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Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore - 721102, India

- *Corresponding Author:
- Anupam Ojha

Department of Applied Mathematics with

Oceanology and Computer Programming

Vidyasagar University, Midnapore–721102, India

**Tel:**3222 276 554

**E-mail:**[email protected]

**Received** August 17, 2013; **Accepted** January 02, 2014; **Published** January 09, 2014

**Citation:** Ojha A, Mondal SK, Maiti M (2014) A Solid Transportation Problem with Partial Non-Linear Transportation Cost. J Appl Computat Math 3: 150. doi: 10.4172/2168-9679.1000150

**Copyright:** © 2014 Ojha A, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

This paper deals with a capacitated, multi-objective and solid transportation problem with imprecise nature of resources, demands, capacity of conveyance and cost. Here transportation cost is inversely varying with the quantity-to be transported from source to destinations in addition with a fixed per unit cost and a small vehicle cost. The transportation problem has been formulated as a constrained fuzzy non-linear programming problem. Next it is transformed into an equivalent crisp multi-objective problem using fuzzy interval approximation and then solved by Interactive Fuzzy Programming Technique (IFPT) and Generalized Reduced Gradient (GRG) method. An illustrative numerical example is demonstrated to find the optimal solution of the proposed model.

Transportation models; Fuzzy numbers; Interval approximation; Interactive fuzzy programming technique

The classical transportation problem (Hitchcock transportation problem) is one of the sub-classes of non-linear programming problem in which all the constraints are of equality type. In many industrial problem, a homogeneous product is delivered from an origin to a destination by means of different modes of transport called conveyances, such as trucks, cargo flights, goods trains, ships, etc. In general, the real life problems are modeled with multi-objective functions which are measured in different respects and they are non-commensurable and conflicting in nature. Furthermore, it is frequently difficult for the decision maker to combine the objective functions in one overall utility function. In a Solid Transportation Problem (STP) more than one objective is normally considered. In many practical situations, it is realistic to assume that the amount which can be sent on any particular route is restricted by the capacity of that route. Moreover, Appa [1] discussed about the different variations in transportation problem.

It is often difficult to estimate the accurate values of transportation cost, delivery time, quantity of goods delivered, demands, availabilities, the capacity of different modes of trans-port between origins to destinations, etc. Depending upon different aspects, these fluctuate due to uncertainty in judgment, lack of evidence, in sufficient information, etc. i.e., it is not possible to get relevant precise data, which are assumed by several researchers Shell [2]. So, a transportation model become more realistic if these parameters are assumed to be flexible/imprecise in nature i.e., uncertain in non-stochastic sense and may be represented by fuzzy numbers. (Here, for the first time a solid transportation problem is considered with fuzzy parameters like, Shell [2], Jimenez and Verdegay [3].

Based on Das et al. [4], the interval number transportation problems were converted into deterministic multi-objective problems. Grzegorzewski [5] approximated the fuzzy number to its nearest interval. Omar and Samir [6] and Chanas and Kuchta [7] discussed the solution algorithm for solving the transportation problem in fuzz environment. Sakawa and Yano [8] proposed an interactive fuzzy decision making method using linear and non-linear membership functions to solve the multi-objective linear programming problem. Gao and Liu [7,9] presented two-phase fuzzy algorithms. Shaocheng [10] discussed about the interval number linear programming. Verma et al. [11], Bit at al. [12,13], Jimenez and Verdegay [14], Li and Lai [15] and Waiel [16] presented the fuzzy compromise programming approach to multi-objective transportation problem.

