Coordinated scheduling of production and air transportation in supply chain management

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, Volume 2, Issue 10, October 2012)

47

Coordinated scheduling of production and air transportation in

supply chain management

A. Tahmasbi-birgani

1

, E. Najafi

2

, R. Soltani

3

, A. Mahmoodi-Rad

4

, S. Molla-Alizadeh-Zavardehi

3

1

Department of Mathematics, Shoushtar Branch, Islamic Azad University, Shoushtar, Iran

2_{Department of Industrial Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran}

3

Department of Industrial Engineering, Islamic Azad University, Masjed Soleyman Branch, Masjed Soleyman, Iran 4_{Department of Mathematics, Islamic Azad University, Masjed Soleyman Branch, Masjed Soleyman, Iran}

Abstract— The efficient coordinated scheduling of production and air transportation becomes a challenging problem as global companies move towards higher collaborative and competitive environments. Therefore, there is a need for a single coordinated optimization model for

generating realistic scheduling for production and

distribution to be used in the supply chain. So we studied the problem of coordinated scheduling of production and air transportation in supply chain management. We consider different assumptions and constraints in our modeling such as due window, delivery tardiness and no delivery tardiness. For these assumptions and policies, a variety of mathematical programming models have been presented to minimize total cost of the supply chain which include distribution, plant, departure time earliness tardiness and delivery earliness tardiness costs.

Keywords— Coordination, Supply chain management, Multi-criteria scheduling, Air transportation, Distribution, Mathematical model.

I. INTRODUCTION

The recent market globalization and merging processes, the development of complex and tightly integrated plants, have encouraged companies to improve their efficiency and increase the planning and scheduling technology not only focused on a plant level, but extended to the optimization of the whole supply chain (SC). Production and transportation operations are the two most important operational functions in a SC. To achieve optimal operational performance in a SC, it is critical to coordinate these two functions and schedule them jointly in a coordinated way.

The coordinated production and transportation processes rely on inventory storing to buffer both activities from each other. However, inventory costs and the trend to operate in a just-in-time (JIT) manner are putting pressure on firms to reduce inventories in their transportation chain. Coordinating production and distribution activities requires the consideration of additional features. In our work, SC is illustrated in Fig. 1.

[image:1.612.324.570.455.548.2]

Within this SC, products are hold in inventory. For delivery of customer’s order finished products are transported to customers using air transportation to meet their due dates. Due to heavy cost of missing a shipment in a scheduled flight, coordination of production and air transportation activities is critical. The earliness costs of departure time may result from the need for holding the customer’s order at the plant (warehouses, inventory cost) or waiting cost or penalties at the airport. Delivery earliness (resp. tardiness) penalty is incurred if an order is completed before its committed due date (resp. after its committed due date). The delivery earliness penalties could result from the need for warehouse in the retailer’s stores. The delivery tardiness cost includes contract penalties, customer dissatisfaction, loss of sales and potential loss of customer goodwill due to late orders.

Fig. 1. Supply chain stages coordination

Optimization of the trade-off between production, distribution, retailer's costs and customer service level is the goal of the decision maker of such systems. So the problem is to find a coordinated schedule that minimize supply chain total cost which includes transportation, plant, delivery earliness tardiness and departure time earliness tardiness costs. We investigate two policies and they are as such: first policy considers delivery tardiness and the second one assumes that no delivery tardiness is authorized.

There are considerable numbers of researches in production-distribution coordination.

Integration Scope Supplier

Supplier

Manufacturer

Distributor

Retailer

Customer Customer

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International Journal of Emerging Technology and Advanced Engineering

48 But all of them have only given consideration to the coordination of production scheduling and vehicle routing and no study has been conducted on the coordination of production scheduling and air transportation. Having reviewed the literature we came to realize that no study has been investigated in this field. Therefore, we were galvanized into presenting a pioneer consolidated model in this area. Also there are a few researches on synchronization of production scheduling and air transportation scheduling. In synchronization the overall problem is decomposed into two coordinated tasks such that the first task is to assign accepted orders to available flights' capacities to minimize the total transportation and delivery earliness tardiness costs. The allocation is restricted by plant such that it should be balanced with plant capacity. The second task is to determine orders sequence or completion time to minimize plant and departure time earliness tardiness costs.

