To preferably comprehend the solving approach to MCCETLTPD, the following usable notations are introduced.

Notation

MCCETLTPD is depicted below. Under the normal case, homogeneous products are to be supplied from m origins including origin i ∈ I = { 1 , 2 , ⋯ , m } to n ˜ destinations including destination j ∈ J = { 1 , 2 , ⋯ , n ˜ } . The shipping capacity for the route from origin i ∈ I (with available supply as s i o ) to destination j ∈ J (with demand as d j o ) is u i j . In a specified limited time T, a large number of sudden increased goods are needed to be supplied from m origins to n ˜ destinations. Within the time-limited T, the shipping capacity and time for the route from origin i ∈ I (with sudden increase supply as s i ) to destination j ∈ J (with sudden increase demand as d j ) are u ′ i j and t i j respectively. Moreover, the expansion cost per unit of the route from origin i ∈ I to destination j ∈ J is b i j e and for the origin i ∈ I and the destination j ∈ J are b i o and b j d respectively. For each ( i , j ) ∈ I × J , t i j > 0 , u i j > 0 , u ′ i j > 0 , b i j e > 0 , b i o > 0 and b j d > 0 . for each i ∈ I , s i o > 0 , s i > 0 ; and for each j ∈ J , d j o > 0 , d j > 0 . Under the aforementioned circumstances, when a large number of sudden increased goods are needed to be transported within time-limited T, the goal is to seek a practicable capacity expanding transportation scheme satisfying the time-limit T as well as all origins’ supply and all destinations’ demands, so that the expanding cost is minimized.

Let x i j be the products quantity transported from origin i ∈ I to destination j ∈ J . Denote the expanding cost by z.Then, MCCETLTPD ’s mathematical model is presented below.

MCCETLTPD : { min z = ∑ ( i , j ) ∈ I × J ( b i j e ( x i j − u i j ) + b i o ( ∑ j ∈ J x i j − s i o ) + b j d ( ∑ i ∈ I x i j − d j o ) ) s .t . { ∑ j ∈ J x i j = s i ∀ i ∈ I ∑ i ∈ I x i j = d j ∀ j ∈ J 0 ≤ x i j ≤ u ′ i j ∀ i ∈ I and ∀ j ∈ J

Due to the high time requirement of transportation in reality, then it is considered that minimize the expanding cost under a certain time-limited T. Because of the advanced science and technology, origins and destinations of the logistics industry usually adopt electronic equipment instead of manual operation [21] , so both the origins and destinations are processed at the same time. Thus the actual transportation time is the maximum shipping time among all routes, that is max { t i j | ∀ i ∈ I , j ∈ J } where t i j is the shipping time from origin i ∈ I to destination j ∈ J . In fact, for any route from origin i ∈ I to destination j ∈ J , the shipping time t i j consists of two parts, i.e., a constant part t i j 1 , and another part t i j 2 that is a linear function of the products quantity x i j transported from origin i ∈ I to destination j ∈ J , denoted as t i j 2 = g ( x i j ) . Therefore, the shipping time t i j ( i , j ) ∈ I × J satisfies the inequality that t i j ≤ T for any ( i , j ) ∈ I × J , where t i j = { t i j 1 + t i j 2 , x i j > 0 0 , x i j = 0 .

For the second part t i j 2 , it is clear that the more a vehicle loads, the slower it moves. In addition, the exact value of time t i j 1 for any route ( i , j ) ∈ I × J can be calculated, if let v i s and v j d denote as the operating speed of origin i ∈ I and destination j ∈ J , respectively, v i j e denotes as the empty vehicle running speed, and d t i j denotes as the distance of the route from origin i ∈ I to destination j ∈ J . Then there are the relationships as follows:

t i j 1 = s i v i s + d j v j d + d t i j v i j e , ∀ i ∈ I and   j ∈ J (1)

t i j 2 = max { 0 , T − t i j 1 } , ∀ i ∈ I and   j ∈ J (2)

g ( x i j e ) ≤ t i j 2 , ∀ i ∈ I and   j ∈ J (3)

Thus, from inequality (3), we can get the maximum transportation quantity x i j e for the route from origin i ∈ I to destination j ∈ J within the time-limited T.

