........................................................................................40
Referencesβ¦....................................................................................................................41
v
LIST OF TABLES
Table 3-1: Numerical Example: Fixed order cost related to parts and suppliers.............19
Table 3-2: Numerical Example: Inventory holding cost and shortage cost related to
parts................................................................................................................20
Table 3-3: Numerical Example: Values of parameters related to suppliers and parts.....20
Table 3-4: Numerical Example: Values of unit price and price break quantity.............. 21
Table 3-5: Numerical Example: Weights of three objectives .........................................21
Table 3-6: Numerical Example: Values of demand for each part for each week .......... 22
Table 3-7 Mean values of demand ..................................................................................22
Table 3-8 Standard deviation values of demand..............................................................22
Table 3-9 Order proportions ............................................................................................26
Table 3-10 Order quantities ............................................................................................27
Table 3-11 Selection of Suppliers....................................................................................27
Table 3-12 Selection of price levels.................................................................................28
Table 3-13 Original values of Standard Deviation of demand........................................30
Table 3-14 New values of Standard Deviation of demand..............................................30
Table 3-15 Order proportions ..........................................................................................31
Table 3-16 Order quantities ............................................................................................31
Table 3-17 Selection of Suppliers....................................................................................32
Table 3-18 Selection of price levels.................................................................................33
Table 3-19 Comparison of values of objectives.............................................................. 33
Table 3-20 Original values of Mean of demand..............................................................34
Table 3-21 New values of Mean of demand................................................................... 34
vi
Table 3-22 Order proportions .........................................................................................35
Table 3-23 Order quantities ............................................................................................35
Table 3-24 Selection of Suppliers....................................................................................36
Table 3-25 Selection of price levels.................................................................................37
Table 3-26 Comparison of values of objectives...............................................................38
vii
ACKNOWLEDGEMENTS
First and foremost, I want to thank my parents, Chintamani Phansalkar and Varsha
Phansalkar, who have always supported me in every step of my life. I cannot imagine
success in any phase of my life without their support. Second, I sincerely thank my thesis
advisor, Dr. M Jeya Chandra, for his guidance throughout my degree and especially during
this research work. He has been an excellent advisor and mentor to me and I have learnt a
lot from him during the course of this degree and the thesis work. I also thank Dr. Vittal
Prabhu for agreeing to be the reader for this thesis.
Finally, I want to thank my family in U.S., Abann Sunny, Sandeep Gagganapalli,
Mohanarangan Tiruppathy, Malav Patel and Abhinav Singh for all the support, memorable
moments and the best two years of my life.
viii
Chapter 1
Introduction
This chapter focuses on presentation of literature review done in the field of multi
criteria optimization for supplier selection and order allocation.
1.1 Multi criteria decision making in Supplier Selection and Order Allocation
Make or buy decisions play a crucial role in any type of industry. If the industry decides
to buy as opposed to make, the important decisions to be made by the buyer are 1) To
decide whether to buy the products from single supplier or multiple suppliers 2) To make
the right choice of suppliers by taking into consideration the buyerβs demand and supplierβs
performance. Davarzani and Norrman (2014) explained the risk of having a single supplier
for a product in their study. The authors concluded that even if a single source may lead to
better quality of the product, buyer may have to compromise on other important criteria
such as product price and lead time of supplier. To make these key decisions, the buyer has
to take into consideration the supplierβs capacity, supplierβs lead time, quality of the
product and price offered per unit product. In case there are multiple suppliers for a single
product, it becomes a multiple criteria decision making problem due to the tradeoff
between cost, quality and lead time. The complexity of the problem increases in case of
supplier selection and order allocation for multiple products and multiple time periods.
Dickson (1966) classified criteria for supplier selection based on their importance. The
criteria such as quality, performance and delivery were classified as extremely important
criteria whereas criteria such as price, technical capability and reputation of supplier were
classified as considerably important. Geographical location of supplier was classified as a
1
criterion of average importance. Different factors are of different level of importance for
every buyer and there is always a tradeoff in a supplier selection problem.
In this thesis, we choose three criteria for supplier selection: 1) The total cost over planning
horizon 2) The number of defective parts procured from supplier 3) Lead time of supplier.
1.2 Multi criteria decision making in Supplier Selection and Order Allocation with
deterministic demand
Mendoza and Ravindran (2008) proposed a multi criteria model for supplier selection
and order allocation to multiple suppliers under deterministic demand. The authors
considered following criteria: the total purchasing expenses, total distance of suppliers
selected, lead time of suppliers selected and percentage of defective parts procured from
suppliers. The multi criteria model is presented in three phases. Phase I concentrates on
screening process for the selection of a small set of suppliers from a large supplier base.
Phase II presents ranking of suppliers based on variety of criteria that are important to the
buyer. Finally, a multi criteria goal programming model is proposed in Phase III. This
model assumes that orders placed over given period of time are equal to the demand. The
optimal solution is obtained using optimization software LINDO.
Feyzan Arikan (2013) proposed a single product multiple supplier multi criteria model
using augmented max-min operator and novel solution approach. The author considered
following criteria for supplier selection: total cost over the planning horizon, weighted
average lead time and weighted average quality defect rate. These objectives were also
used by Ziqi Ding (2014) in her multi criteria model with single product and multiple
suppliers. Demand was assumed to be deterministic in this model and shortages are not
2
allowed. The author takes into account supplierβs capacity constraints and buyerβs demand
constraints and proposes a weighted objective method approach for model formulation.
Mohsen, Majid and Ali (2011) proposed a supplier selection and order allocation model
by considering different transportation alternatives (TAs) used by suppliers. This is a single
product multiple supplier multiple period model under assumption of deterministic
demand. The authors choose a single criteria for supplier selection: total cost over planning
horizon. The total cost is presented as the sum of ordering cost, transportation cost and
inventory cost. Another multi criteria model was proposed using fuzzy multi-attribute
utility approach by Cuiab and Makab (2014). The authors chose following criteria for
supplier selection in their model: total cost of purchasing, quality of products purchased
from suppliers and delivery time of suppliers. Further, the authors proposed a mixed integer
nonlinear programming model using constraint programming to provide optimal solution.
1.3 Multi criteria decision making in Supplier Selection and Order Allocation with
stochastic demand
Guo (2014) considered a supplier selection and order allocation model under stochastic
demand. The author proposed a single product, single buyer and multiple supplier model
by considering different levels viz. suppliers, warehouses and retailers. In this model,
supplier selection is done by warehouses. These warehouses supply items to retailers. This
model chose a single criteria for supplier selection: total cost over planning horizon. Yishan
Sun (2015) developed a multiple product, multiple supplier and multiple time period model
by assuming the demand to follow uniform distribution. Shortages are allowed in this
model. The author considered three objectives for supplier selection: total cost over
3
planning horizon, weighted average lead time and weighted average quality defect rate.
The optimal solution is calculated using weighted average method.
Another supplier selection and order allocation model considering lead time and
demand uncertainty was developed by Andreas Thorsen (2014). The primary objective of
this research paper was to study inventory control under demand and lead time
uncertainties. Fariborz and Yazdian (2011) proposed a multi criteria goal programming
model for multiple products and multiple suppliers for multiple time periods. The authors
also proposed a fuzzy TOPSIS to evaluate the suppliers based on buyerβs criteria in the
first phase. Further, mixed integer linear programming model is developed for two
objectives: periodic budget and total purchasing cost.
Kazemi, Ehsani, Glock and Schwindl (2015) proposed a mathematical programming
model for supplier selection and order allocation. This research paper aimed at studying
the degree of buyerβs satisfaction level using a set of metric distance functions and
comparing the results with an earlier study that used weighted max-min approach. Kannan
et al (2013) provided a fuzzy multi criteria decision making approach to the supplier
selection and order allocation problem. This model assumes that the demand is stochastic.
The mathematical programming model considers two criteria for supplier selection: total
cost of purchasing and total value of purchasing.
1.4 Multi criteria decision making in Supplier Selection and Order Allocation with
price discounts
Price discounts are offered by suppliers depending on the order quantity from the buyer.