In this paper, a capacitated-multi-objective, solid transportation problem is formulated in fuzzy environment with non-linear varying transportation charge and an extra cost for transporting the amount to an interior place through small vehicles (like rickshaw, auto etc.). In the non-linear varying transportation charge, one part is linearly and another nonlinearly proportional to the transported amount. Here, the non-linear cost increases with the increase of transported amount but the rate of increase decreases. The fuzzy quantities and parameters are replaced by equivalent nearest interval numbers and thus a fuzzy multiobjective, capacitated and solid transportation problem is transformed to corresponding crisp multi-objective transportation problems. Membership function is formulated for each objective function from their individual minimum and maximum. These membership functions may be of different types and it may depend upon the decision maker’s (DM) choice/past data, if available. The main contribution of the paper is the mathematical formation of the above mentioned innovative transportation problem in fuzzy environment and its solution. Here for the first time, here man-machine interaction has been introduced in the transportation system. In this way, multi-objective transportation problem is solved using IFPT through GRG. This model is illustrated with an example. In particular, results of some specific transportation models are presented.

**Definition 1:** Fuzzy Number [17]:

If X is a collection of objects denoted generically by x then a fuzzy set Ã in X is a set of ordered pairs:

is called the membership function of x in which maps X to the membership space [0,1].

(i) A is normal, i.e.the supremum of

(ii) A is fuzzy convex, i.e. μA(x) is 1 ∀x∈S and

(iii) A is fuzzy convex, i.e.

**Definition 2:** α-Cut of a fuzzy number [17] (**Figure 1**):

A α-cut of a fuzzy number A is defined as crisp set

A_{α}={x : μ ˜(x) ≥ α, x ∈ X} where α ∈ [0, 1]

**Definition 3:** Interval Approximation [17]:

Aα is a non-empty bounded closed interval contained in X and it can be denoted by A_{α}=[AL(α), AR(α)]. A_{L}(α) and A_{R}(α) are the lower and upper bounds of the closed intervalrespectively. A fuzzy number A with α_{1}, α_{2}-cut A_{α1}=[A_{L}(α1), A_{R}(α1)],

A_{α2}=[A_{L}(α_{2}), A_{R}(α_{2})] and if α_{2} ≥ α_{1}, then A_{L}(α_{2}) ≥ A_{L}(α_{1}) and A_{R}(α_{1}) ≥ A_{R}(α_{2}).

**Definition 4:** Nearest interval approximation of a fuzzy number [18]:

Here a fuzzy number is approximated by a corresponding crisp interval. Suppose A and B, two fuzzy numbers with α-cuts are [A_{L}(α), A_{R}(α)] and [B_{L}(α), B_{R}(α)] respectively. Then the distance between and is

Given A is a fuzzy number. We find a closed interval which is nearest to with respect to metric d. It can be done since each interval is also a fuzzy number with constant α-cut for all α € [0, 1]. Hence (C_{d}(A))α=[C_{L}, C_{R}]. Now we have to minimize

with respect to CL and CR. In order to minimize , it is sufficient to minimize the function . The first partial derivatives are

Solving,

we get,

and

Again since,

and

So D(C_{L}, C_{R}) i.e. is global minimum. Therefore the interval

is nearest interval approximation of fuzzy number with respect to metric d.

Let be a fuzzy number. The α-level interval of A is defined as .

When is a trapezoidal fuzzy numbers then A_{L}(α)=a_{1}+α(a_{2}−a_{1}) and A_{R}(α)=a_{3}−α(a_{4}−a_{3}), 0<α ≤ 1.

By nearest interval approximation method lower and upper limits of the interval are respectively

and

Therefore, the nearest crisp interval number considering as a trapezoidal fuzzy number is [(a_{1}+a_{2})/2, (a_{3}+a_{4})/2].

We consider an interval A=[a_{L}, a_{R}]={a:a_{L} ≤ a ≤ a_{R}, a ∈ R}, where a_{L} and a_{R} are the left and right limits of A respectively. The interval A is also denoted by its centre and width as:

Where a_{C}=(a_{R}+a_{L})/2 and a_{W}=(a_{R}−a_{L})/2 are the center and half-width of A respectively.

Definition 5 [18]: The order relation, ≤RC between A=[a_{L}, a_{R}] and B=[b_{L}, b_{R}] is defined as

Where a_{C}=(a_{R}+a_{L})/2 and a_{W}=(a_{R}−a_{L})/2 are the center and half-width of A respectively.

