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COST REDUCTION IN LOGISTICS
An Application of Linear Optimisation Modeling
Author: Anthony Mok
Date: 16 Nov 2023
Email: xxiaohao@yahoo.com
AGENDA
3
Linear Optimisation Modeling - An
Introduction
4 Project’s Primary Goals
5 Context
6 Dataset
9 Strategies For Modelling
11 Data Analysis
13 SE Queries - Findings & Conclusion
LINEAR OPTIMISATION MODELING
Linear optimisation modeling is a technique
that involves creating a mathematical model
of a problem which defines the:
• Decision variables: Things that can be
controlled, like the amount of furniture to
produce or number of employees to hire
• Objective function: What are to be
optimised, like maximising profit or
minimising cost
• Constraints: Limitations imposed on the
decision variables, like available resources
The model is solved by using specialised
algorithms to uncover the values of the
decision variables that best achieve objective
function while satisfying all the constraints.
Linear Optimisation Modeling 3 2023
PROJECT’S
PRIMARY GOALS
Finding Best Method of Transporting Goods from
Social Enterprise’s Outlets to Retailers
CONTEXT
Linear Optimisation Modeling 5 2023
Background: Invited by a local Social Enterprise (SE) to provide, as skill-base
volunteerism, to solve a logistical problem
Problem: Determine optimal quantity of Product X to be delivered from each SE’s outlets
to different retailers at minimum transportation cost
SE’s Outlets: Delivers from 3 outlets - in Jurong, Alexandra, & Tuas
Retailers: Delivers to 6 retailers - in Jurong, Alexandra, Tampines, Yishun, Changi,
Bishan, & Woodlands
Request: Applies linear optimisation modeling to find optimal quantity of Project X to be
delivered where it will be able to minimise the transportation cost significantly, which will
result in increased profitability
2023
Linear Optimisation Modeling 6
Retailers SE’s Outlets
Jurong Alexandra Tuas
Jurong 2 5 4
Alexandra 8 3 9
Tampines 7 15 5
Yishun 11 16 10
Changi 6 7 5
Bishan 7 16 12
Woodlands 10 14 6
Transportation Cost
Table below shows the average transportation costs per carton of Product X moving from SE’s outlets
to retailers:
Table 1- Average transportation costs per carton of Product X (All units in S$)
DATASET
2023
Linear Optimisation Modeling 7
Storage Capacity
Table below shows the capacity of the different outlets of the Social Enterprise:
Table 2 - Capacity of different outlets of the Social
Enterprise (All units in number of cartons)
DATASET
SE’s Outlets
Jurong 3980
Alexandra 1785
Tuas 4856
2023
Linear Optimisation Modeling 8
Demand
Table below shows the average demand of different retailers
Table 3 - Average demand of different retailers (All units are
in number of cartons)
DATASET
Retailers
Jurong 1168
Alexandra 1560
Tampines 1439
Yishun 986
Changi 1658
Bishan 2035
Woodlands 1159
SET-UP OF LINEAR OPTIMISATION MODEL
Linear Optimisation Modeling 9 2023
SET-UP OF SOLVER PARAMETERS
Linear Optimisation Modeling 10 2023
OUTCOMES FROM SOLVER APPLICATION
Linear Optimisation Modeling 11 2023
Subjected to all storage capacity (Slide 7) and demand constraints (Slide 8), the optimal values of
the Decision Variables on the number of cartons of Product X to be transported from the SE’s
outlets to the retailers are:
$53,972 is the minimised value of the Objective Function
OUTCOMES FROM SOLVER APPLICATION
Linear Optimisation Modeling 12 2023
Binding Constraints
Demand requirements for all seven retailers
are binding constraints: the optimal solution
must satisfy these demand requirements
exactly; reduce the demand for any of these
retailers, the optimal solution would change
Non-binding Constraints
Two of the demand requirements are non-
binding: the optimal solution does not need to
satisfy these requirements exactly; even if
quantities are reduced at these two outlets,
the optimal solution might not change. The
Jurong outlet has a slack of 389, which means
that it has 389 more units of Product X than
are required to meet the demand
QUERY (1) FROM SE - SENSITIVITY ANALYSIS
Linear Optimisation Modeling 13 2023
The sensitivity range for the demand requirements for the retailer at Bishan is between 0 and 2,424
cartons (2035 – Allowable Decrease @ 2035, & 2035 + Allowable Increase @ 389, respectively).
Since the decrease is to 1,735 cartons, from 2,035 cartons, it continues to fall within this range.
Given that the Shadow Price is +$7 within this range, a saving in costs of $2,100 ($7/carton * 300
cartons) will have to be subtracted from the current total cost of $53,972, which is $51,872. Since
less cartons are transported, there should be a fall in the current total cost of transportation. The
Shadow Price does not suggest any lose owing to the fall in economies of scale now that the bulk of
the transportation has become smaller
What would be the impact on the model if the average demand at Bishan (B6) is reduced from 2035 to
1735 cartons while keeping other parameters the same?
