Multi-depot Vehicle Routing Problem with Split Delivery and Time Windows

  Bulletin of Mathematics Vol. 08, No. 02 (2016), pp. 169–177.

  

Multi-depot Vehicle Routing Problem with Split

Delivery and Time Windows

_Abstract.

Shaflina Izar, Saib Suwilo, and Sutarman

Multi-depot vehicle routing problem with split delivery and time win-

dows (MDVRPSDTW) is a model that minimize out-cost by the company and im-

proves quality of service period. We built this model from a previous model that is a

multi-depot split-delivery vehicle routing problem model (MDVRPSD). This model

developed by the addition of delivery deadline or delivery time previously agreed be-

tween company and there customer so that this model becomes a multi-depot vehicle

routing problem with split delivery and time windows. On the time contradiction

we bring up penalty charge, where the company contravene the agreed deadline.

1 INTRODUCTION

  The classic Vehicle Routing Problem (VRP) model is a distributed model sourced from one depot to many customers. Every customer has an order must be served by exactly one vehicle where the total number of orders delivered by a vehicle must not exceed the capacity of a carrier vehicle, every vehicle is assumed to have a capacity almost the same, the route traveled by the vehicle must begin and end in the same depot. The objective function of this model is to minimize the cost of the vehicle travels. This model was introduced by Danzig and Ramser [3]. Many variations VRP models have been introduced, some of them are savings by split delivery Received 25-11-2016, Accepted 23-12-2016.

  2000 Mathematics Subject Classification: 15A23, 15A48

Key words and Phrases: Multi-Depot, Vehicle Routing Problems, Split-Delivery, Time Windows.

  Shaflina, Suwilo & Sutarman – Vehicle Routing Problems

  routing transpor Dror and Trudeu [6], lagrangian duality applied to the vehicle routing problems with time windows Kallehauge [5], vehicle routing problem split delivery Archetti and Speranza [2], multi-depot vehicle routing problem with time windows considering delivery and installation vehicle Bae and Moon [4], etc. Most of researchers focusing his research on minimizing the cost of all the routes. In this article we intend to develop one of the variations of the VRP model with calculate the delivery time by vehicle to the consumer so that there is time improvement service.

2 VEHICLE ROUTING PROBLEM

  Vehicle Routing Problem (VRP) is used to present or determine the optimal route which vehicle through to serve all customers, in this case cus- tomers and there requests have been know for some time, where all existing customers are already registered and the starting point of departure is a warehouse or depot. The constraint is the capacity of each vehicle. Its ob- jective function is to minimize the total cost (function of capacity and total route or time) in serving customers. This VRP model is able to reduce the cost issued at 5 − 20% of the total cost of transportation normally issued [9]. Set:

  I : Set of delivery nodes

  J : Set of depot nodes N

  : Set of all nodes, N = I ∪ J K : Set of delivery vehicle k

  : index of vehicle Parameters:

  C abk : fixed cost of the delivery person for vehicle k from a to b d

  a : demand at customer a

  x : number of boolean ( 1 if vehicle k delivery the demand

  abk

  from a to b) _{X X} min C x

  ab ab (1) _P a∈N b∈N

  x

  ab = 1 ∀ b ∈ N

  (2)

  a∈N _P

  x = 1 ∀ a ∈ N

  ab

  (3)

  b∈N Shaflina, Suwilo & Sutarman – Vehicle Routing Problems _P

  x a = K (4)

  a∈N _P

  x K =

  0b

  (5) _{X X} b∈N x ≥ r(i) ∀ I ⊆ N {0}, I 6= φ (6)

  ab a∈N b∈N

  The objective function (1) indicate the total cost of the route, con- straint (2) and (3) indicate each customers must be served by one vehicle, constrain (4) and (5) indicate that vehicles K must be arrive to and depart from the depot, and (6) indicate the minimal number of vehicles required to serve the set of customer.