Incremental price discount is offered by some suppliers i.e. price discount is offered on the
4
order quantity above the price break quantity. However, some suppliers offer all unit
discounts i.e. price discount is offered on all units of the order quantity if the order quantity
is above the price break quantity. Bundling type of discount is offered by suppliers, in case
two or more types of products are ordered together. Also, business volume type of discount
is offered by suppliers in case the total volume of business (measured in dollars) is handed
to them. In case of a multiple product, multiple supplier and multiple time period model,
price discount offered by suppliers can play a significant role in supplier selection and order
allocation. This thesis assumes that all unit type of discount is offered by each supplier.
Fariborz, Mahsa and Hamidreza (2013) proposed a multi-product multi supplier and
multi period model assuming price discount for deterministic demand. This model does not
allow shortages. Goal programming is used by authors to solve for optimal solution. Total
cost of purchase is the criteria for supplier selection in this model. Shakouri, Javadi and
Karamati (2013) considered an interactive decision making process for the supplier
selection and order allocation problem. The authors developed a fuzzy multi objective
linear programming model and applied piecewise linear membership functions to represent
the following objectives: total ordering cost, total number of defective parts and total
number of late delivered items by suppliers.
Jafar and Elham (2010) developed a multi item supplier selection and lot size planning
model with deterministic demand by considering two types of price discounts viz. business
volume discount and incremental discount. This model has three objectives: total ordering
cost, weighted average lead time and weighted average quality defect rate. Adeinat and
Ventura (2015) proposed a supply chain problem with multiple suppliers where each
supplier offers a price discount. A mathematical programming model is developed
5
considering the following criteria for supplier selection: total replenishment and inventory
cost by taking into consideration supplierβs quality and supplierβs capacity constraints.
Lee and Kang (2013) considered an integrated multi criteria model for supplier
selection with price discounts and proposed a genetic algorithm to tackle the mixed integer
programming problem. The model considers following types of cost in the objective
function: ordering cost, holding cost, transportation cost and purchasing cost. Jadidi and
Zolfaghari (2014) models the multi criteria supplier selection problem considering three
criteria: total cost, number of quality rejects and lead time of supplier. Further, the research
paper provides a comparative analysis of compromise programming and weighted goal
programming techniques, used to solve the problem. A mathematical programming model
with same criteria was modeled by Ayhan and Kilic (2015). Suppliers are first selected
using buyerβs criteria and Analytical Hierarchy Process and a mixed integer linear
programming model is proposed to solve the single period multi product multi supplier
problem. Demand is assumed to be deterministic in this research. All unit type of discount
is assumed by the authors for this model.
Cebi and Otay (2016) developed a single period multi product and multi supplier
programming model for supplier selection and order allocation with price discounts offered
by each supplier. The authors focused on the following criteria of supplier selection: total
procurement costs, total number of late deliveries and total number of defective items.
Fuzzy goal programming is used to optimize the order quantities for selected suppliers.
Many of the previous studies in the research area of supplier selection and order
allocation using multiple criteria decision making focus on products with deterministic
demand. Further, shortages are not considered by most of the studies. Also, many supplier
6
selection and order allocation models are developed for single period planning. The aim of
this thesis is to develop a multiple product, multiple supplier and multiple time period
supplier selection and order allocation model. Further, this thesis also aims to develop a
model in which each supplier offers an all unit price discount for each part. Many studies
have a single objective of minimizing the total cost over the planning horizon. However,
this thesis aims at developing a multi objective supplier selection and order allocation
model with following objectives: minimizing the total cost, minimizing the number of
defective parts and minimizing the weighted average lead time.
Chapter 2 includes the formulation of the model along with all objectives and all
constraints. Chapter 3 presents a numerical example along with the sensitivity analysis.
Conclusions and scope for future research are presented in chapter 4.
7
Chapter 2
Model Formulation
In this chapter, we present a multi-criteria model to solve the supplier selection and order
allocation problem. Multi-criteria optimization techniques are used to solve the problem
that has three objective functions viz. minimize the total cost, minimize the number of
defective parts and minimize the weighted average lead time.
2.1 Model Assumptions
The multi criteria model has the following assumptions;
β’ A single buyer purchases m different parts from n suppliers in T time periods.
β’ Buyer can purchase a part from single or multiple suppliers in each period
β’ Demand for part i in period t is Dit, which follows normal distribution with a mean Β΅it
and a standard deviation Οit.
β’ Fixed order cost fij is constant for supplier j supplying part i in any period.
β’ Supplier j has a constant quality rejection rate qij, lead time lij and unit transportation
cost cij for part i during the planning horizon.
β’ Inventory holding cost and shortage cost for part i are assumed to be hi and sci
respectively.
β’ Capacity of supplier j for part i in each period is CAPij.
β’ Price discount is assumed to be offered by every supplier. Every supplier offers p price
levels for each part.
8
β’ Model Indices:
i : part (i = 1,2,β¦.., m). There are m types of parts.
j : supplier (j = 1,2,β¦.., n ). There are n suppliers for each part.
t : time period (t= 1,2,β¦.., T). There are T time periods in planning horizon.
k : price level (k= 1,2,β¦β¦, p). There are p price levels for each supplier for each part.
β’ Model Parameters:
lij : lead time of supplier j for part i (in weeks).
Dit : Demand for part i in period t.
Β΅it : mean of demand Dit.
Οit : standard deviation of demand Dit.
Ξ¦(Dit) : probability density function for demand Dit.
Cij : unit shipping cost of supplier j for part i.
CAPij : capacity of supplier j for part i in each period.
fij : fixed order cost for supplier j for part i.
qij : quality rejection rate of part i for supplier j.
pijk : price per unit for part i from supplier j at price level k.
bijk : quantity at which price break k occurs for part i from supplier j.
hi : unit holding cost for part i.
sci : unit shortage cost for part i.
β’ Decision variables
xijtk : proportion of part i purchased from supplier j in period t.
Oit : order quantity for part i in period t.
Iit : inventory level for part i at the end of period t.
9
Sit : shortage quantity for part i at the end of period t.
Ξ΄ijt : 1, if order is placed to supplier j for part i in period t.
: 0, otherwise.
yijkt : 1, if order is placed with supplier j for part i at price level k for period t.
: 0, otherwise.
2.2 Objective functions
The multi criteria model has three objective functions: to minimize total cost, to minimize
number of defective parts and to minimize weighted average lead time.
2.2.1 To minimize the total cost over the planning horizon
In this model, total cost comprises of following five components.
β’ Fixed cost
β’ Purchasing cost
β’ Shipping cost
β’ Inventory holding cost
β’ Shortage cost
Each component of total cost is expressed below.