**Definition 5 [18]:** The order relation, ≤_{RC} between A=[a_{L}, a_{R}] and B=[b_{L}, b_{R}] is defined as

The order relation ≤_{RC} represents the decision maker’s preference for the alternative with the lower minimum value, i.e., if A ≤_{RC} B, then A is preferred to B for the minimization problem.

**Theorem: **The order relation ≤_{RC} satisfied transitive law.

**Proof:** Let A=[a_{L}, a_{R}], B=[b_{L}, b_{R}] and C=[c_{L}, c_{R}] such that

Then subtracting we have a_{L} ≤ b_{L} ≤ c_{L} and

A ≤_{RC} B & B ≤_{RC} C ⇒ A ≤_{RC} C

If possible

Moreover a_{L}=c_{L} and a_{R}=c_{R}

which is the contradiction. Therefore . Thus the theorem proved.

The solid transportation problem (STP) stated by Shell [2] is a generalization of classical transportation problem. Here, we consider m origins(or sources) Oi (i=1,2,....m) , n desti-nations(i.e. demands) Dj (j=1,2,....n) and K conveyances Ek (k=1,2,....K). Let be the fuzzy amount of a homogeneous product available at i-th origin, be the fuzzy demand at j-th destination and represents the fuzzy amount of product which can be carried by k-th conveyance. The fuzzy penalty is associated with transportation cost (for p=1), distance, time etc, (for p=2,3..P) of one unit of a product to transport from i-th source to j-th destination by means of the k-th conveyance for p-th criterions. be the inversely varying cost for different quantity to transport from i-th source to j-th destination by means of the k-th conveyance. The penalty may represent transportation cost, delivery time, quantity of goods delivered etc. The vehicle carrying cost F (x_{ijk}) for the quantity x_{ijk} from i-th source O_{i} to j-th destination D_{j} via k-th conveyance is defined as:

m=[x_{ijk}/v_{c}], v_{c}=vechicle capacity and v=vehicle cost.

Let r_{ijk} be the capacity restrictions on route (i,j) by means of k-th conveyance. Therefore, the fuzzy-capacitated -constrained multicriteria solid transportation problem represented as:

(1)

(2)

0 <γ_{ijk}< 1

subject to the constraints

(3)

(4)

(5)

(6)

Where are fuzzy numbers which may be represented by triangular, trapezoidal or other fuzzy numbers.

Following ($4) fuzzy numbers are approximated to

respectively. Then the earlier transportation model takes the following form:

(7)

(8)

(9)

(10)

(11)

(12)

subject to the constraints

(13)

(14)

(15)

(16)

The constraints (13),(14),(15) and (16) can be written in the following form:

(17)

(18)

(19)

(20)

**Step-1:** Optimize (minimized & maximized) each objective function individually at a time ignoring the others. Let, and be the solutions for the objective functions and (p=1,2,3,...,P) respectively [8].

**Step-2:** Determine the corresponding values for every objective at each solution derived and form a pay-off (**Table 1**).

x | Z_{R}^{1} |
Z_{C}^{1} |
Z_{R}^{2} |
Z_{C}^{2} |
- | - | Z_{C}^{P} |

x_{R}^{1} |
Z_{R}^{1}(x_{R}^{1}) |
Z_{C}^{1} (x_{R}^{1}) |
Z_{R2(xR1}) |
Z_{C}^{2} (x_{R}^{1}) |
- | - | Z_{C}^{P} (x_{R}^{1}) |

x_{C}^{1} |
Z_{R}^{1}(x_{C}^{1}) |
Z_{C}^{1}(x_{C}^{1}) |
Z_{R}^{2}(x_{C}^{1}) |
Z_{C}^{2}(x_{C}^{1}) |
- | - | Z_{C}^{P} (x_{C}^{1}) |

x_{R}^{2} |
Z_{R}^{1}(x_{R}^{2}) |
Z_{C}^{1}(x_{R}^{2}) |
Z_{R}^{2}(x_{R}^{2}) |
Z_{C}^{2}(x_{R}^{2}) |
- | - | Z_{C}^{P}(x_{R}^{2}) |

x_{C}^{2} |
Z_{R}^{1}(x_{C}^{2}) |
Z_{C}^{1}(x_{C}^{2}) |
Z_{R}^{2}(x_{C}^{2}) |
Z_{C}^{2}(x_{C}^{2}) |
- | - | Z_{C}^{P} (x_{C}^{2}) |