QUERY (2) FROM SE - SENSITIVITY ANALYSIS
Linear Optimisation Modeling 14 2023
The sensitivity range for the storage capacity for the Social Enterprise’s outlet at Tuas is between
4,467 and 5,244 cartons (4856 – Allowable Decrease @ 389, & 4856 + Allowable Increase @ 388,
respectively). As the increase is 5,306 cartons, from 4,856 cartons, it is 62 cartons more than the
allowable 5,244 cartons. This is outside this sensitivity range. There is a need to recalculate the current
Linear Optimisation Model to find the new Decision Variables and Objective Function
What would be the impact on the model if the available capacity of the Tuas Outlet (A3) is increased
from 4856 to 5306 cartons while keeping the other parameters the same?
Linear Optimisation Modeling 15 2023
The optimal values of the Decision Variables on the number of cartons to be transported from the 2 SE outlets at Jurong and Tuas to the
retailer at Changi have changed. The SE outlet at Jurong will not ship the 388 cartons of Product X to the retailer at Changi, while the SE outlet
at Tuas will ship more cartons to completely make up the 388 cartons not being transported by the SE outlet at Jurong. This is because it is
cheaper, by $1 per carton, to move products from Tuas to Changi. This is to minimise the total transportation costs
By doing recalculation using the Solver, the following are uncovered:
Owing to this, $53,584 is now the new minimised value of the Objective
Function; a saving of transportation costs of $388 (a saving of $1/carton *
388 cartons shipped) from the original minimised value of the Objective
Function of $53,972
QUERY (2) FROM SE - SENSITIVITY ANALYSIS
QUERY (3) FROM SE - SENSITIVITY ANALYSIS
Linear Optimisation Modeling 16 2023
Based on the Sensitivity Report of the original Linear Optimisation Model; the allowable decrease of
the objective coefficient, that is the average cost of transporting one carton from the SE outlet at
Alexandra to the retailer at Yishun, is $5. The average transportation cost per carton for this route now
falls from $16 to $7, which is a drop of $9. This is beyond the allowable decrease in average shipping
cost of $5. So, there is a need to conduct a recalculation of the original Linear Optimisation Model
What would be the impact on the model if the average transportation costs per carton of Product X
from Alexandra (A2) to Yishun (B4) decreases from $16 to $7: what is the optimal quantity (number
of cartons) of the product transferred from Alexandra (A2) to Yishun (B4)? How is the total cost
affected?
QUERY (3) FROM SE - SENSITIVITY ANALYSIS
Linear Optimisation Modeling 17 2023
Here are the findings from this rework:
The new minimised value of the Objective Function is different from the original; from $53,972 to
$53,072, a saving of costs of $900
QUERY (3) FROM SE - SENSITIVITY ANALYSIS
Linear Optimisation Modeling 18 2023
This change in the Objective Function is caused by 4 shifts in the optimal values of the Decision
Variables:
• Transfer of cartons of Product X from Alexandra (A2) to Yishun (B4) has increased from 0 cartons to
225 cartons
• Shipment from Tuas (A3) to Yishun (B4) has dropped from 989 cartons to 764 cartons
• Jurong (A1) is now moving 163 cartons to Changi (B5) instead of 388 cartons
• Finally, Tuas’s (A3) transfers 1,495 cartons to Changi (B5) instead of 1,270
QUERY (3) FROM SE - SENSITIVITY ANALYSIS
Linear Optimisation Modeling 19 2023
• Finally, given these shifts, the storage capacity of Jurong (A1) is now 3,366 cartons, instead of 3,591,
and the storage capacity of Alexandra (A2) is 1,785 cartons where previously this was 1,560; a total of
225 cartons have been switched from Jurong to Alexandra. While the storage capacity at Tuas (A3)
has not changed, there are switches in number of cartons shipped from this company warehouse to
Yishun and Changi
• All in all, it is more compelling to ship from Alexandra (A2) to Yishun (B4) and Tuas (A3) to Changi
(B5) because of this fall in transportation cost from Alexandra
• The Solver is suggesting these adjustments because of significant drop in cost of transportation from
Alexandra (A2) to Yishun (B4); a savings of $9 ($16 - $7)
• From these, the contribution of the saving of $900 (= $(+1,575 - 2,250 -1,350 + 1,125) from the new
minimised value of the Objective Function, ie. to be deducted from the original minimised optimal
value of the original Objective Function
COST REDUCTION IN LOGISTICS
An Application of Linear Optimisation Modeling
Author: Anthony Mok
Date: 16 Nov 2023
Email: xxiaohao@yahoo.com

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Optimal Decision Making - Cost Reduction in Logistics

  • 1. COST REDUCTION IN LOGISTICS An Application of Linear Optimisation Modeling Author: Anthony Mok Date: 16 Nov 2023 Email: xxiaohao@yahoo.com