β’ Fixed cost: If supplier j is used for part i in period t, fixed cost is charged by supplier
j to the buyer. Fixed cost is given by
οΏ½οΏ½οΏ½ππππππ
ππ
π‘π‘=1
β πΏπΏπππππ‘π‘
ππ
ππ=1
ππ
ππ=1
10
β’ Purchasing cost: Purchasing cost over planning horizon is given by
οΏ½οΏ½οΏ½οΏ½π₯π₯πππππ‘π‘ππ β ππππππππ β πππππ‘π‘
ππ
ππ=1
ππ
π‘π‘=1
ππ
ππ=1
ππ
ππ=1
β’ Shipping cost: Shipping cost is paid by the buyer. It is given by
οΏ½οΏ½οΏ½π₯π₯πππππ‘π‘ππ β ππππππ β πππππ‘π‘
ππ
π‘π‘=1
ππ
ππ=1
ππ
ππ=1
β’ Inventory holding cost: Total inventory holding cost is given by
= οΏ½οΏ½ οΏ½ βππ
πΌπΌππ,π‘π‘β1
0
ππ
π‘π‘=1
ππ
ππ=1
β οΏ½πΌπΌππ,π‘π‘β1 β π·π·πππ‘π‘ οΏ½ β ππ (π·π·πππ‘π‘) β πππ·π·πππ‘π‘
= οΏ½οΏ½βππ β πΌπΌππ,π‘π‘β1 οΏ½ ππ (π·π·πππ‘π‘) β πππ·π·πππ‘π‘
πΌπΌππ,π‘π‘β1
0
ππ
π‘π‘=1
ππ
ππ=1
β βππ οΏ½ π·π·πππ‘π‘ β ππ (π·π·πππ‘π‘) β πππ·π·πππ‘π‘
πΌπΌππ,π‘π‘β1
0
= οΏ½οΏ½βππ β πΌπΌππ,π‘π‘β1 οΏ½ππβ
(π·π·πππ‘π‘ β Β΅πππ‘π‘)2
πππππ‘π‘2
πππππ‘π‘ β β2ππβ πππ·π·πππ‘π‘
πΌπΌππ,π‘π‘β1
0
ππ
π‘π‘=1
ππ
ππ=1
β βππ οΏ½ π·π·πππ‘π‘ β ππβ
(π·π·πππ‘π‘ β Β΅πππ‘π‘)2
πππππ‘π‘2
πππππ‘π‘ β β2ππβ πππ·π·πππ‘π‘
πΌπΌππ,π‘π‘β1
0
(2.1)
Let (π·π·πππ‘π‘ β Β΅πππ‘π‘)/πππππ‘π‘ = π£π£
Therefore, π·π·πππ‘π‘ = πππππ‘π‘ β π£π£ + Β΅πππ‘π‘ and πππ·π·πππ‘π‘ = πππππ‘π‘ β πππ£π£
Substituting in (2.1),
= οΏ½οΏ½βππ βπΌπΌππ,π‘π‘β1β2ππ
οΏ½ ππβπ£π£2/2 β πππ£π£
πΌπΌππ,π‘π‘β1
0
ππ
π‘π‘=1
ππ
ππ=1
β βππβ2ππ
οΏ½ (πππππ‘π‘ β π£π£ + Β΅πππ‘π‘
πΌπΌππ,π‘π‘β1
0
) β ππβπ£π£22
β πππ£π£ (2.2)
11
= οΏ½οΏ½βππ βπΌπΌππ,π‘π‘β1β2ππ
οΏ½ ππβπ£π£22
πππ£π£
πΌπΌππ,π‘π‘β1
0
ππ
π‘π‘=1
ππ
ππ=1
β βππβ2ππ
οΏ½πππππ‘π‘ οΏ½ π£π£ππβπ£π£22
πΌπΌππ,π‘π‘β1
0
πππ£π£ + Β΅πππ‘π‘ οΏ½ ππβπ£π£22
πΌπΌππ,π‘π‘β1
0
πππ£π£οΏ½ (2.3)
First, we solve β« ππβπ£π£2/2 πππ£π£πΌπΌππ,π‘π‘β10
Let πΌπΌ = β« ππβπ£π£2/2 πππ£π£πΌπΌππ,π‘π‘β10
πΌπΌ2 = οΏ½ ππβπ£π£2/2 πππ£π£ β οΏ½ ππβπ£π£2/2 πππ£π£
πΌπΌππ,π‘π‘β1
0
πΌπΌππ,π‘π‘β1
0
= οΏ½ ππβπ£π£2/2 πππ£π£ β οΏ½ ππβπ’π’2/2 ππππ
πΌπΌππ,π‘π‘β1
0
πΌπΌππ,π‘π‘β1
0
= οΏ½ οΏ½ ππβ(π’π’2+π£π£2 )/2 ππππ πππ£π£
πΌπΌππ,π‘π‘β1
0
πΌπΌππ,π‘π‘β1
0
Using polar co-ordinate system,
ππ = ππππππππππ , π£π£ = ππ πππ π π π ππ
πΌπΌ2 = οΏ½ οΏ½ ππβππ2/2 ππππππππππ2ππ
0
β
0
πΌπΌ2 = 2ππ οΏ½ ππβππ2/2 ππππππ β
0
Let ππ2 = π€π€
Therefore, πππ€π€ = 2ππππππ π π . ππ. ππππππ = πππ€π€/2
12
πΌπΌ2 = 2ππ οΏ½ ππβπ€π€/2 πππ€π€/2 β
0
πΌπΌ2 = [β2ππβπ€π€2 ]0β = β2(0β1)
2= 1
β΄ πΌπΌ = β« ππβπ£π£2/2 πππ£π£πΌπΌππ,π‘π‘β10 = 1 (2.4)
Second, we solve β« π£π£ππβπ£π£2
2
πΌπΌππ,π‘π‘β10 πππ£π£
Let π£π£2
2= π₯π₯2
β΄ π₯π₯ = π£π£β2
πππ π ππ πππ₯π₯ = πππ£π£β2
π π . ππ. πππ£π£ = β2πππ₯π₯
β« π£π£ππβπ£π£2
2
πΌπΌππ,π‘π‘β10 πππ£π£ = β« β2π₯π₯πΌπΌππ,π‘π‘β1
0 β ππβπ₯π₯2β β2πππ₯π₯
= 2 οΏ½ π₯π₯
πΌπΌππ,π‘π‘β1
0
ππβπ₯π₯2πππ₯π₯
= [ππβπ₯π₯2]0πΌπΌππ,π‘π‘β1
= [ππβπ£π£2/2]0πΌπΌππ,π‘π‘β1
β« π£π£ππβπ£π£2
2
πΌπΌππ,π‘π‘β10 πππ£π£ = [1 β ππ
βπΌπΌππ,π‘π‘β12
2 ] (2.5)
Substituting (2.4) and (2.5) in (2.3), inventory holding cost is given by
οΏ½οΏ½βππβ2ππ
ππ
π‘π‘=1
ππ
ππ=1
οΏ½ πΌπΌπ π ,π‘π‘β1 β πππππ‘π‘ οΏ½1β ππβπΌπΌπ π ,π‘π‘β1
2
2 οΏ½ + Β΅πππ‘π‘ οΏ½ (2.6)
13
β’ Shortage cost: Total shortage cost is given by
οΏ½οΏ½ οΏ½ ππππππ
πΌπΌππ,π‘π‘β1
0
ππ
π‘π‘=1
ππ
ππ=1
β οΏ½π·π·πππ‘π‘ β πΌπΌππ,π‘π‘β1 οΏ½ β ππ (π·π·πππ‘π‘) β πππ·π·πππ‘π‘
= οΏ½οΏ½ππππππ οΏ½ π·π·πππ‘π‘ β ππ (π·π·πππ‘π‘) β πππ·π·πππ‘π‘
πΌπΌππ,π‘π‘β1
0
ππ
π‘π‘=1
ππ
ππ=1
β ππππππ β πΌπΌππ,π‘π‘β1 οΏ½ ππ (π·π·πππ‘π‘) β πππ·π·πππ‘π‘
πΌπΌππ,π‘π‘β1
0
= οΏ½οΏ½ππππππ οΏ½ π·π·πππ‘π‘ βππβ(π·π·πππ‘π‘ β Β΅πππ‘π‘)2
πππππ‘π‘2
πππππ‘π‘ β β2ππβ πππ·π·πππ‘π‘
πΌπΌππ,π‘π‘β1
0
ππ
π‘π‘=1
ππ
ππ=1
β ππππππ β πΌπΌππ,π‘π‘β1 οΏ½ ππβ(π·π·πππ‘π‘ β Β΅πππ‘π‘)2
πππππ‘π‘2
πππππ‘π‘ β β2ππβ πππ·π·πππ‘π‘
πΌπΌππ,π‘π‘β1
0
(2.7)
Let (π·π·πππ‘π‘ β Β΅πππ‘π‘)/πππππ‘π‘ = π£π£
Therefore, π·π·πππ‘π‘ = πππππ‘π‘ β π£π£ + Β΅πππ‘π‘ and πππ·π·πππ‘π‘ = πππππ‘π‘ β πππ£π£
Substituting in (2.7), we get
οΏ½οΏ½ππππππβ2ππ
οΏ½ (πππππ‘π‘ β π£π£ + Β΅πππ‘π‘
πΌπΌππ,π‘π‘β1
0
) β ππβπ£π£22
β πππ£π£ β ππππππ β πΌπΌπ π ,π‘π‘β1
β2ππ οΏ½ ππβπ£π£2/2 β πππ£π£
πΌπΌππ,π‘π‘β1
0
ππ
π‘π‘=1
ππ
ππ=1
(2.8)
= οΏ½οΏ½ππππππβ2ππ
ππ
π‘π‘=1
ππ
ππ=1
οΏ½πππππ‘π‘ οΏ½ π£π£ππβπ£π£22
πΌπΌππ,π‘π‘β1
0
πππ£π£ + Β΅πππ‘π‘ οΏ½ ππβπ£π£22
πΌπΌππ,π‘π‘β1
0
πππ£π£οΏ½ β ππππππ β πΌπΌππ,π‘π‘β1β2ππ
οΏ½ ππβπ£π£22
πππ£π£
πΌπΌππ,π‘π‘β1
0
Using results from (2.4) and (2.5), we get shortage cost as,
οΏ½οΏ½ππππππβ2ππ
ππ
π‘π‘=1
ππ
ππ=1
οΏ½ πππππ‘π‘ οΏ½1β ππβπΌπΌπ π ,π‘π‘β1
2
2 οΏ½ + Β΅πππ‘π‘ β πΌπΌππ,π‘π‘β1οΏ½ (2.9)
14
β’ Total cost: Total cost is the sum of fixed cost, purchasing cost, shipping cost, inventory
