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MULTI-STAGE SUPPLY NETWORK WITH CONSIDERATION OF CROSS-BORDER

IMPACTS - DELAY TIME AT CROSS BORDER

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Course

Date

Abstract

Since the formulation of NAFTA agreement trade has been blooming between Canada

and the United States, however, there is still a gap that requires fulfillment. The gap is because of

the buffer time at the cross-border between the two countries, which has resulted in high costs

incurred by all the stakeholders involved in the supply chain network. In this research, an integer

linear programming mathematical model is designed with the aim of bringing the gap and come

up with the suitable solutions. The model seeks to determine whether construction of new

facilities or dismantling of some will optimize the supply chain costs extensively. The outcomes

of the research with the incorporation of multiple parameters recommended that dismantling of

the facilities would be the possible solution in optimizing the costs. The report will go into

details on how each parameter was calculated, the findings and all the recommendations drawn

from the results.

Background Brief

Ever since the passage of the CUSFTA (Canada-U.S. Free Trade Agreement) as well as

NAFTA (North American Free Trade Agreement), trade has experienced explosive growth. So

far, the many objectives of both agreements have been met especially when it comes to

eliminating the barriers to trade before the two agreements. The latter was the main hindrance to

trade between the two countries. Some of the benefits of the agreement according to NAFTA is

that it has helped accelerated growth regarding the economy as well as enhancing the purchasing

powers of different suppliers as well as purchasers of the goods and services. The other objective

is that it has created a more comprehensive market competition while at the same time bringing

forth diversification of products and services.

According to the gap illustrated in the abstract or in other words the cross-border buffer

time problem, it is important to note that trucks are the primary mode of transportation for the

NAFTA goods. It accounts for about 60 percent of the shipment, which makes them a vital part

of the entire supply chain system. The buffer time at the borders, however, has been increasing

day after day due to various factors one of them being the events of September 11, 2001. This led

to more scrutiny at the borders increasing the buffer time. The delays have therefore held cost

for both suppliers and the manufacturers affecting their operation. It is estimated that the costs

related to both postponements as well as uncertainty at the cross-border clearance are $4.02

billion.

It is important to note that the effects are experienced by the truckers too who incur

Taylor, John C., Douglas R. Robideaux, and George C. Jackson. "US-Canada transportation

and logistics: border impacts and costs, causes, and possible solutions." Transportation

Journal (2004): 5-21.

costs amounting to almost $ 290 million per year, this is because the delays incur them nearly 33

minutes per each shipment job they take. The losses incurred during the cross-border

transportation is high, and therefore the mathematical model developed for this problem will

seek to find out the demand of goods to be supplied to come up with the best supply management

design regarding facility setup or closing of some.

Research Objective and Mathematical Model Formulation

The buffer time at the cross-border between Canada and the United States has accelerated

the number of losses that the manufacturers, suppliers, as well as truckers, incur. For this

reasons, the primary objective of this study is to optimize the total cost incurred by the supply

chain network that exists at the cross-border. The approach that will be effective in tackling the

objective is the Integer Linear Programming mathematical model.

The approach is significant

and will be used since it will help in deciding the optimal quantities of goods that should be

produced by each facility as well as shipped by them. The approach is also essential in

determining the production capacity to be installed at each service at a particular period; this is

because the model is capacitated dynamic facility location allocation.

In any multi-period problem, it is important to note that the demand of any customer is

dynamic and as a result may change with time. For this reason, the mathematical model

Manzini, Riccardo, and Elisa Gebennini. "Optimization models for the dynamic facility

location and allocation problem." International Journal of Production Research 46, no. 8 (2008):

2061-2086.

illustrates above can develop as well as dismantle any installed production capacity with the goal

of optimizing the production capacity from any given facility. The model is also relevant since it

affects the cost components of the entire supply chain network. These costs components include

cross-border as well as domestic transportation costs and costs associated with the dismantling as

well as the construction of the capacities. The excel solver considers the supply chain cost

parameters that are essential in determining the optimized quantities of production. The model

will operate within the permissible international values and limit the output of carbon IV oxide.

The above costs, as well as constraints, are the most vital portions that must be considered to

have a final report.

Parameters

The problem under study requires two main stages, which include identifying the various

relevant parameters, setting up of decision variables and developing an objective function, which

will optimize the cross-border supply chain network.