  • 2. AGENDA 3 Linear Optimisation Modeling - An Introduction 4 Project’s Primary Goals 5 Context 6 Dataset 9 Strategies For Modelling 11 Data Analysis 13 SE Queries - Findings & Conclusion
  • 3. LINEAR OPTIMISATION MODELING Linear optimisation modeling is a technique that involves creating a mathematical model of a problem which defines the: • Decision variables: Things that can be controlled, like the amount of furniture to produce or number of employees to hire • Objective function: What are to be optimised, like maximising profit or minimising cost • Constraints: Limitations imposed on the decision variables, like available resources The model is solved by using specialised algorithms to uncover the values of the decision variables that best achieve objective function while satisfying all the constraints. Linear Optimisation Modeling 3 2023
  • 4. PROJECT’S PRIMARY GOALS Finding Best Method of Transporting Goods from Social Enterprise’s Outlets to Retailers
  • 5. CONTEXT Linear Optimisation Modeling 5 2023 Background: Invited by a local Social Enterprise (SE) to provide, as skill-base volunteerism, to solve a logistical problem Problem: Determine optimal quantity of Product X to be delivered from each SE’s outlets to different retailers at minimum transportation cost SE’s Outlets: Delivers from 3 outlets - in Jurong, Alexandra, & Tuas Retailers: Delivers to 6 retailers - in Jurong, Alexandra, Tampines, Yishun, Changi, Bishan, & Woodlands Request: Applies linear optimisation modeling to find optimal quantity of Project X to be delivered where it will be able to minimise the transportation cost significantly, which will result in increased profitability
  • 6. 2023 Linear Optimisation Modeling 6 Retailers SE’s Outlets Jurong Alexandra Tuas Jurong 2 5 4 Alexandra 8 3 9 Tampines 7 15 5 Yishun 11 16 10 Changi 6 7 5 Bishan 7 16 12 Woodlands 10 14 6 Transportation Cost Table below shows the average transportation costs per carton of Product X moving from SE’s outlets to retailers: Table 1- Average transportation costs per carton of Product X (All units in S$) DATASET
  • 7. 2023 Linear Optimisation Modeling 7 Storage Capacity Table below shows the capacity of the different outlets of the Social Enterprise: Table 2 - Capacity of different outlets of the Social Enterprise (All units in number of cartons) DATASET SE’s Outlets Jurong 3980 Alexandra 1785 Tuas 4856
  • 8. 2023 Linear Optimisation Modeling 8 Demand Table below shows the average demand of different retailers Table 3 - Average demand of different retailers (All units are in number of cartons) DATASET Retailers Jurong 1168 Alexandra 1560 Tampines 1439 Yishun 986 Changi 1658 Bishan 2035 Woodlands 1159
  • 9. SET-UP OF LINEAR OPTIMISATION MODEL Linear Optimisation Modeling 9 2023
  • 10. SET-UP OF SOLVER PARAMETERS Linear Optimisation Modeling 10 2023
  • 11. OUTCOMES FROM SOLVER APPLICATION Linear Optimisation Modeling 11 2023 Subjected to all storage capacity (Slide 7) and demand constraints (Slide 8), the optimal values of the Decision Variables on the number of cartons of Product X to be transported from the SE’s outlets to the retailers are: $53,972 is the minimised value of the Objective Function
  • 12. OUTCOMES FROM SOLVER APPLICATION Linear Optimisation Modeling 12 2023 Binding Constraints Demand requirements for all seven retailers are binding constraints: the optimal solution must satisfy these demand requirements exactly; reduce the demand for any of these retailers, the optimal solution would change Non-binding Constraints Two of the demand requirements are non- binding: the optimal solution does not need to satisfy these requirements exactly; even if quantities are reduced at these two outlets, the optimal solution might not change. The Jurong outlet has a slack of 389, which means that it has 389 more units of Product X than are required to meet the demand
  • 13. QUERY (1) FROM SE - SENSITIVITY ANALYSIS Linear Optimisation Modeling 13 2023 The sensitivity range for the demand requirements for the retailer at Bishan is between 0 and 2,424 cartons (2035 – Allowable Decrease @ 2035, & 2035 + Allowable Increase @ 389, respectively). Since the decrease is to 1,735 cartons, from 2,035 cartons, it continues to fall within this range. Given that the Shadow Price is +$7 within this range, a saving in costs of $2,100 ($7/carton * 300 cartons) will have to be subtracted from the current total cost of $53,972, which is $51,872. Since less cartons are transported, there should be a fall in the current total cost of transportation. The Shadow Price does not suggest any lose owing to the fall in economies of scale now that the bulk of the transportation has become smaller What would be the impact on the model if the average demand at Bishan (B6) is reduced from 2035 to 1735 cartons while keeping other parameters the same?
