Assessment 2 (Individual Assignment) Assessment 2 (Individual Assignment) There are a number of typical models in the Operations Research field which can be applied to a wide range of supply chain...

Table of Contents
1 Introduction    2
2 Description    2
2.1 FORMULATING PROBLEM    2
3 Background    3
3.1 Problem Statement    4
3.2 Solution    4
3.3 Balancing an Unbalanced Transportation Problem    5
3.3.a Excess Supply    5
3.3.b Unmet Demand    5
3.4    Finding Best Feasible Solution (BFS)    6
3.4.a Loop    6
4 Design and analysis of supply chain    8
5 Summary    9
6 REFERENCES    9
1 Introduction
An optimization and integer programming combinatorial problem known as the vehicle routing problem (VRP) was proposed by Dantzig and Ramser in 1959. The main purpose was to service many customers with a vehicles fleet. It is a benchmark problem used to find solutions in the areas of transportation, distribution, and logistics.
2 Description
The Vehicle Routing Problem (VRP) is one of the better known operational research problem where supply of customers’ demands is satisfied through one or several depots. Discovering a set of delivery routes fulfilling certain requirements or constraints and at minimal total cost is the main objective. The VRP plays a vital role in planning of distribution systems transportation and logistics for many sectors like garbage collection, mail delivery, or task sequencing.
There are many examples of VRP like the one for school bus routing. Here,a fleet of n vehicles serves m schools whose students are scattered over a location. They need to be picked from their stops and dropped at their respective schools. We have a common starting point for vehicles.
Vehicle routes are formed in such a way that:
1. Minimizes the distance (or time) travelled (variable costs);
2. Minimizes the quantity of the vehicles (fixed costs);
3. Minimizes combination of quantity of the vehicles used and total distance (or time) travelled
With the constraints that
· a vehicle visits each stop exactly once;
· number of students never exceed vehicle carrying capacity;
2.1 FORMULATING PROBLEM
The information specified by the transportation problem:
A set of m supply points from which a good/service is shipped. Supply point i can supply at most si units
A set of n demand points to which the good/service is shipped. Demand point j must receive at least dj units.
Each unit produced at supply point i and shipped to demand point j incurs a variable cost of cij.
The relevant data can be formulated in a transportation tableau:
    
    
    
    
    
    
    
    Demand Point 1
    Demand Point 2
    
    Demand Point n
    SUPPLY
    Supply Point 1
    c11
    c12
     
    c1n
    s1
    Supply Point 2
    c21
    c22
     
    c2n
    s2
    
     
     
     
     
    
    Supply Point m
    cm1
    cm2
     
    cmn
    sm
    DEMAND
    d₁
    d₂
    
    dn
    
If, total supply = total demand
then the problem is said to be a balanced transportation problem.
Let xij = number of units shipped from supply point i to demand point j
Decision variable xij: number of units shipped from supply point i to
demand point j
then the general LP representation of a transportation problem is
min Σi∑j cijxij
such that.
Σi xij ≤ si (i=1,2, ..., m)         Supply constraints
and
Σi xij ≥ dj (j=1,2, ..., n)         Demand constraints
Also,
xij ≥ 0
If a problem has the constraints given above then it is a transportation maximization problem
3 Background
Relating to current scenario of nCOVID-19 pandemic and the case of lockdowns consultancy study was undertaken on a hypothetical large mail order company (Let’s assume as ABCorp.) which operated by taking postal or telephonic orders from the general public in Australia and doorstep delivery of the goods was done either by:
· the manufacturer; or
· national postal service; or
· from company depots by self-operated delivery vehicles.
A large proportion of the deliveries were through company operated vehicles (from a number of depots) and it was this portion of the company's operation that was examined. For the purposes of our study 3 depots in Melbourne were chosen as a target for closer analysis.
As per the current situation, the company needs to deliver to every home. Thus, the cost of providing services to people has increased and attempting to design vehicle routes at the individual home level would incur more overheads.
However in Australia, as in many other countries, the country has been divided up into a number of cities, based on alpha-numeric postcodes. For example the postcode for RMIT is VIC 3053. Postcodes are widely used throughout Australia and so aggregation was done on that basis.
3.1 Problem Statement
ABCorp has three depots that supply the needs of four cities. Each depot can supply the following numbers of packages: depot 1, 35 K; depot 2, 50 K; and depot 3, 40 k. The peak delivery demands in these cities is as follows (in K's): city 1, 45 K; city 2, 20 K; city 3, 30 K; city 4, 30 K. The costs of sending 1000 packages from depot to city is given in the table below.
    From
    To
    
    City 1
    City 2
    City 3
    City 4
    Depot 1
    A$8
    A$6
    A$10
    A$9
    Depot 2
    A$9
    A$12
    A$13
    A$7
    Depot 3
    A$14
    A$9
    A$16
    A$5
3.2 Solution
To minimize the cost of meeting each city’s peak demand, a balanced transportation problem was formulated in a transportation tableau and represented as a LP model.
Representation of the problem as a LP model
xij: number of (in 1000’s) packages received at depot i and sent to city j.
min z = 8(x11) + 6(x12) + 10(x13) + 9(x14) + 9(x21) + 12(x22) + 13(x23) + 7(x24) + 14(x31) + 9(x32) + 16(x33) + 5(x34)
Such that, the supply constraints are
x11 + x12 + x13 + x14 ≤ 35
x21 + x22 + x23 + x24 ≤ 50
x31 + x32 + x33 + x34 ≤ 40
and the demand constraints are
x11 + x21 + x31 ≥ 45
x12 + x22 + x32 ≥ 20
x13 + x23 + x33 ≥ 30
x14 + x24 + x34 ≥ 30
and
xij ≥ 0 (i = 1,2,3 and j = 1,2,3,4)
Formulating the transportation problem
    
    City 1
    City 2
    City 3
    City 4
    SUPPLY
    Depot 1
    8
    6
    10
    9
    35
    Depot 2
    9
    12
    13
    7
    50
    Depot 3
    14
    9
    16
    5
    40
    DEMAND
    45
    20
    30
    30
    125
Since total supply = total demand, it is a balanced transportation problem.
3.3 Balancing an Unbalanced Transportation Problem
3.3.a Excess Supply
In such a situation when total supply exceeds total demand, a penalty is often associated with unmet demand. The transportation problem than can be balanced by creating a dummy demand point that has a demand equal to the amount of excess supply. Since shipments to the dummy demand point are not real shipments, they are assigned a cost of zero. These shipments indicate unused supply capacity.
3.3.b Unmet Demand
If total demand exceeds total supply then there is no feasible solution.

3.3.b.(i) Scenario 1 Modifed ABCorp for Excess Supply
Supposing the demand for city 1 is 40K. Thus, formulating a balanced transport problem.
Total demand = 120K and
Total supply = 125K
To balance the problem, we need to add a dummy point with a demand of
125K – 12K = 5K packages
From each depot, the cost of shipping 1K packages to the dummy is 0. So, the updated table is as follows
    
    City 1
    City 2
    City 3
    City 4
    Dummy
    SUPPLY
    Depot 1
    8
    6
    10
    9
    0
    35
    Depot...