3 MULTI DEPOT SPLIT DELIVERY

VEHICLE ROUTING PROBLEM

  Multi depot split delivery vehicle routing problem (MDSDVRP) is a variant of VRP, this model is a distributed model from many depot to many customer and the customer can be served by more than one vehicles. Constrain of this model is depot assignment, route and the total vehicle have to operate. Problems of vehicle routing problem is usually applied to complete graph. In a complete directed graph G = (V, E) on the transport network, it should be noted c is the initial cost matrix input from the

  ab

  finance function (depends on various parameters). For each pair of points, _{′ ′ ′} we extend the transport network on the graph from G to G = (V , E ). _′ Where the placement of point 0 in the transport of free graph so V = V ∪ {0}. This way has been introduced in 2011 by Yu [1]. Point 0 used as the virtual point must be chosen carefully as it is the depot point, part of the same problem as route model. Added new path between point 0 and customer is a set of two pairs of the value of the cost (c ao and c ob ) the establishment of fees for point b is overview (EC b )where EC b = c ob . From this we will obtain the objective function c ab as follows: _ c _{′ } ab if {a, b} ⊂ V c = EC b if a = 0 and b ∈ V (7)

  ab _

  c

  ao if b = 0 and a ∈ V Shaflina, Suwilo & Sutarman – Vehicle Routing Problems

  In the formulation of the problem there are three parameters to be considered in among them are:

  1 Costs of setting up depots at the point a referred to as EC a

  2 The level of demand at each point a are d a ∀ a ∈ V

  3 Maximum available vehicles are K and every vehicle has capacity C (k = 1, 2, . . . , K)

  k

  The purpose of this model is to:

  1 Determine the optimal number of depots and their locations where this has not yet been set.

  2 Calculate the optimal cost of the entire customers.

  3 Evaluate the optimal number of vehicles used.

  4 Combine the route for each vehicle on duty. For a fee optimal delivery. The variables are:

  X

  abk ∈ {0, 1} is a bolean variable to determine the route

  (1 if (a, b) traversed by vehicle k) Y N

  ak ∈ is an integer number of resources stored in node a by vehicle k.

  w a ∈ {0, 1} is 1 if the point a was depot The Integer Linear Programing model formulation as follows (Ray and

  Soeanu [7]): _{X X X X}

• min EC j w j C ab x abk , ∀ a 6= b (8)

  a∈V a∈V b∈V

  Subject to: Flow conservation: _{X X ′} x ≥ 1∀ b ∈ V , a 6= b (9)

  abk a∈V k∈K _{X X}

  x ≤ |K| (10)

  

obk

b∈V k∈K

  x _X oak = x aok ∀ a ∈ V and k ∈ K (11) _{X ′} x x _{′ ′} apk = pbk ∀ p ∈ V and k ∈ K, a, b 6= p (12) a∈V b∈V

  Route Allowance (Route Reduction): _{X X X} x − x ≤ |S| − 1, S ⊆ V, |S| ≥ 2 (13)

  abk obk Shaflina, Suwilo & Sutarman – Vehicle Routing Problems

  Capacity Limits: _X y

  ak ≤ C k ∀ k ∈ K (14) _X a∈V

  y = d w , ∀ a ∈ V (15)

  ak a j k∈K _X

  y ≤ d x , ∀ a ∈ V and k ∈ K (16)

  ak a abk _′ b∈V

  Depot Assignment: _X x ≥ w , ∀ a ∈ V (17)

  oak a k∈K

  x ,

  oak ≤ w a ∀ a ∈ V, k ∈ K (18)

  Variabel _′ x , a

  abk ∈ {0, 1}; where a, b ∈ V 6= b, k ∈ K

  w a ∈ {0, 1}; where a ∈ V y

  ak ≥ 0; where a ∈ V, a 6= b, k ∈ K

  The objective function (9) indicate the costs of setting up depots at the point a and the total cost of the route, constraint (10) indicate each customer must be visited by the vehicles X abk = 1 if (a, b) is traversed by vehicle k, constraint (11) means set maximum vehicle limits that can be used and ensure all vehicles which is used back to point 0, constraint (12) is a modification of the common sub-tour constraint in 2002 Toth [10] in some accommodations for start and end at point 0 and through the determining depot before reaching depot 0,(13) Forced to serve customers at the agreed place, the route has been set and the total on the specified route should not overload capacity, (15),(16) and (17)indicated split delivery constrain and ensure all customer demands will be met and the last constraint (18) and (19)ensure that each vehicles must start and end its work on the same depot .

  Shaflina, Suwilo & Sutarman – Vehicle Routing Problems

  

4 MULTI-DEPOT VEHICLE ROUTING

PROBLEM WITH SPLIT DELIVERY AND

TIME WINDOWS

  Multi depot vehicle routing problem with split delivery and time win- dows (MDVRPSDTW) is a model developed from existing models formerly the multi depot vehicle routing problem split delivery Ray [8], in this models we added time windows for the constrain and the objective functions. In the formulation of our problems consider four parameters: 1 Costs of setting up depots at the point j is referred to as EC j .