  • 14. QUERY (2) FROM SE - SENSITIVITY ANALYSIS Linear Optimisation Modeling 14 2023 The sensitivity range for the storage capacity for the Social Enterprise’s outlet at Tuas is between 4,467 and 5,244 cartons (4856 – Allowable Decrease @ 389, & 4856 + Allowable Increase @ 388, respectively). As the increase is 5,306 cartons, from 4,856 cartons, it is 62 cartons more than the allowable 5,244 cartons. This is outside this sensitivity range. There is a need to recalculate the current Linear Optimisation Model to find the new Decision Variables and Objective Function What would be the impact on the model if the available capacity of the Tuas Outlet (A3) is increased from 4856 to 5306 cartons while keeping the other parameters the same?
  • 15. Linear Optimisation Modeling 15 2023 The optimal values of the Decision Variables on the number of cartons to be transported from the 2 SE outlets at Jurong and Tuas to the retailer at Changi have changed. The SE outlet at Jurong will not ship the 388 cartons of Product X to the retailer at Changi, while the SE outlet at Tuas will ship more cartons to completely make up the 388 cartons not being transported by the SE outlet at Jurong. This is because it is cheaper, by $1 per carton, to move products from Tuas to Changi. This is to minimise the total transportation costs By doing recalculation using the Solver, the following are uncovered: Owing to this, $53,584 is now the new minimised value of the Objective Function; a saving of transportation costs of $388 (a saving of $1/carton * 388 cartons shipped) from the original minimised value of the Objective Function of $53,972 QUERY (2) FROM SE - SENSITIVITY ANALYSIS
  • 16. QUERY (3) FROM SE - SENSITIVITY ANALYSIS Linear Optimisation Modeling 16 2023 Based on the Sensitivity Report of the original Linear Optimisation Model; the allowable decrease of the objective coefficient, that is the average cost of transporting one carton from the SE outlet at Alexandra to the retailer at Yishun, is $5. The average transportation cost per carton for this route now falls from $16 to $7, which is a drop of $9. This is beyond the allowable decrease in average shipping cost of $5. So, there is a need to conduct a recalculation of the original Linear Optimisation Model What would be the impact on the model if the average transportation costs per carton of Product X from Alexandra (A2) to Yishun (B4) decreases from $16 to $7: what is the optimal quantity (number of cartons) of the product transferred from Alexandra (A2) to Yishun (B4)? How is the total cost affected?
  • 17. QUERY (3) FROM SE - SENSITIVITY ANALYSIS Linear Optimisation Modeling 17 2023 Here are the findings from this rework: The new minimised value of the Objective Function is different from the original; from $53,972 to $53,072, a saving of costs of $900
  • 18. QUERY (3) FROM SE - SENSITIVITY ANALYSIS Linear Optimisation Modeling 18 2023 This change in the Objective Function is caused by 4 shifts in the optimal values of the Decision Variables: • Transfer of cartons of Product X from Alexandra (A2) to Yishun (B4) has increased from 0 cartons to 225 cartons • Shipment from Tuas (A3) to Yishun (B4) has dropped from 989 cartons to 764 cartons • Jurong (A1) is now moving 163 cartons to Changi (B5) instead of 388 cartons • Finally, Tuas’s (A3) transfers 1,495 cartons to Changi (B5) instead of 1,270
  • 19. QUERY (3) FROM SE - SENSITIVITY ANALYSIS Linear Optimisation Modeling 19 2023 • Finally, given these shifts, the storage capacity of Jurong (A1) is now 3,366 cartons, instead of 3,591, and the storage capacity of Alexandra (A2) is 1,785 cartons where previously this was 1,560; a total of 225 cartons have been switched from Jurong to Alexandra. While the storage capacity at Tuas (A3) has not changed, there are switches in number of cartons shipped from this company warehouse to Yishun and Changi • All in all, it is more compelling to ship from Alexandra (A2) to Yishun (B4) and Tuas (A3) to Changi (B5) because of this fall in transportation cost from Alexandra • The Solver is suggesting these adjustments because of significant drop in cost of transportation from Alexandra (A2) to Yishun (B4); a savings of $9 ($16 - $7) • From these, the contribution of the saving of $900 (= $(+1,575 - 2,250 -1,350 + 1,125) from the new minimised value of the Objective Function, ie. to be deducted from the original minimised optimal value of the original Objective Function
  • 20. COST REDUCTION IN LOGISTICS An Application of Linear Optimisation Modeling Author: Anthony Mok Date: 16 Nov 2023 Email: xxiaohao@yahoo.com