To efficiently solve MCCETLTPD model, we give following definition.

Definition 1. As shown in Figure 1, a network with lower and upper arc capacities G = ( V , s , t ; A ; C _ , C ¯ ; B ) is constructed for MCCETLTPD model. Node set V = { 1 , 2 , ⋯ , m , m + 1 , ⋯ , m + 2 , ⋯ , m + n ˜ + 1 , m + n ˜ + 2 } . Source s = m + n ˜ + 1 . Sink t = m + n ˜ + 2 . Arc set A = { ( i , m + k ) | i ∈ I , j ∈ J } ∪ { ( s , k ) | k ∈ I } ∪ { ( m + k , t ) | k ∈ J } . The lower and upper capacity of arc ( s , k ) ( k ∈ I ) in set A are s k . The lower and upper capacity of arc ( m + k , t ) ( k ∈ J ) in set A are d k . The lower and upper capacity of arc ( i , m + k ) ( i ∈ I , k ∈ J ) in set A are zero and u i k , respectively. The number of nodes in G is n = m + n ˜ + 2 . As per Definition 1 in [22] , lower and upper arc capacity matrices C _ = ( C _ i j ) n × n and C ¯ = ( C ¯ i j ) n × n are determined by following stipulations. If i = s and j = k ( k ∈ I ), then both C _ i j and C ¯ i j are set as s k . If i = m + k ( k ∈ J ) and j = t , then both C _ i j and C ¯ i j are set as d k . If i ∈ I and j = m + k ( k ∈ J ), then C _ i j and C ¯ i j are set as zero and u i k , respectively. For other situation, if i = j , then C _ i j and C ¯ i j are set as zero and positive infinity ( + ∞ ) respectively, else both C _ i j and C ¯ i j are set as zero. And the expansion cost per unit matrix B = ( B i j ) n × n is determined by following stipulations. If i = s and j = k ( k ∈ I ), then B i j is set as b k o . If i = m + k ( k ∈ J ) and j = t , then B i j is set as b k d . If i ∈ I and j = m + k ( k ∈ J ), then B i j is set as b i k e . For other situation, if i = j , then B i j is set as zero.

Obviously, in the constructed network G by Definition 1, if there is a feasible flow from source s to sink t, then for each ( i , k ) ∈ I × J , let x i k be the flow on arc ( i , m + k ) , hence we derive MCCETLTPD model’s a feasible solution as x = { x i j | ( i , j ) ∈ I × J } . So we get following Proposition 1.

Proposition 1. The constructed network G by Definition 1 reflects the structure of the MCCETLTPD model’s feasible solutions. That is to say, there exists one-to-one correspondence between the feasible solution of the MCCETLTPD model and the feasible flow in the constructed network G by Definition 1, i.e., a feasible flow in the constructed network G by Definition 1 corresponds to a feasible solution of the MCCETLTPD model, and vice versa. Furthermore, whether there exists a feasible flow in the constructed network G by Definition 1 equals whether there exists a feasible solution of the MCCETLTPD model.

For the route ( i , k ) from origin i ∈ I to destination k ∈ J in network G, if feasible flow f i j ( j = m + k ) on arc ( i , m + k ) ( ∀ i ∈ I , k ∈ J ) is no more than u i j , it means that arc ( i , m + k ) of the network G is allowed to transport flow f i j ( j = m + k ); otherwise arc ( i , m + k ) of G needs to be expended.

The MCCETLTPD is actually equivalent to the minimum cost maximum flow problem of G constructed by Definition 1, so we can get Theorem 1 as follows.

Theorem 1. The algorithm called MCCETLTPD-A judges the existence of the MCCETLTPD model’s feasible solution and finds the MCCETLTPD model’s optimum solution if existing.