Li et al. [2] studied the synchronization of single machine assembly and air transportation considering single destination. The overall problem is decomposed into air transportation problem and single machine assembly scheduling problem. They formulated two problems and then presented a backward heuristic algorithm for production of products in the single machine environment. Li et al. [3] developed their previous work to consider multiple destinations in the distribution activity. Li et al. [4] demonstrated the allocation of orders to flights have the structure of regular transportation problem, while the scheduling of single machine assembly problem is NP-hard. They also proposed a forward heuristic and a then backward heuristic for production scheduling [5]. Li et al. [6] developed their previous work by considering parallel machine assembly environment in production. The problem was modeled as a parallel machine assembly with only departure time earliness penalties or with no tardiness policy. They also demonstrated the scheduling of parallel machine assembly problem is NP-Complete and a simulated annealing algorithm was presented to determine the sequence of orders in parallel machine problem.

Zandieh and Molla-Alizadeh-Zavardehi [7] extended the proposed model by Li et al. [3] and proposed Synchronized scheduling models considering due window and two type flight capacities. Zandieh and Molla-Alizadeh-Zavardehi [8] extended their work considering various capacities with different transportation cost and also charter flights (commercial flights). Rostamian Delavar et al. [9] proposed a coordinated production and air transportation scheduling and two genetic algorithm are developed to solve the problem.

Due to the dependency of most metaheuristic algorithms on the correct choice of operators and parameters, a Taguchi experimental design method is applied to set and estimate the proper values of proposed algorithms parameters to improve their performance. In this paper, we extended the works by Zandieh and Molla-Alizadeh-Zavardehi [7] and [8] to coordination of these two task in a single optimization model that include various capacities, delivery tardiness, no delivery tardiness and due window.

Also many papers about of production scheduling and vehicle routing problem coordination in the case of road transportation can be seen in(e.g., Blumenfeld et al., [10]; Fumero and Vercellis, [11]; Chen, [12];Lee and Chen, [13]; Chang and lee, [14]; Chen and Vairaktarakis, [15]; Li et al., [16]; Soukhal et al., [17]; Li and Ou, [18]; Wang and lee, [19]; Wang and Cheng, [20]; Zhong et al., [21]; Chen and Lee, [22]). And there is a survey on integrated analysis of production-distribution activities (Vidal and Goetschalckx, [23]; Erenguc et al., [24]; Sarmiento and Nagi, [25]; Goetschalckx et al., [26]).

The remainder of this paper is organized as follows. In section 2, the related works are discussed. In section 3, general assumptions considered throughout the paper are defined. In sections (3) the coordinated mathematical model considering delivery tardiness is presented. Then, considering Due window, the presented mathematical model are extended and a numerical example is solved. In section (4) we considered the no delivery tardiness policy and the related coordinated model is presented and also numerical example is solved. Finally the conclusions are given and the potential future works are presented in the last section.

II. GENERAL ASSUMPTIONS

The problem is modeled considering on the following assumptions;

1. The allocation of air transportation and scheduling of production are for the accepted orders in the previous planning period.

2. Order fulfillment is achieved when the order reaches the destination airport on time.

3. There are several flights in the planning period with different departure and arrival time and other specifications such as capacity, cost, etc.

4. Business processing cost and time, together with loading time and cost for flights are included in the transportation cost and transportation time.

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International Journal of Emerging Technology and Advanced Engineering

49 III. WITH DELIVERY TARDINESS

The model allocates orders to the existing transportation capacities and determines the sequence and completion time for the allocated orders in production. This requires solving a scheduling problem to ensure that allocated orders catch their flights so that total cost of supply chain is minimized. The required notation to present the model is as follows:

j j i

i,, ,  The order / job index, _i₁_,₂_,...,_N_;

f

f,  The flight index, f 1,2,...,N;

k The destination index, k1,2,...,K;

f

D The departure time of flight f at the local airport;

f

A The arrival time of flight f at the destination;

i

Q The quantity of order

i

;

i



The delivery earliness penalty cost (/unit/h) of order

i

;

i

 The delivery tardiness penalty cost (/unit/h) of order i;

i

d The due date of order ;i

i

Des The order i's destination;

f

des The flight f' destination; s

tf

Cap The available tth type capacity of flight f thatt1,...,T; f1,...,F;

tf

Tc The transportation cost for per unit product when allocated to tth type capacity of flight f;

tif

q

The quantity of portion of order i allocated to type t capacity of flight f;

i

p

The processing time of order i;

i

c

The completion time of order / job i;

i

 The per hour earliness penalty of order / job i for production;

p

p,  The position or sequence of order

i

p1,...,N;

ip

u 1 if order

i

be in position p, 0 otherwise;