holding cost and shortage cost. The first objective is given by
πππ π π π ππ1 = β β β ππππππ β πΏπΏπππππ‘π‘πππ‘π‘=1
ππππ=1
ππππ=1 + β β β β π₯π₯πππππ‘π‘ππ β ππππππππ β πππππ‘π‘
ππππ=1
πππ‘π‘=1
ππππ=1
ππππ=1
+ β β β π₯π₯πππππ‘π‘ππ β ππππππ β πππππ‘π‘πππ‘π‘=1
ππππ=1
ππππ=1 +
β β βππβ2ππ
πππ‘π‘=1
ππππ=1 οΏ½ πΌπΌππ,π‘π‘β1 β πππππ‘π‘ οΏ½1 β ππ
βπΌπΌππ,π‘π‘β12
2 οΏ½ + Β΅πππ‘π‘ οΏ½ +
β β πππππ π β2ππ
πππ‘π‘=1
πππ π =1 οΏ½ πππ π π‘π‘ οΏ½1 β ππ
βπΌπΌππ,π‘π‘β12
2 οΏ½ + Β΅π π π‘π‘ β πΌπΌπ π ,π‘π‘β1 οΏ½ (2.10)
2.2.2 To minimize the number of defective parts
The product of quality rejection rate and quantity supplied is summed over all products,
suppliers and time periods. The number of defective parts should be minimum. The second
objective is given by
πππ π π π ππ2 = οΏ½οΏ½οΏ½ππππππ β π₯π₯πππππ‘π‘ππ β πππππ‘π‘
ππ
ππ=1
ππ
ππ=1
ππ
π‘π‘=1
(2.11)
2.2.3 To minimize weighted average lead time
The product of lead time of each part and quantity supplied is summed over all products,
suppliers and time periods and further divided by total order quantity to give weighted
average lead time. The weighted average lead time should be minimum. The third objective
is given by
πππ π π π ππ3 =β β β ππππππ β π₯π₯πππππ‘π‘ππ β πππππ‘π‘ππ
ππ=1ππππ=1
πππ‘π‘=1
β β πππππ‘π‘ππππ=1
πππ‘π‘=1
15
2.3 Constraints
Capacity constraints:
Capacity constraints ensure that the quantity ordered from each supplier in each period is
not more than supplierβs capacity. This constraint also ensures that binary variable Ξ΄ijt =1
in case supplier j is chosen in period t.
οΏ½π₯π₯πππππ‘π‘ππ β πππππ‘π‘ β€ πΆπΆπΆπΆππππππ β πΏπΏπππππ‘π‘ ππππππ ππππππ π π = 1, . . ,ππ ππ = 1, . . ,π π π‘π‘ = 1, . . ,ππππ
ππ=1
(2.13)
Price break constraints:
Price break constraints ensure that price discount is applied when the order quantity from
supplier j exceeds the price break quantity bijk for product i and price level k. This constraint
also ensures that binary variable yijk = 1 if price level k is chosen for part i for supplier j.
0 β€ π₯π₯πππππ‘π‘ππ β πππππ‘π‘ β€ ππππππππ β π¦π¦πππππ‘π‘ππ ππππππ ππππππ π π = 1, .ππ ππ = 1, . . ,π π π‘π‘ = 1, . . ,ππ ππ = 1, . . ,ππ (2.14)
οΏ½π¦π¦πππππ‘π‘ππ β€ 1 ππππππ ππππππ π π = 1, . . ,ππ ππ = 1, . . ,π π ππ
ππ=1
π‘π‘ = 1, . . ,ππ (2.15)
Demand constraints:
The following constraint is derived from the condition that inventory holding cost cannot
be negative.
16
πΌπΌππ,π‘π‘β1 β πππ π π‘π‘ οΏ½1 β ππβπΌπΌππ,π‘π‘β12
2 οΏ½ + Β΅π π π‘π‘ β₯ 0
(πΌπΌπ π ,π‘π‘β1 + Β΅πππ‘π‘ β πππππ‘π‘)πππππ‘π‘
β₯ β ππβπΌπΌπ π ,π‘π‘β1
2
2 (2.16)
The following constraint is derived from the condition that shortage cost cannot be
negative.
πππππ‘π‘ οΏ½1β ππβπΌπΌπ π ,π‘π‘β1
2
2 οΏ½+ Β΅πππ‘π‘ β πΌπΌππ,π‘π‘β1 β₯ 0
ππβπΌπΌππ,π‘π‘β12
2 β€ οΏ½πππ π π‘π‘ + Β΅π π π‘π‘ β πΌπΌπ π ,π‘π‘β1οΏ½
πππ π π‘π‘ (2.17)
Other constraints:
These constraints ensure that all decision variables are non-negative and all binary
variables are well defined.
οΏ½οΏ½π₯π₯πππππ‘π‘ππ = 1 ππππππ ππππππ π π = 1, . . ,ππ π‘π‘ = 1, β¦ ,ππ (2.18)ππ
ππ=1
ππ
ππ=1
π₯π₯πππππ‘π‘ππ, πΌπΌπππ‘π‘ , πππππ‘π‘ β₯ 0, πΏπΏπππππ‘π‘ ,π¦π¦πππππππ‘π‘ β (0,1) (2.19)
2.4 Weighted Objective Method
The idea behind formulation of this multi criteria problem is to assign weights to each
objective to transform the problem into a single weighted objective problem. An example
of formulation of weighted objective problem is given below (Masud and Ravindran, 2008)
17
πππ π π π ππ = β ππππππππ(π₯π₯)ππππ=1
πππππππππππππ‘π‘ π‘π‘ππ π₯π₯ β ππ
β ππππ = 1.ππππ=1
ππππ β₯ 0
where Ξ»i is the weight assigned to objective ππππ(π₯π₯)
and S = {x | gi (x) β€ 0, j= 1, 2,β¦.,m}
where gj (x) β€ 0 represents constraint j.
Under this method, the multi objective problem becomes a single objective problem
as follows,
πππ π π π ππ = π€π€1 β ππ1 + π€π€2 β ππ2 + π€π€3 β ππ3
where w1 = weight assigned to objective ππ1 (minimize total cost)
π€π€2 = weight assigned to objective ππ2 (minimize number of defective parts)
π€π€3 = weight assigned to objective ππ3 (minimize weighted average lead time)
and π€π€1 + π€π€2 + π€π€3 = 1
18
Chapter 3
Numerical Example and Sensitivity Analysis
3.1 Problem description:
A company wants to purchase 3 different parts for its assembly. There are 3 suppliers for
each part and all these suppliers offer 2 levels of price discounts for each part. The
following values are used as inputs.