- | - | - | - | - | - | - | - |

- | - | - | - | - | - | - | - |

x_{R}^{P} |
Z_{R}^{1}(x_{R}^{P}) |
Z_{C}^{1}(x_{R}^{P}) |
Z_{R}^{2}(xRP) |
Z_{C}^{2}(x_{R}^{P}) |
- | - | Z_{C}^{P}(x_{R}^{P}) |

x_{C}^{P} |
Z_{R}^{1}(x_{C}^{P}) |
Z_{C}^{1}(x_{C}^{P}) |
Z_{R}^{2}(x_{C}^{P}) |
Z_{C}^{2}(x_{C}^{P}) |
- | - | Z_{C}^{P} (x_{C}^{P} ) |

**Table 1:** Pay-off Matrix for the objective functions.

Also consider and be the upper and lower bounds of functions (i=1,2,3,....,P) from **Table 1** respectively.

**Step-3:** Let and be the membership functions of and respectively. The decision maker can select the above membership functions in a subjective manner from among the following four types of functions: linear, exponential, hyperbolic and hyperbolic inverse functions.

**Different types of membership functions:** Let Z_{U} and Z_{L} be the upper and lower bounds of the function Z(x) respectively. For the function Z(x), the corresponding membership functions are defined as follows:

(i) (Type-I) Linear membership function:

μ_{Z} (x)=[Z_{U}−Z(x)]/[Z_{U}−Z_{L}](21)

(ii) (Type-II) Exponential membership function:

μ_{Z} (x)=a[1−exp{−b([Z_{U}−Z(x)]/[Z_{U}−Z_{L}])}](22)

(iii) (Type-III) Hyperbolic membership function:

(23)

where c=tanh^{-1}1.

(iv) (Type-IV) Hyperbolic inverse membership function:

(24)

where

**Step-4:** After determining the membership functions for each of the objective functions, we adopt the maximizing decision proposed by Bellman and Zadeh [19], the resulting problem to be solved is:

Max (min of all μ(x))

subject to x ∈ S where S is the feasible region of problem.

By introducing an auxiliary variable β, the above problem can be transformed into the following equivalent conventional problem.

Maximize β,(25)

(26)

(27)

where 0 ≤ β ≤ 1 and x ∈ S (the set of all feasible solutions)

However, with the four types of membership functions given by (21),(22),(23) and (24), the resulting problem is a non-linear programming problem. The constraint like μ_{z}(x) ≥ β takes the following form:

(28)

Where for Types -I, II, III and IV respectively. The solution of this final problem gives β=β* and the corresponding x=x∗ is the required optimum solution.

To illustrated proposed model numerically, we a set of input data for two origins and destination, three different conveyance and criteria (**Table 2**).