 The per hour plant costs (including machine cost, operator wages and other production variable costs which is completely related to the length of working hours;

i

I The idle time before order

i

in the schedule;

max

C The maximum completion time of orders that is equal to shut down time of shop;

LN

A large positive number;



A number between (0, 1);

Li et al. (2006a and 2007) defined two type capacities in each flight with two different transportation cost. For extension of their work and generating more realistic schedule we assumed that in many industries, we may have only one or more than two type capacities in each flight. Therefore, we considered T type capacities by

f t

Cap_, in the notation. Also if for a given flight f, we have h type capacities, the

f t

Cap, that

t



h



1 ...,

T

will be zero. The

mathematical programming formulation of the model is shown as follow:

  





  

   i i f tif F f N i T t T t N i F f tif

tfq d A q

TC *max(0, )*

min

1 1 1 1 1 1

  



 

 i f i tif F f N i T t q d

A )*

, 0 max( * 1 1 1  max 1 1 1 1 * ) (

* D_f C_i q_tif C

i F f N i T t    



   (1) s.t: F f N i des Des

q_tif _i _f

T t ,..., 1 ; ,..., 1 0 ) ( * ) ( 1    



 (2)

F

f

T

t

Cap

q

_tif _tf

N i

,...,

1 ;

,...,

1







 (3)

N

i

Q

q

_tif _i

F f T t

,...,

1

1 1





  (4)

N

i

u

_ip N p

,...,

1





 (5)

N

p

u

ip N i

,...,

1





 (6)



P I



c i N

u I

p

u ip i i i

N i p p i i ip N p ,..., 1 1 1 1 1                            



(7) F f D p q q f i f ti T t f f f ti T t f f N i ..., , 2 , 1 5 . 0 5 . 0 , 0 max 1 1 1 1 1                                             





(8) max 1

C

c

u

_iN _i N i





 (9) N i

Ii0 1,..., (10)

N p

N i

uip{0,1} 1,..., ; 1,..., (11)



tif

(4)

International Journal of Emerging Technology and Advanced Engineering

50 The decision variables are

q

_tif

,

c

_i

,

I

_i

,

u

_ip and

C

_max

.

The objective is to minimize total cost which consists of total transportation cost for the orders allocated to type 1 to T capacity, total delivery earliness tardiness penalties, total departure time earliness penalties of jobs and plant cost. Constraint sets (2) ensures that if order

i

and flight f have different destinations, order

i

cannot be allocated to flight f. Constraint sets (3) that the capacity 1 to T of flight f is not exceeded. Constraint set (4) ensures that order i is completely allocated. Constraint sets (5) and (6) state that each job has to be assigned to a position, and each position has to be covered by a job. Constraint set (7) calculates completion time of jobs, considering inserted idle times among jobs. Since all jobs must catch their scheduled flights, constraint set (8) ensures that order

i

catches all of its departure times or the completion time of qtif has to be less than or equal to their related flight departure times

allocated to qtif. Constraint set (9) calculates

C

_max and can be replaced by the constraint set (13).

N i C

Cmax i 1,..., (13)

Equations simplification: some of the equations can be further simplified as follows.

Constraint set (8):

1. c D i N f F

q q

f i f

ti T

t f ti T

t

,..., 1 ; ,..., 1 5

. 0

5 . 0 ,

0 max

1

1 _ _ _

    

 

    

 

      

   

 

      

 



(14)

As obvious in above the constraint set (14) is simpler but their numbers F times increase compared to constraint set (8). Also constraint set (14) can be simpler as follows:

2. q_ti_f



c_i D_f



i N f F

T

t

,..., 1 ; ,..., 1 0

1

 

     

 

 



(15)

Due window:Usually the customers accept flexibility from due date, as they tolerate a small degree of uncertainty on the supplier's side. This uncertainty might come about as a result of some problems such as production lead time, raw material defect, machine broken or traffic jams, problems with delivery itself such as flight's delay etc. It is commonly agreed to assent small deviations from a due date and thus a delivery or due window is arranged as shown in Fig. 2. [27]. The required notation is as follows:

i i

l

e

,

The due window of order i, where

e

_i is the earliest due date and l_i is the latest due date;

The earliness time of order

i

is equal to max (0,e_iA_f) and the tardiness time of order

i

is equal to max(0,Af li). Hence the objective function is transformed as follows:

min







 

   



 i i f tif F

f N i T t tif tf F f N i T t

q A e q

TC *max(0, )*

1 1 1 1

1 1



 



 

 i f i tif

F

f N

i T

t

q l

A )*

, 0 max( * 1 1 1



max 1

1 1

* ) (

* Df Ci qtif C

i F

f N

i T

t



  



  

(16)

[image:4.612.333.525.351.472.2]

The models with due window are generalized case of models with delivery date, because when both e_i and l_i be equal to d_i the problem is transformed to models with delivery date.