β’ Number of parts=3 , m=3, i= {1,2,3}
β’ Number of suppliers=3, n=3, j= {1,2,3}
β’ Number of weeks in planning horizon=5, T=5, t= {1,2,3,4,5}
β’ Number of price levels offered by each supplier of each part=2, p=2, k= {1,2}
The following inputs are assumed and used in the numerical example
Product Supplier Fixed order
cost i j fij ($) 1 1 500 1 2 400 1 3 600 2 1 500 2 2 400 2 3 600 3 1 500 3 2 400 3 3 600
Table 3-1: Numerical Example: Fixed order cost related to parts and suppliers
19
Part
Inventory holding
cost Shortage
cost i hi ($) sci ($) 1 3 6 2 4.5 7.5 3 5 8
Table 3-2: Numerical Example: Inventory holding cost and shortage cost related to parts
Part Supplier Defect rate Capacity
Lead time
(weeks) Shipping cost ($)
i j qij CAPij lij Cij 1 1 0.04 1000 0.5 1.5 1 2 0.02 800 0.7 2 1 3 0.01 800 0.7 2.1 2 1 0.05 800 0.7 2 2 2 0.03 1000 0.6 1.8 2 3 0.02 600 0.8 1.5 3 1 0.04 600 0.5 2.5 3 2 0.03 900 0.6 1.9 3 3 0.02 700 0.5 2.5
Table 3-3: Numerical Example: Values of parameters related to suppliers and parts
20
Table 3-4: Numerical Example: Values of unit price and price break quantity
Table 3-5: Numerical Example: Weights of three objectives
Product Supplier Price break
level Price
break qty
Unit Price
($) i j k (bijk) Pijk 1 1 1 500 25 2 1000 20 1 2 1 500 22 2 800 19 1 3 1 500 30 2 800 25 2 1 1 450 18 2 800 15 2 2 1 600 22
2 1000 18 2 3 1 350 20 2 600 16 3 1 1 350 30 2 600 26 3 2 1 550 33 2 900 30 3 3 1 400 27
2 700 24
W1 W2 W3 0.7 0.2 0.1
21
Product Week Demand i t Dit 1 1 1500 1 2 1750 1 3 2000 1 4 1900 1 5 1600 2 1 1700 2 2 2100 2 3 2400 2 4 2200 2 5 2100 3 1 1800 3 2 2000 3 3 1800 3 4 2150 3 5 2000
Table 3-6: Numerical Example: Values of demand for each part for each week
Mean demand ΞΌ1t ΞΌ2t ΞΌ3t
1750 2100 1950
Table 3-7: Mean values of demand
Std. dev.of demand Ο1t Ο2t Ο3t 206 255 150
Table 3-8: Standard deviation values of demand
22
3.2 Mathematical model:
Decision variables:
xijtk : proportion of part i purchased from supplier j in period t at price level k, i=1, 2, 3 and j=1, 2, 3, t=1, 2,., 5 and k=1, 2
Oit : order quantity for part i in period t, i= 1, 2, 3 and t=1, 2, 3, 4, 5.
Iit : inventory level for part i at the end of period t, i=1, 2, 3 and t=1, 2, 3, 4, 5.
Sit : shortage quantity for part i at the end of period t, i=1, 2, 3 and t=1, 2, 3, 4, 5.
Ξ΄ijt =1, if order is placed for part i to supplier j for period t. i=1, 2, 3 and j=1, 2, 3 and t=1,2,3,4, 5.
=0, otherwise.
yijtk = 1, if price break level k is selected for part i from supplier j for period t.
=0, otherwise for i=1, 2, 3 and j=1, 2, 3 and t=1, 2, 3, 4, 5 and k=1, 2
Objective functions:
1) Minimize total cost over planning horizon
ππππππ ππ1 = οΏ½{5
π‘π‘=1
500 β (πΏπΏ11π‘π‘ + πΏπΏ21π‘π‘+ πΏπΏ31π‘π‘) + 400 β (πΏπΏ12π‘π‘
+ πΏπΏ22π‘π‘ + πΏπΏ32π‘π‘) + 600 β (πΏπΏ13π‘π‘ + πΏπΏ23π‘π‘ + πΏπΏ33π‘π‘} +
β [{(25 + 1.5)π₯π₯11π‘π‘1 + (20 + 1.5)π₯π₯11π‘π‘2 + (22 + 2)π₯π₯12π‘π‘1 5π‘π‘=1 + (19 + 2)π₯π₯12π‘π‘2 +
(30 + 2.1)π₯π₯13π‘π‘1+(25 + 2.1)π₯π₯13π‘π‘2}ππ1π‘π‘ + {(18 + 2)π₯π₯21π‘π‘1 + (15 + 2)π₯π₯21π‘π‘2 +(22 + 1.8)π₯π₯22π‘π‘1 + (18 + 1.8)π₯π₯22π‘π‘2 + (20 + 1.5)π₯π₯23π‘π‘1 + (16 + 1.5)π₯π₯23π‘π‘2}ππ2π‘π‘ + {(30 + 2.5)π₯π₯31π‘π‘1+(26 + 2.5)π₯π₯31π‘π‘2 + (33 + 1.9)π₯π₯32π‘π‘1 + (30 + 1.9)π₯π₯32π‘π‘2 +
(27 + 2.5)π₯π₯33π‘π‘1 + (24 + 2.5)π₯π₯33π‘π‘2}ππ3π‘π‘] + β [ 3β2ππ
5π‘π‘=1 οΏ½ πΌπΌ1,π‘π‘β1 β 206οΏ½1 β
ππβπΌπΌ1,π‘π‘β1
2
2 οΏ½ + 1750οΏ½ + 4.5β2ππ
οΏ½ πΌπΌ2,π‘π‘β1 β 255 οΏ½1 β ππβπΌπΌ2,π‘π‘β1
2
2 οΏ½ + 2100οΏ½ +
5β2ππ
οΏ½ πΌπΌ3,π‘π‘β1 β 150 οΏ½1 β ππβπΌπΌ3,π‘π‘β1
2
2 οΏ½ + 1950οΏ½] + β [ 6β2ππ
5π‘π‘=1 οΏ½ 206 οΏ½1 β
23
ππβπΌπΌ1,π‘π‘β1
2
2 οΏ½ + 1750 β πΌπΌ1,π‘π‘β1 οΏ½ + [ 7.5β2ππ
οΏ½ 255οΏ½1 β ππβπΌπΌ2,π‘π‘β1
2
2 οΏ½ + 2100 β πΌπΌ2,π‘π‘β1 οΏ½
+ 8β2ππ
οΏ½ 150οΏ½1 β ππβπΌπΌ3,π‘π‘β1
2
2 οΏ½ + 1950 β πΌπΌ3,π‘π‘β1 οΏ½]
2) Minimize number of defective parts
ππππππ ππ2 = οΏ½[{(π₯π₯11π‘π‘1 + π₯π₯11π‘π‘2)0.04 + (π₯π₯12π‘π‘1 5
π‘π‘=1
+ π₯π₯12π‘π‘2)0.02 + (π₯π₯13π‘π‘1
+ π₯π₯13π‘π‘2)0.01}ππ1π‘π‘ + {π₯π₯21π‘π‘1 + π₯π₯21π‘π‘2)0.05 + (π₯π₯22π‘π‘1 + π₯π₯22π‘π‘2)0.03
+ (π₯π₯23π‘π‘1 + π₯π₯23π‘π‘2)0.02}ππ2π‘π‘ + {(π₯π₯31π‘π‘1 + π₯π₯31π‘π‘2)0.04 + (π₯π₯32π‘π‘1
+ π₯π₯32π‘π‘2)0.03 + (π₯π₯33π‘π‘1
+ π₯π₯33π‘π‘2)0.02} ππ3π‘π‘]
3) Minimize weighted average lead time
ππππππ ππ3 = οΏ½[{(π₯π₯11π‘π‘1 + π₯π₯11π‘π‘2)0.5 + (π₯π₯12π‘π‘1 5
π‘π‘=1
+ π₯π₯12π‘π‘2)0.7 + (π₯π₯13π‘π‘1
