Case:
Eight supermarkets in different popular locations in Dubai are willing to sign a contract for one year with Nestle Factory. They request a specific demand to be delivered weekly on Saturday directly to their stores with a particular time window. Nestle factory management in Dubai received the requests and assigned a team to study and fulfill the requirements.
The team started by finding the distance and time matrix for the eight locations: Al Mamzar, Mirdif, Al Barsha, Port Saeed, Jumeirah, Al Quoz, and Al Khwaneej and Al Muraqqabat.
The team assigned four homogenous vehicles to fulfill the customers’ demand with a capacity of 215 water bottles for each vehicle.
The distance matrix is as follows:
■(Nestle Factory@Al Mamzer@Mirdif@Al Barsha@Port Saeed@Jumeirah@Al Quoz@Al Khawaneej@Al Muraqqabat) ■(0@1@2@3@4@5@6@7@8) (■(0&51&49&24&45&36&29&57&46@51&0&12&28&6&15&22&18&5@49&12&0&25&10&18&21&9&11@24&28&25&0&22&13&6&33&23@45&6&10&22&0&10&16&18&2@36&15&18&13&10&0&7&27&11@29&22&21&6&16&7&0&29&17@57&18&9&33&18&27&29&0&19@46&5&11&23&2&11&17&19&0))
Distance in Km Depot Al Mamzar Mirdif Al Barsha Port Saeed Jumeirah Al Quoz Al Khwaneej Al Muraqqabat
Depot 0 51 49 24 45 36 29 57 46
Al Mamzar 51 0 12 28 6 15 22 18 5
Mirdif 49 12 0 25 10 18 21 9 11
Al Barsha 24 28 25 0 22 13 6 33 23
Port Saeed 45 6 10 22 0 10 16 18 2
Jumeirah 36 15 18 13 10 0 7 27 11
Al Quoz 29 22 21 6 16 7 0 29 17
Al Khwaneej 57 18 9 33 18 27 29 0 19
Al Muraqqabat 46 5 11 23 2 11 17 19 0
The time matrix is as follows:
■(Nestle factory@Al Mamzer@Mirdif@Al Barsha@Port Saeed@Jumeirah@Al Quoz@Al Khawaneej@Al Muraqqabat) ■(0@1@2@3@4@5@6@7@8) (■(0&38&37&18&34&27&22&43&35@38&0&9&21&5&11&17&14&4@37&9&0&19&8&14&16&7&8@18&21&19&0&17&10&5&25&17@34&5&8&17&0&8&12&14&2@27&11&14&10&8&0&5&20&8@22&17&16&5&12&5&0&22&13@43&14&7&25&14&20&22&0&14@35&4&8&17&2&8&13&14&0))
Time in minutes Depot Al Mamzar Mirdif Al Barsha Port Saeed Jumeirah Al Quoz Al Khwaneej Al Muraqqabat
Depot 0 38 37 18 34 27 22 43 35
Al Mamzar 38 0 9 21 5 11 17 14 4
Mirdif 37 9 0 19 8 14 16 7 8
Al Barsha 18 21 19 0 17 10 5 25 17
Port Saeed 34 5 8 17 0 8 12 14 2
Jumeirah 27 11 14 10 8 0 5 20 8
Al Quoz 22 17 16 5 12 5 0 22 13
Al Khwaneej 43 14 7 25 14 20 22 0 14
Al Muraqqabat 35 4 8 17 2 8 13 14 0
The demand and time window of each customer is as follows:
Locations Demand ( bottles) Time window
Al Mamzer 355 08:00 am to 10:00 am
Mirdif 925 08:00 am to 10:00 am
Al Barsha 150 10:00 am to 12:00 pm
Port Saeed 332 10:00 am to 12:00 pm
Jumeirah 263 12:00 pm to 02:00 pm
Al Quoz 418 12:00 pm to 02:00 pm
Al Khawaneej 306 02:00 pm to 04:00 pm
Al Muraqqabat 810 02:00 pm to 04:00 pm
Linear Programming Formulation
Conceptual goal
Nestle’s team goal is to minimize the traveled distance by each vehicle to fulfill the customer demand at their time window.