Fig. 2. The penalty function around

e

_iand

i

l

Numerical example:In order to validate and verify the proposed models, a common small problem is solved by the Lingo 8 software in all models. The parameters values for test problem are as follows:

2



T , N2, F4, Q110, Q28, p12,

2

2

p , e111, l112,

e

2



12

, l212



12,

3

2





,



14,



2



5

,



12,



21,



10

1

1

Des , Des2 2, des12, des21, des31,

2

4

des , D13, D25, D37, D4 12, A14,

5 . 6

2

A , A38.5, A413, Cap113, Cap124,

5

13

Cap , Cap₁₄2, Cap₂₁7, Cap₂₂10, 0

23

Cap , Cap₂₄6,

Tc

₁₁



5

, Tc123, Tc134,

4

14



Tc

,

Tc

₂₁



7

, Tc225, Tc230, Tc248.

i

e

i

l

Penalties

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International Journal of Emerging Technology and Advanced Engineering

51 The results obtained from solving the test problem are as follows:

0

111



q

, q1124, q1135, q1140, q1210,

0

122



q

,

q

₁₂₃



0

,

q

₁₂₄



2

,

q

₂₁₁



0

,

q

212



1

,

0

213

q ,

q

214



0

,

q

221



0

,

q

222



0

,

q

223



0

,

6

224



q

,

c

1



6

,

c

2



8

, Cmax 8,

I

1



4

, I2 0,

1

11



u

, u120,

u

21



0

, u221.

IV. NO DELIVERY TARDINESS

Since no tardiness is authorized, the objective function excludes the delivery tardiness penalties and minimizes the total transportation costs, delivery and departure time earliness penalties and plant cost. Therefore, constraint set (18) ensures that the arrival time of all flights allocated to the order

i

is less than or equal to its delivery due date. The problem under study can be formulated as follows:

tif f i i F f N i T t tif tf F f N i T t q A d q

Tc *( )* min 1 1 1 1 1 1  



     



max 1 1 1

*

)

(

*

D

_f

C

_i

q

_tif

C

i F f N i T t













   (17) s.t: F f N i d A q q i f tif T t tif T t ,..., 1 ; ,..., 1 0 ) ( 5 . 0 ) 5 . 0 , 0 max( 1

1 _ _ _ _

                         



  (18)

The other constraints of the model are the same as constraint sets (2), (3), (4), (5), (6), (7), (8), (9), (10), (11) and (12).

Due window: The constraints sets (17) and (18) is changed as follows:



i f



tif

F f i N i T t tif F f tf N i T t q A e q

Tc *max0, * min 1 1 1 1 1 1  



      





max

1 1 1

*

D

_f

C

_i

q

_tif

C

F f i N i T t













   (19) F f N i l A q q i f tif T t tif T t ,..., 1 ; ,..., 1 0 ) ( 5 . 0 ) 5 . 0 , 0 max( 1

1 _ _ _ _

                         



  (20)

Numerical example. The solutions are as follows:

0

111



q

,

q

₁₁₂



4

,

q

113



0

,

q

114



0

,

q

121



3

,

0

122

q ,

q

₁₂₃



0

,

q

124



0

,

q

211



0

,

q

212



6

,

0

213



q

,

q

214



0

,

q

221



4

,

q

222



0

,

q

223



0

,

1

224



q

,

c

₁



10

,

c

₂



3

,

C

_max



10

,

I

₁



5

,

1

2



I

,

u

₁₁



0

,

u

₁₂



1

,

u

₂₁



1

,

u

₂₂



0

.

V. CONCLUSION AND FUTURE WORKS

In this paper, we studied supply chain coordination problem. We have presented several models with different policies. Numerical examples were implemented using Lingo to validate and verify the proposed mathematical models. Since there is the first and is not another research about coordination of air transportation and production scheduling in supply chain, many research can develop this subject. Further research can be conducted by all assumption that studied in production scheduling and transportation allocation research such as stochastic processing time, non-split in transportation allocation, etc. In addition, metaheuristics can be applied to solve the proposed models considering production configuration such as, single machine, parallel machine, flow shop, job shop, etc.

Acknowledgment

This study was partially supported by Islamic Azad University, Shoushtar Branch. The authors are grateful for this financial support.

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International Journal of Emerging Technology and Advanced Engineering

52

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