+ π₯π₯13π‘π‘2)0.7}ππ1π‘π‘ + {π₯π₯21π‘π‘1 + π₯π₯21π‘π‘2)0.7 + (π₯π₯22π‘π‘1 + π₯π₯22π‘π‘2)0.6 + (π₯π₯23π‘π‘1+ π₯π₯23π‘π‘2)0.8}ππ2π‘π‘ + {(π₯π₯31π‘π‘1 + π₯π₯31π‘π‘2)0.5 + (π₯π₯32π‘π‘1 + π₯π₯32π‘π‘2)0.6 + (π₯π₯33π‘π‘1+ π₯π₯33π‘π‘2)0.5} ππ3π‘π‘] /
οΏ½{ππ1π‘π‘
5
π‘π‘ =1
+ ππ2π‘π‘ + ππ3π‘π‘ }
Constraints:
1) Capacity constraints:
οΏ½π₯π₯πππππ‘π‘ππ β πππππ‘π‘ β€ πΆπΆπΆπΆπΆπΆππππ β πΏπΏπππππ‘π‘ ππππππ ππππππ ππ = 1,2,3 ππ = 1,2,3 π‘π‘ = 1,2,3,4,52
ππ=1
24
2) Price break constraints:
0 β€ π₯π₯πππππ‘π‘ππ β πππππ‘π‘ β€ ππππππππ β π¦π¦πππππ‘π‘ππ ππππππ ππππππ ππ = 1,2,3 ππ = 1,2,3 π‘π‘ = 1,2,3,4,5 ππ= 1,2
οΏ½π¦π¦πππππ‘π‘ππ β€ 1 ππππππ ππππππ ππ = 1,2,3 ππ = 1,2,3 2
ππ=1
π‘π‘ = 1,2,3,4,5
3) Demand constraints:
From equation (2.16), we have
(πΌπΌππ,π‘π‘β1 + Β΅πππ‘π‘ β πππππ‘π‘)
πππππ‘π‘ β₯ β ππ
βπΌπΌππ,π‘π‘β122 ππππππ ππππππ ππ = 1,2,3 ππππππ π‘π‘ = 1,2,3,4,5
For t=1, i=1, we have Β΅πππ‘π‘ = 1750 andπππππ‘π‘ = 206. Initial inventory is assumed to
be 0.
(0+1750-206)/ (206) β₯ βππ0 . This constraint is satisfied.
From equation (2.17), we have
ππβπΌπΌππ,π‘π‘β12
2 β€ οΏ½πππππ‘π‘ + Β΅πππ‘π‘ β πΌπΌππ,π‘π‘β1οΏ½
πππππ‘π‘ ππππππ ππππππ ππ = 1,2,3 ππππππ π‘π‘ = 1,2,3,4,5
ππβπΌπΌππ,π‘π‘β12
2 β€ οΏ½πππππ‘π‘ + Β΅πππ‘π‘ β πΌπΌππ,π‘π‘β1οΏ½
πππππ‘π‘ ππππππππππππππ
For t=1, i=1, we have Β΅πππ‘π‘ = 1750 andπππππ‘π‘ = 206. ππ0 β€ (206 + 1750-0) / (206). This constraint is satisfied.
25
4) Other constraints:
οΏ½ οΏ½π₯π₯πππππ‘π‘ππ = 1 ππππππ ππππππ ππ = 1,2,3 ππππππ π‘π‘2
ππ=1
3
ππ=1
= 1,2,3,4,5.
π₯π₯πππππ‘π‘ππ , πΌπΌπππ‘π‘, πππππ‘π‘ β₯ 0, πΏπΏπππππ‘π‘,π¦π¦πππππππ‘π‘ β (0,1)
Solutions:
For order proportion xijtk
x1111 x1121 x1131 x1141 x1151 x1112 x1122 x1132 x1142 x1152 0 0 0 0 0 0.47 0.54 0.5 0.53 0.5
x1211 x1221 x1231 x1241 x1251 x1212 x1222 x1232 x1242 x1252 0 0 0 0 0 0.53 0.46 0.4 0.42 0.5
x1311 x1321 x1331 x1341 x1351 x1312 x1322 x1332 x1342 x1352 0 0 0.1 0.05 0 0 0 0 0 0
x2111 x2121 x2131 x2141 x2151 x2112 x2122 x2132 x2142 x2152 0 0 0 0 0 0.44 0.38 0.33 0.36 0.38
x2211 x2221 x2231 x2241 x2251 x2212 x2222 x2232 x2242 x2252 0 0 0 0 0 0.35 0.33 0.42 0.36 0.33
x2311 x2321 x2331 x2341 x2351 x2312 x2322 x2332 x2342 x2352 0 0 0 0 0 0.21 0.29 0.25 0.28 0.29
x3111 x3121 x3131 x3141 x3151 x3112 x3122 x3132 x3142 x3152 0 0 0 0 0 0.31 0.3 0.31 0.27 0.3
x3211 x3221 x3231 x3241 x3251 x3212 x3222 x3232 x3242 x3252 0 0 0 0 0 0.31 0.35 0.31 0.4 0.35
x3311 x3321 x3331 x3341 x3351 x3312 x3322 x3332 x3342 x3352 0 0 0 0 0 0.38 0.35 0.38 0.33 0.35
Table 3-9: Order proportions
26
For order quantity Oit
Table 3-10: Order quantities
For binary variable Ξ΄ijt
Table 3-11: Selection of suppliers
O11 O12 O13 O14 O15 1500 1750 2000 1900 1600 O21 O22 O23 O24 O25
1700 2100 2400 2200 2100 O31 O32 O33 O34 O35
1800 2000 1800 2150 2000
Ξ΄111 Ξ΄112 Ξ΄113 Ξ΄114 Ξ΄115 1 1 1 1 1 Ξ΄121 Ξ΄122 Ξ΄123 Ξ΄124 Ξ΄125 1 1 1 1 1 Ξ΄131 Ξ΄132 Ξ΄133 Ξ΄134 Ξ΄135 0 0 1 1 0 Ξ΄211 Ξ΄212 Ξ΄213 Ξ΄214 Ξ΄215 1 1 1 1 1 Ξ΄221 Ξ΄222 Ξ΄223 Ξ΄224 Ξ΄225 1 1 1 1 1 Ξ΄231 Ξ΄232 Ξ΄233 Ξ΄234 Ξ΄235 1 1 1 1 1 Ξ΄311 Ξ΄312 Ξ΄313 Ξ΄314 Ξ΄315 1 1 1 1 1 Ξ΄321 Ξ΄322 Ξ΄323 Ξ΄324 Ξ΄325 1 1 1 1 1 Ξ΄331 Ξ΄332 Ξ΄333 Ξ΄334 Ξ΄335 1 1 1 1 1
27
For binary variable y ijtk
y 1111 y1121 y 1131 y 1141 y 1151 y 1112 y 1122 y 1132 y 1142 y 1152
0 0 0 0 0 1 1 1 1 1
y 1211 y1221 y 1231 y 1241 y 1251 y 1212 y 1222 y 1232 y 1242 y 1252
0 0 0 0 0 1 1 1 1 1
y 1311 y1321 y 1331 y 1341 y 1351 y 1312 y 1322 y 1332 y 1342 y 1352
0 0 1 1 0 0 0 0 0 0
y 2111 y2121 y 2131 y 2141 y 2151 y 2112 y 2122 y 2132 y 2142 y 2152
0 0 0 0 0 1 1 1 1 1
y 2211 y2221 y 2231 y 2241 y 2251 y 2212 y 2222 y 2232 y 2242 y 2252
0 0 0 0 0 1 1 1 1 1
y 2311 y2321 y 2331 y 2341 y 2351 y 2312 y 2322 y 2332 y 2342 y 2352
0 0 0 0 0 1 1 1 1 1 y 3111 y3121 y 3131 y 3141 y 3151 y 3112 y 3122 y 3132 y 3142 y 3152
0 0 0 0 0 1 1 1 1 1 y 3211 y3221 y 3231 y 3241 y 3251 y 3212 y 3222 y 3232 y 3242 y 3252
0 0 0 0 0 1 1 1 1 1 y 3311 y3321 y 3331 y 3341 y 3351 y 3312 y 3322 y 3332 y 3342 y 3352
0 0 0 0 0 1 1 1 1 1
Table 3-12: Selection of price levels
28
Objective 1: Z1 =Total cost = $ 815116.6
Objective 2: Z2 =Number of defective parts= 914
Objective 3: Z3 =Weighted Average lead time: 3.05 weeks
From table 3-5, we have weights for each objective as follows
W1 = 0.7, W2 = 0.2, W3 = 0.1
Objective function value= W1 * Z1 + W2 * Z2 + W3 * Z3
= (0.7*815116.6) + (0.2 * 914) + (0.1*3.05)
= 570764.6
29
3.3 Sensitivity analysis
1) When the values for standard deviation are decreased by 10 % and the mean is kept the
same as original example, we get following input values.