MCCETLTPD-A:

Step 1: For any route ( i , j ) ∈ I × J , input all the known parameters under normal conditions, such as ( s i o , d j o , u i j ) . Then construct network G = ( V , s , t ; A ; C _ , C ¯ ) by Definition 1. For each i , j ∈ { 1 , 2 , ⋯ , n } , set C i j = C ¯ i j to keep matrix C ¯ in C = ( C i j ) .

Step 2: For any route ( i , j ) ∈ I × J , input all the known parameters about capacity expansion, such as ( s i , d j , T , b i o , b j d , b i j e , d t i j , v i s , v j d , v i j e ) , then find the time value of t i j 1 and t i j 2 by formulas (1) and (2) respectively, and then find the maximum transportation quantity x i j e by inequality (3) under the time-limited T, and let u ′ i j = x i j e . For each ( i , j ) ∈ I × J , set u i j = u ′ i j to keep variable u ′ i j as u i j , then construct new network G = ( V , s , t ; A ; C _ , C ¯ ; B ) by Definition 1. Set Y = 0 .

Step 3: Call minimum cost maximum flow algorithm MCMF-A in [19] for currently gotten new network G = ( V , s , t ; A ; C _ , C ¯ ; B ) . If MCMF-A in [19] finds a feasible flow’s flow matrix as F = ( f i j ) n × n for currently gotten new network G, then consecutively operate as below: “If the flow amount ∑ ( s , i ) ∈ A f s i − ∑ ( i , s ) ∈ A f i s is equal to the total expanding goods amounts ∑ i ∈ I s i (or equal to ∑ j ∈ J d j ) then the transportation scheme with time-limited T is satisfied, set Y = 1 and go Step 4”; otherwise, it means that the transportation task cannot be completed within time-limited T, then stop;

Step 4: By the flow matrix F = ( f i j ) n × n , the capacity expansion matrix R = ( r i j ) n × n is obtained as follows: For any i , j ∈ { 1 , 2 , ⋯ , n } , if f i j > C i j , let r i j = f i j − C i j ; else let r i j = 0 . Here in, for any i , j ∈ { 1 , 2 , ⋯ , n } , if r i j > 0 , then the capacity for the route ( i , j ) needs to be expend. So, if R = ( r i j ) n × n is not a zero matrix, according to capacity expansion matrix R = ( r i j ) n × n and the expansion cost per unit matrix B = ( B i j ) n × n , the minimum expansion cost z ′ = ∑ i , j ∈ V B i j r i j is obtained. then go to Step 5;

Step 5: If Y = 1 , then set x i j = f i j for each ( i , j ) ∈ I × J , set z = z ′ , output MCCETLTPD model’s optimum solution x = { x i j | ( i , j ) ∈ I × J } and optimum objective value z, and output Y = 1 to indicate that MCCETLTPD model’s optimum solution and optimum objective value are found. Otherwise, output Y = 0 to indicate that feasible solution is not existent for MCCETLTPD model.

To preferably comprehend the solving algorithm MCCETLTPD-A, an algorithm flow chart of MCCETLTPD-A is shown in Figure 2. The algorithm flow chart clearly shows the idea of algorithm MCCETLTPD-A.

Consider an MCCETLTPD instance of 3 origins and 4 destinations, i.e., 3 × 4 problem [21] , which was first discussed by LI Zhen-ping et al., but we make improvements to illustrate the effectiveness of our algorithm MCCETLTPD-A. In the paper of [21] , within time-limited T = 15 h , the transportation scheme was not completed under the normal case. However, we can not only complete the transportation scheme within time-limited T = 15 h by expanding the route capacity, but also the number of products increases suddenly.

Firstly, we give the data ( s i o , d j o , u i j ) shown in Table 1 under the normal case. Then we give the goods sudden increase data ( s i , d j , d t i j ), ( b i o , b j d , b i j e ), and ( v i s , v j d , v i j e ) shown in Table 2, Table 3 and Table 4, respectively, i.e., the goods sudden increase and need to capacity expand within time-limited T. By running MCCETLTPD-A based MATLAB computational programs, we apply MCCETLTPD-A to solve 3 × 4 problem, and validate the efficiency for MCCETLTPD-A to solve MCCETLTPD.