  2 The level of demand at each point a is d a ∀ a ∈ N .

  3 Maximum available vehicles are K and every vehicle has capacity C (K = 1, 2, ... K .

  k

  4 The penalty fee charged for the delay at point a by vehicle k is referred to as ED .

  ak

  Set: I : Set of delivery nodes J

  : Set of depot nodes N : Set of all nodes, N = I ∪ J K

  : Set of delivery vehicle k : index of vehicle Parameters:

  EC : fixed cost for develop depot j

  j

  CF

  k : fixed cost for vehicle k

  CT

  k : Transportation cost for vehicle k per unit time

  CR k : Labor cost of the delivery person for vehicle k per unit time ED

  ak : fixed cost for delay time of delivery vehicle k at customer a

  t : Transportation time between node a and b unity of time

  ab

  e

  ak : earliest time at customer a by vehicle k

  l : latest time at customer a by vehicle k

  ak

  r

  k : maximum route time allowed for vehicle k

  d : demand at customer a

  a

  w

  j : number of boolean ( 1 if j is a depot)

  q : capacity of vehicle k

  k

  y ak : amount of resource deposited at node a by vehicle k M

  : a sufficiently large number

  Shaflina, Suwilo & Sutarman – Vehicle Routing Problems

  Variables: at ak : Arrival time of the delivery vehicle k at customer a at

  bk : Arrival time of delivery vehicle k at depot b

  wd ak : Waiting time of the delivery vehicle at customer a The Integer Linear Programing model formulation as follows: _{X X X X}

• EC w CT t x

  j j k ab abk j∈J a∈N b∈N k∈K _{X X X X X}

• CF x CR a
• k abk k bk

  k∈K a∈J b∈I b∈J k∈K _X

• ED ak z a (19)

  a∈N

  Subject to: Flow conservation: _{X X} x abk = x bak , ∀ a ∈ N, k ∈ K (20)

  a∈N a∈N _{X X}

  x = 1, ∀ b ∈ I (21)

  abk _X k∈K a∈N _X

  x

  abk ≤ 1, ∀ k ∈ K (22) a∈N b∈I

  Capacity: _X y ,

  ak ≤ q k ∀ k ∈ K (23) _X a∈N

  y ak ≤ d a w j , ∀ a, j ∈ N (24)

  k∈K _X

  y x ,

  ak ≤ d a abk ∀ a ∈ N and k ∈ K (25) b∈N

  Depot assignment: _X x ≥ w , ∀ a ∈ N (26)

  oak a k∈K

  x ,

  oak ≤ w a ∀a ∈ N, k ∈ K (27) Shaflina, Suwilo & Sutarman – Vehicle Routing Problems

  Time windows: _X at x

  bk ≥ at ak + wd ak + t ab + M ( abk − 1), ∀ a ∈ N, b ∈ N (28)

k∈K

  r ≥ at ≥ at + wd + t + M (x − 1), ∀ a ∈ N, b ∈ N, k ∈ K (29)

  k bk ak ak ab abk

  e ,

  ak ≤ at ak + wd ak ≤ l ak ∀ a ∈ N (30)

  1 if l ak − (at ak + wd ak ) > 0 z = (31)

  ak

  0 if l ak − (at ak + wd ak ≤ 0 x ∈ {0, 1}; where a, b ∈ N, k ∈ K (32)

  abk

  w ∈ {0, 1}; where j ∈ J (33)

  j

  y ≥ 0; where a ∈ I, a 6= b , k ∈ K (34)

  ak

  The objective function (20) are the sum of fixed cost for develop depot a, fixed cost for vehicle k, Transportation cost for vehicle k per unit time, La- bor cost of the delivery person for vehicle k per unit time, and fixed cost for delay time of delivery vehicle k at customer a if the company breaking the agreement. constraint (21), (22) and (23) indicate each customer must be visited by the vehicles x abk = 1 if (a, b) is traversed by vehicles k and the orders of the customer can be served by more than one vehicles. constraint (24), (25) and (26) means forced to serve customers at the agreed place, the route has been set and the total on the specified route should not overload capacity and delivery of demand can be used split delivery. constraint (27) and (28) indicate the depot assignment at point a and ensure all vehicles that depart from depot a must be back on depot a (each vehicles must start and end its work on the same depot). (29), (30),(31),(32),(33),(34) and (35) indicate the times windows constraint, constraint (32) ensure the penalty charged if the company breaking the agreement (arrival time of the delivery vehicle k at customer a more than latest time at customer a) .

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