Std. dev.of demand Ο1t Ο2t Ο3t 206 255 150
Table 3-13: Original values of Standard deviation of demand
Table 3-14: New values of Standard deviation of demand
All other inputs are kept the same. The optimal solution is as follows,
For order proportion xijtk
x1111 x1121 x1131 x1141 x1151 x1112 x1122 x1132 x1142 x1152
0 0 0 0 0 0.54 0.54 0.48 0.53 0.52 x1211 x1221 x1231 x1241 x1251 x1212 x1222 x1232 x1242 x1252
0 0 0 0 0 0.46 0.46 0.4 0.42 0.48 x1311 x1321 x1331 x1341 x1351 x1312 x1322 x1332 x1342 x1352
0 0 0.12 0.05 0 0 0 0 0 0 x2111 x2121 x2131 x2141 x2151 x2112 x2122 x2132 x2142 x2152
0 0 0 0 0 0.42 0.38 0.36 0.39 0.42 x2211 x2221 x2231 x2241 x2251 x2212 x2222 x2232 x2242 x2252
0 0 0 0 0 0.32 0.33 0.36 0.32 0.32
Std. dev.of demand Ο1t Ο2t Ο3t 185 230 135
30
x2311 x2321 x2331 x2341 x2351 x2312 x2322 x2332 x2342 x2352
0 0 0 0 0 0.26 0.29 0.28 0.27 0.26 x3111 x3121 x3131 x3141 x3151 x3112 x3122 x3132 x3142 x3152
0 0 0 0 0 0.31 0.3 0.28 0.29 0.26 x3211 x3221 x3231 x3241 x3251 x3212 x3222 x3232 x3242 x3252
0 0 0 0 0 0.31 0.35 0.4 0.31 0.33 x3311 x3321 x3331 x3341 x3351 x3312 x3322 x3332 x3342 x3352
0 0 0 0 0 0.38 0.35 0.32 0.4 0.41
Table 3-15: Order proportions
For order quantity Oit
Table 3-16: Order quantities
For binary variable Ξ΄ijt
Ξ΄111 Ξ΄112 Ξ΄113 Ξ΄114 Ξ΄115 1 1 1 1 1 Ξ΄121 Ξ΄122 Ξ΄123 Ξ΄124 Ξ΄125 1 1 1 1 1 Ξ΄131 Ξ΄132 Ξ΄133 Ξ΄134 Ξ΄135 0 0 1 1 0
O11 O12 O13 O14 O15 1750 1750 2100 1900 1650
O21 O22 O23 O24 O25 1900 2100 2200 2050 1900
O31 O32 O33 O34 O35 1800 2000 2150 1750 1700
31
Ξ΄211 Ξ΄212 Ξ΄213 Ξ΄214 Ξ΄215 1 1 1 1 1 Ξ΄221 Ξ΄222 Ξ΄223 Ξ΄224 Ξ΄225 1 1 1 1 1 Ξ΄231 Ξ΄232 Ξ΄233 Ξ΄234 Ξ΄235 1 1 1 1 1 Ξ΄311 Ξ΄312 Ξ΄313 Ξ΄314 Ξ΄315 1 1 1 1 1 Ξ΄321 Ξ΄322 Ξ΄323 Ξ΄324 Ξ΄325 1 1 1 1 1 Ξ΄331 Ξ΄332 Ξ΄333 Ξ΄334 Ξ΄335 1 1 1 1 1
Table 3-17: Selection of suppliers
For binary variable y ijtk
y 1111 y1121 y 1131 y 1141 y 1151 y 1112 y 1122 y 1132 y 1142 y 1152
0 0 0 0 0 1 1 1 1 1
y 1211 y1221 y 1231 y 1241 y 1251 y 1212 y 1222 y 1232 y 1242 y 1252
0 0 0 0 0 1 1 1 1 1
y 1311 y1321 y 1331 y 1341 y 1351 y 1312 y 1322 y 1332 y 1342 y 1352
0 0 1 1 0 0 0 0 0 0
y 2111 y2121 y 2131 y 2141 y 2151 y 2112 y 2122 y 2132 y 2142 y 2152
0 0 0 0 0 1 1 1 1 1
y 2211 y2221 y 2231 y 2241 y 2251 y 2212 y 2222 y 2232 y 2242 y 2252
0 0 0 0 0 1 1 1 1 1
32
y 2311 y2321 y 2331 y 2341 y 2351 y 2312 y 2322 y 2332 y 2342 y 2352
0 0 0 0 0 1 1 1 1 1
y 3111 y3121 y 3131 y 3141 y 3151 y 3112 y 3122 y 3132 y 3142 y 3152
0 0 0 0 0 1 1 1 1 1
y 3211 y3221 y 3231 y 3241 y 3251 y 3212 y 3222 y 3232 y 3242 y 3252
0 0 0 0 0 1 1 1 1 1
y 3311 y3321 y 3331 y 3341 y 3351 y 3312 y 3322 y 3332 y 3342 y 3352
0 0 0 0 0 1 1 1 1 1
Table 3-18: Selection of price levels
Objective 1: Z1 =Total cost = $ 699834.1
Objective 2: Z2 =Number of defective parts= 811
Objective 3: Z3 =Weighted Average lead time: 3.01 weeks
Objective Original value New value Percentage
decrease
Total cost $815,116.60 $699,834.10 14.14% Total number of quality defects
914 811 11.26%
Weighted average lead
time 3.05 weeks 3.01 weeks 1.31%
Table 3-19: Comparison of values of objectives
33
From table 3-5, we have weights for each objective as follows
W1 = 0.7, W2 = 0.2, W3 = 0.1
Objective function value= W1 * Z1 + W2 * Z2 + W3 * Z3
= (0.7*699834.1) + (0.2 * 811) + (0.1*3.01)
= 490046.4
The percentage decrease in the value of objective function is 570764.6β490046.4570764.6
= 14.12 %
This decrease in the value of objective function is probably because of the decrease in the
standard deviation of demand. Therefore, decrease in the values of standard deviation of
demand by 10% leads to decrease in the value of objective function by 14.12 %, keeping
all other input parameters constant.
2) When the input values for mean of demand are increased by 10% and values for
standard deviation are kept same as original example, we have the following inputs,
Mean demand ΞΌ1t ΞΌ2t ΞΌ3t
1750 2100 1950
Table 3-20: Original Values of Mean of demand
Table 3-21: New values of Mean of demand
Mean demand ΞΌ1t ΞΌ2t ΞΌ3t
1925 2310 2145
34
All other inputs are maintained same. The solution changes as follows.