Solution: As per Table 1, the known parameters are below. Origins’ number m = 3. Destinations’ number n ˜ = 4 . Under the normal case, the origin supply vector S o = ( s 1 o , s 2 o , s 3 o ) , destination demand vector D o = ( d 1 o , d 2 o , d 3 o , d 4 o ) and the route shipping capacity matrices U = ( u i j ) are below.

S o = ( 7 , 4 , 9 ) ; D o = ( 3 , 6 , 5 , 6 ) ; U = [ 9 5 8 8 10 5 3 8 2 5 11 5 ]

As per Table 2, Table 3 and Table 4, the origin expanding supply vector S = ( s 1 , s 2 , s 3 ) , destination expanding demand vector D = ( d 1 , d 2 , d 3 , d 4 ) , the route shipping distance matrices D t = ( d t i j ) , origin capacity expansion cost per unit vector B s = ( b 1 o , b 2 o , b 3 o ) , destination capacity expansion cost per unit vector B d = ( b 1 d , b 2 d , b 3 d , b 4 d ) , the route capacity expansion cost per unit matrices B e = ( b i j e ) , the origin operating speed vector V s = ( v 1 s , v 2 s , v 3 s ) , the destination operating speed vector V d = ( v 1 d , v 2 d , v 3 d , v 4 d ) , and the route empty vehicle running speed matrices V e = ( v i j e ) are below. The time-limited T = 15   h . The t i j 1 and t i j 2 (see Table 5) are obtained by Equations (1) and (2), respectively. Next, by inequality (3) of g ( x i j e ) = 1 2 x i j e ≤ t i j 2 , the route maximum shipping quantity x i j e under the time-limited T is obtained shown in Table 6.

S = ( 17 , 14 , 19 ) ; D = ( 13 , 16 , 5 , 16 ) ; B s = ( 1 , 2 , 3 ) ;

B d = ( 2 , 1 , 3 , 1 ) ; V s = ( 10 , 10 , 10 ) ; V d = ( 10 , 10 , 10 , 10 )

D t = [ 10 2 8 3 10 3 9 4 6 12 10 4 ] ; B e = [ 1 2 2 1 1 3 2 1 1 1 1 2 ] ; V e = [ 1 1 1 1 1 1 1 1 1 1 1 1 ]

By calling MCCETLTPD-A based MATLAB computational programs and using the above known parameters as input, the optimum transportation scheme (i.e., maximum flow f i j ) is obtained, see Table 7, and from Table 1 and Table 7, the route needed capacity expansion quantity r i j is obtained shown in Table 8.

So, the transportation scheme can be completed within time-limited T = 15   h . Then, from Table 3 and Table 8, the expanding cost is obtained, that is z = 136.2.

Similarly to the problem of 3 × 4 problem, we apply MCCETLTPD-A to 10 × 10

problem given in Tables 9-14. For 10 × 10 problem, we consider the normal case (i.e., the products are not sudden increase), but the products need to be transported within the specified time-limited T = 36   h . By inequality (3) of g ( x i j e ) = 1 2 x i j e ≤ t i j 2 , the route maximum shipping quantity x i j e under the time-limited T = 36   h is obtained shown in Table 15.

For 10 × 10 problem, by calling MCCETLTPD-A based MATLAB computational programs and using the above known parameters as input, the optimum shipping scheme (i.e., maximum flow f i j ) is obtained, see Table 16, and from Table 9 and Table 16, the needed capacity expansion quantity r i j is obtained shown in Table 17. So, the transportation scheme can be completed within time-limited T = 36   h . From Table 11 and Table 17, the expanding cost is obtained, that is z = 44 .

Although there is no sudden increase in goods, the capacity expansion may be required to complete the transportation scheme within the specified time-limited T, so our method can also solve such problems. Furthermore, if we know the transportation cost per unit, we also can get the total cost of completing the transportation scheme, i.e., the sum of transportation cost and expanding cost.