The second stage, on the other hand, will

include the development of the mathematical model that will help in the acquisition of the

quantity of goods that require transportation between the facilities and the customers. The same

will cover the corresponding costs associated with both. The model as already mentioned earlier

will help in the calculation of the total expenses related to the construction as well as the

dismantling of the facilities for an optimized solution. The model develops; therefore, consist of

Anderson, Bill. "The border and the Ontario economy." Cross-Border Transportation Centre,

University of Windsor, Windsor, ON (Accessed online 23-12-2017 at: www. uvvindsor.

ca/crossborder/system/...[I'he_Border_Reprot2012. pdf)(2012).

different locations denoted by V and consumers of the products indicated by U. The two are

presumed to spread throughout Canada and U.S.

The locations mentioned are believed to have a default production capacity indicated by

. It is of importance to note that the capacity specified can be constructed or dismantled

depending on the predefined capacities, which are based on the ever-changing demands of the

customers. However, during the construction and dismantling of any capacity costs are incurred.

The demand parameters under the model are believed to be deterministic and have the capability

of changing from time to time (Note than only demand and not supply is deterministic and that

having the capability to change does not mean it changes; it only means that it is flexible if need

be). During this times demand is fixed and therefore the supply must always be equal or higher

than demand itself. The parameter does not only incorporate the above parameters but also take

into considerations the scenario parameter, for example, should an orange alert exist in

Ambassador Bridge. Ps. therefore denotes the probability of this scenario to occur. The model is

thus developed in a manner that any given buffer time contains several scenarios (S). Buffer time

can be a single day or even a year depending on the period under study, however, for our model,

the buffer time is not fixed and therefore the users many include whichever periods seem fit for

them. Hence the following parameters are used to denote the bufffer time for customer demand,

i, t.

The goal of the thesis is to optimize the quantity of goods to be produced as well as

shipped. At the same time, it seeks to minimize the entire cost of the supply chain network and

find solutions whether a capacity requires construction or dismantling. The parameter used for

shipping and which are necessary for Z

j, I, s, t.

Other parameters that are useful in the developed model are d

j, I, t.

That represents the

transportation cost and production value of a single unit in a year from location j to a customer I

for a buffer time t. E1 which is a set of all domestic routes is represented by the following

parameter {(j, i): j ∈ 1 V, i ∈ 𝑈1 𝑜𝑟 j ∈ 2 V, I ∈ U2} while that of cross border routes is E2 =

{(j, i): j ∈ 1 V, i ∈ U2 𝑜𝑟 j ∈ 2 V, i ∈ U1}. E

on the other hand is the parameter for increase in

terms of cost when crossing border at s scenario. Fixed cost required for setting up a facility at a

given location is a

j, k, t,

and that of dismantling is ấ

j, k, t.

Decision Variables

Binary variables are defined as follows:

a) Z

j, I, s, t.

= this is the quantity of customer (j) demand transported from the plant in S

scenario, taking into consideration buffer time t. ∀ 𝑖 ∈𝑈, 𝑗 ∈ 𝑉, s ∈ 𝑆, t ∈ 𝑇

b)Y

j, k, t

= (1: set up of plant k, at location j, taking in consideration buffer time at the cross

border t, t,∀ j ∈V, k ∈ K, t ∈ 𝑇, 0: otherwise)

c)𝑦 ̂𝑗,𝑘,𝑡= (1: Dismantling a plant at location j taking into consideration the buffer time at the

cross border t, t,∀ j ∈V, k ∈ K, t ∈ 𝑇, 0: otherwise)