For order proportion xijtk
x1111 x1121 x1131 x1141 x1151 x1112 x1122 x1132 x1142 x1152 0 0 0 0 0 0.43 0.54 0.38 0.56 0.46
x1211 x1221 x1231 x1241 x1251 x1212 x1222 x1232 x1242 x1252 0 0 0 0 0 0.57 0.46 0.38 0.44 0.54
x1311 x1321 x1331 x1341 x1351 x1312 x1322 x1332 x1342 x1352 0 0 0 0 0 0 0 0.24 0 0
x2111 x2121 x2131 x2141 x2151 x2112 x2122 x2132 x2142 x2152 0 0 0 0 0 0.44 0.42 0.36 0.44 0.62
x2211 x2221 x2231 x2241 x2251 x2212 x2222 x2232 x2242 x2252 0 0 0 0 0 0.35 0.32 0.36 0.33 0
x2311 x2321 x2331 x2341 x2351 x2312 x2322 x2332 x2342 x2352 0 0 0 0 0 0.21 0.26 0.28 0.23 0.38
x3111 x3121 x3131 x3141 x3151 x3112 x3122 x3132 x3142 x3152 0 0 0 0 0 0.24 0.32 0.28 0.29 0.42
x3211 x3221 x3231 x3241 x3251 x3212 x3222 x3232 x3242 x3252 0 0 0 0 0.1 0.33 0.32 0.4 0.31 0
x3311 x3321 x3331 x3341 x3351 x3312 x3322 x3332 x3342 x3352 0 0 0 0 0 0.43 0.36 0.32 0.4 0.48
Table 3-22: Order proportions
For order quantity Oit
Table 3-23: Order quantities
O11 O12 O13 O14 O15 1400 1750 2100 1800 1500
O21 O22 O23 O24 O25 1700 1900 2200 1800 1300
O31 O32 O33 O34 O35 1650 1900 2150 1750 1450
35
For binary variable Ξ΄ijt
Ξ΄111 Ξ΄112 Ξ΄113 Ξ΄114 Ξ΄115 1 1 1 1 1
Ξ΄121 Ξ΄122 Ξ΄123 Ξ΄124 Ξ΄125 1 1 1 1 1
Ξ΄131 Ξ΄132 Ξ΄133 Ξ΄134 Ξ΄135 0 0 1 0 0
Ξ΄211 Ξ΄212 Ξ΄213 Ξ΄214 Ξ΄215 1 1 1 1 1
Ξ΄221 Ξ΄222 Ξ΄223 Ξ΄224 Ξ΄225 1 1 1 1 0
Ξ΄231 Ξ΄232 Ξ΄233 Ξ΄234 Ξ΄235
1 1 1 1 1
Ξ΄311 Ξ΄312 Ξ΄313 Ξ΄314 Ξ΄315
1 1 1 1 1
Ξ΄321 Ξ΄322 Ξ΄323 Ξ΄324 Ξ΄325
1 1 1 1 1
Ξ΄331 Ξ΄332 Ξ΄333 Ξ΄334 Ξ΄335
1 1 1 1 1
Table 3-24: Selection of suppliers
36
For binary variable y ijtk
y 1111 y1121 y 1131 y 1141 y 1151 y 1112 y 1122 y 1132 y 1142 y 1152 0 0 0 0 0 1 1 1 1 1
y 1211 y1221 y 1231 y 1241 y 1251 y 1212 y 1222 y 1232 y 1242 y 1252 0 0 0 0 0 1 1 1 1 1
y 1311 y1321 y 1331 y 1341 y 1351 y 1312 y 1322 y 1332 y 1342 y 1352 0 0 0 0 0 0 0 0 0 0
y 2111 y2121 y 2131 y 2141 y 2151 y 2112 y 2122 y 2132 y 2142 y 2152 0 0 0 0 0 1 1 1 1 1
y 2211 y2221 y 2231 y 2241 y 2251 y 2212 y 2222 y 2232 y 2242 y 2252 0 0 0 0 0 1 1 1 1 1
y 2311 y2321 y 2331 y 2341 y 2351 y 2312 y 2322 y 2332 y 2342 y 2352 0 0 0 0 0 1 1 1 1 1
y 3111 y3121 y 3131 y 3141 y 3151 y 3112 y 3122 y 3132 y 3142 y 3152 0 0 0 0 0 1 1 1 1 1
y 3211 y3221 y 3231 y 3241 y 3251 y 3212 y 3222 y 3232 y 3242 y 3252 0 0 0 0 1 1 1 1 1 0
y 3311 y3321 y 3331 y 3341 y 3351 y 3312 y 3322 y 3332 y 3342 y 3352 0 0 0 0 0 1 1 1 1 1
Table 3-25: Selection of price levels
Objective 1: Z1 =Total cost = $ 928322.2
Objective 2: Z2 =Number of defective parts= 1004
Objective 3: Z3 =Weighted Average lead time: 3.11 weeks
37
Objective Original value New value Percentage
increase
Total cost $815,116.60 $928,322.20 13.88%
Total number of quality defects
914 1004 9.84%
Weighted average lead
time
3.05 weeks 3.11 weeks 1.96%
Table 3-26: Comparison of values of objectives
From table 3-5, we have weights for each objective as follows
W1 = 0.7, W2 = 0.2, W3 = 0.1
Objective function value= W1 * Z1 + W2 * Z2 + W3 * Z3
= (0.7*928322.2) + (0.2 * 1004) + (0.1*3.11)
= 650026.6
The percentage increase in the value of objective function is 650026.6β570764.6570764.6
= 13.85 %
This increase in the value of objective function is probably because of the increase in the
mean values of demand. Therefore, the value of objective function increases by 13.85%
when values of mean of demand are increased by 10%, keeping all other input parameters
constant.
38
Chapter 4
Summary, Conclusions and scope for future research
In this chapter, we present a summary of the thesis and conclusions from the research
work. This research work can be extended by changing the assumptions or by adding new
parameters to the existing problem to increase the complexity. Scope for future research is
also presented in this chapter.
4.1 Summary and Conclusions
In this thesis, a multi criteria decision making model is developed for supplier selection
and order allocation. The primary objective of this model is to help the buyer make
decisions regarding choice of suppliers in case of procured products. The multi criteria
model is developed for the case of multiple products, multiple suppliers and multiple time
periods procurement activity. Each supplier is assumed to provide all-unit type of discount
for each part. The problem is formulated by considering three objectives: the total cost over
planning horizon, the number of defective parts procured from the supplier and the
weighted average lead time of the supplier. This problem is solved using weighted
objective method.
A numerical example is solved following the model formulation to illustrate the
functioning of the model. The supplier selection and order allocation problem is solved by
assuming that there are three parts to be procured. Each part can be supplied by three
different suppliers. This problem is solved by assuming that the buyer has to plan for five
weeks of procurement i.e. there are five weeks in the planning horizon. Further, each
supplier is assumed to provide two price levels for each part based on the quantity ordered.
39
It is observed that the total cost, the number of quality defects and the weighted average
lead time decreased when the standard deviation of the demand was decreased, keeping all
other input parameters constant. Next, there is increase in the total cost, the number of
quality defects and the weighted average lead time when the mean values of the demand
increased. There is no change in the selection of suppliers with the changes in above
mentioned inputs. A change is observed in selection of price levels offered by the suppliers
when the demand for each week changed. It is observed that the model chose to procure
maximum quantity from the supplier that offered the best price before choosing the next
best supplier.
4.2 Scope for future research
The following can be considered to extend the work done in this thesis.
β’ Each supplier is assumed to provide all-unit type of discount. However, in reality,
some other types of discounts such as business volume discount and incremental
discount are offered by the suppliers.
β’ In reality, each supplier has different transportation alternatives (TA) for shipping
the parts to the buyer. Each transportation alternative offers different cost and
different values for lead time. The model can be extended by considering
transportation alternatives for each supplier.
β’ Selection of suppliers that are located globally increases the supply chain risk due
to delays in shipping or due to delays as a result of occasional natural hazards. The
multi criteria model can be extended by adding an objective of minimizing the
supply chain risk.
40
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