Indices

i: Customers, i ∈ U

j: Locations , j ∈ V

s: Scenarios, s ∈ S

k: Capacities, k ∈ K

t: Buffer time at cross-border, t ∈ T

Constraints

∑

∈

𝑍

j, s,t

≥ ℎ

i,t

× 𝑝

∀ 𝑖 ∈ 𝑈,𝑡 ϵ 𝑇 ,𝑠 ϵ 𝑆

∑

∈

∑

∈

j,I,s,t

≤ 𝐻 ̂

+∑

t ′ ≤ t

∑

k ϵ K

𝑦

𝑗

𝑘

𝑡

′

𝐻

𝑘

-∑

t ′ ≤ t

∑

k ϵ K

𝑦 ̂

𝑗

𝑘

𝑡

𝐻

𝑘

∀𝑗 𝜖 𝑉,t 𝜖 𝑇

∑

i ϵ

𝑈

∑

s ϵ S

𝑍

𝑗

𝑖

𝑠

𝑡

× 𝑒

𝑗

𝑖

≤ Ω ∀ 𝑗 𝜖 𝑉, 𝑡 𝜖 𝑇

∑

𝑘

∈𝐾

𝑦 ̂

𝑗

𝑡

≤ 1 ∀ 𝑗 𝜖 𝑉, 𝑡 𝜖 𝑇

∑

𝑘∈𝐾

𝑦

𝑗

𝑡

≤ 1 ∀ 𝑗 𝜖 𝑉, 𝑡 𝜖 𝑇

𝑍

𝑗

𝑠

𝑡

≥ 0 ∀ 𝑖 𝜖 𝑈, 𝑗 𝜖 𝑉, s 𝜖 𝑆 , 𝑡 𝜖 𝑇

𝑦

𝑗

𝑡

∈ {0,1} ∀ 𝑗 𝜖 𝑉, 𝑡 𝜖 𝑇,𝑘 𝜖 𝐾

𝑦 ̂

𝑗

𝑡

∈ {0,1} ∀ 𝑗 𝜖 𝑉, 𝑡 𝜖 𝑇,𝑘 𝜖 𝐾

∑

𝑘

∈𝐾

𝑦 ̂

𝑗

𝑡

×𝐻

𝑘

≤ 𝐻 ̂

𝑗

∀ 𝑗 ∈ 𝑉

Objective Mathematical Function

Construction or Dismantling of Facilities

∑

𝑡

𝜖

𝑇

{ ∑

𝑗

𝜖

𝑉

∑

𝑘

𝜖

𝐾

𝑎

𝑗

𝑘

𝑡

𝑦

𝑗

𝑘

𝑡

+ ∑

𝑘

𝜖

𝐾

∑

𝑗

𝜖

𝑉

𝑗

𝑘

𝑡

𝑦 ̂

𝑗

𝑘

𝑡

+ ∑

(

𝑗

𝑖

)ϵ

𝐸

∑

s ϵ S

𝑑

𝑗

𝑖

𝑡

𝑍

j,i, s,t

+ ∑

(

𝑗

𝑖

)ϵ

𝐸

∑

s ϵS

𝑑

𝑗

𝑖

𝑡

𝑍

𝑗

𝑖

𝑠

𝑡

1+𝑒

)}+ ∑

i ϵ

𝑈

∑

s ϵ S

∑

𝑗

ϵ V

∑

𝑡

ϵ T

𝑍

𝑗

𝑖

𝑠

𝑡

× 𝑒

𝑗

𝑖

×𝛼 - 𝐸 ×𝛼

Solution Procedures

The procedures indicated here have one primary objective, and that is to explain the

application of the model in real life and how it helps the supply chain managers. The first

solution with gives a brief solution on how to meet the demand for any given scenario at the

cross-border. Some of the assumptions made include the duration of each scenario and the period

under study. These are one day and one year respectively. The total number of scenarios is,

therefore, s1 + s2 = S which is equivalent to the number of days N.

The expected number of days in a year for a probable scenario =. N*Ps

Daily Supply = ∑ 𝑍𝑗, 𝑖, 𝑠, 𝑡 𝑗∈𝑉/ N ∗ P𝑠

Daily Demand = ℎ

t / 𝑁

Since supply is equal or greater than demand, the following statement is correct:

∑ 𝑍𝑗, 𝑖, 𝑠, 𝑡 𝑗∈𝑉/ N ∗ P𝑠 ≥ ℎ

t / 𝑁

At any given scenario, it is important to note that scenario demand constraint in added, and this

results in the equation below

∑ Z

𝑗

𝑠

𝑡

ϵ V ≥ ℎ

t × Ps

In conclusion to the above brief solution, it is therefore of importance that each scenario

attracts different costs when crossing the border regarding buffer time. The equation, therefore,

includes the parameter e

which is the additional costs. The model develops comes up with a

solution on the number of goods that should be produced and shipped for each probable scenario.

Bibliography

Anderson, Bill. "The border and the Ontario economy." Cross-Border Transportation

Centre, University of Windsor, Windsor, ON (Accessed online 23-12-2017 at:

www. uvvindsor. ca/crossborder/system/...[I'he_Border_Reprot2012.

pdf)(2012).

Manzini, Riccardo, and Elisa Gebennini. "Optimization models for the dynamic facility

location and allocation problem." International Journal of Production

Research 46, no. 8 (2008): 2061-2086.

Taylor, John C., Douglas R. Robideaux, and George C. Jackson. "US-Canada

transportation and logistics: border impacts and costs, causes, and possible

solutions." Transportation Journal (2004): 5-21.

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