p = 1 | p = 2 | p = 3 | |||

(1, 3, 5, 7) | (2, 5, 6,9) | (6, 9, 10,12) | (1, 2, 6, 7) | ||

(3, 7, 9, 11) | (7, 9, 12,15) | (6, 9, 11,14) | (4, 5, 7, 12) | ||

(3, 6, 9, 11) | (6, 9, 13,17) | (5, 9, 11,13) | (3, 7, 9, 12) | ||

(2, 5, 7, 12) | (5, 8, 11,15) | (5, 7, 9, 12) | (5, 9, 11, 13) | ||

(2, 3, 5, 9) | (1, 5, 9, 10) | (3, 7, 10,13) | (3, 5, 9, 12) | ||

(4, 6, 8, 12) | (4, 7, 8, 14) | (7, 9, 11,15) | (6,10,12,15) | ||

(3, 7, 9, 12) | (2, 7, 10,13) | (6, 9, 13,15) | (4, 6, 8, 10) | ||

(5, 8, 10, 13) | (3, 8, 12, 16) | (4, 7, 9, 12) | (6, 9, 11, 14) | ||

(4, 5, 7, 9) | (4, 8, 12, 14) | (1, 3, 5, 7) | (2, 4, 6, 10) | ||

(5, 9, 12, 14) | (3, 7, 11, 13) | (3, 5, 8, 12) | (3, 7, 10, 13) | ||

(6, 8, 13, 15) | (6, 9, 12, 15) | (2, 5, 9, 10) | (3, 5, 9, 13) | ||

(3, 5, 9, 13) | (5, 9, 12, 15) | (5, 9, 13, 16) | (4, 6, 10, 12) |

**Table 2:** Input data for transportation costs.

v=12, V_{c}=5 and

Let the route capacities, r_{ijk}’s are

r_{111}=16; r_{121}=19; r_{211}=14; r_{221}=15; r_{112}=17; r_{122}=19; r_{212}=16; r_{222}=14;

r_{113} =16 ; r_{123}=19 ; r_{213}=14 ; r_{223}=16;

γ_{111}=0.3; γ_{121}=0.7; γ_{211}=0.45; γ_{221}=0.75;

γ_{112}=0.25; γ_{122}=0.5; γ_{212}=0.25; γ_{222}=0.35;

γ_{113}=0.75; γ_{123}=0.8; γ_{213}=0.95; γ_{223}=0.85;

We now approximate the fuzzy numbers to their nearest intervals, then form the objective functions , , , , , find their individual solutions , , , , , , minimizing , , , , , and separately.

The individual minimum and maximum of , , , , , are shown is pay-off table in **Table 2**. Now solving the problem (25), with all membership functions as linear, we get **Table 3**.

Obj. | ||||||

Max. | 817.548 | 1036.145 | 582.368 | 817.282 | 630.437 | 772.24 |

Min. | 718.0256 | 936.0466 | 522 | 751 | 467.812 | 615.5 |

**Table 3:** Pay-off table.

Following IFPT, the auxiliary variable β is maximized and the optimal solutions are displayed in **Tables 3 and 4** and corresponding decision variable are again if we take the membership functions μ^{1}_{C}, μ^{2}_{C} and μ^{3}_{R} as exponential and others linear, then the solution and the objective functions are shown in **Tables 5 and 6**, and corresponding decision variable are shown in **Table 7**.

β* | Z_{C}^{1}* |
Z_{R}^{1}* |
Z_{C}^{2}* |
Z_{R}^{2}* |
Z_{C}^{3}* |
Z_{R}^{3}* |
Z_{av} |

0.426653 | 775.086 | 988.212 | 556.612 | 789.001 | 561.052 | 693.894 | 727.31 |

**Table 4:** Optimal auxiliary variable.

Conv-1 | Conv-2 | Conv-3 | |||
---|---|---|---|---|---|

12.418 | 0 | 5.903 | 6.676 | 11 | 0 |

0 | 6.582 | 3.676 | 8.244 | 0 | 11.497 |

**Table 5:** The transported amounts.

β* | Z_{C}^{1}* |
Z_{R}^{1}* |
Z_{C}^{2}* |
Z_{R}^{2}* |
Z_{C}^{3}* |
Z_{R}^{3}* |
Z_{av} |

0.25486E-06 | 804.934 | 1021.602 | 551.658 | 775.665 | 615.786 | 757.824 | 754.58 |

**Table 6:** Optimal auxiliary variable.

β* | Z_{C}^{1}* |
Z_{R}^{1}* |
Z_{C}^{2}* |
Z_{R}^{2}* |
Z_{C}^{3}* |
Z_{R}^{3}* |
Z_{av} |

0.394097 | 766.902 | 976.196 | 570.267 | 793.062 | 550.606 | 698.779 | 725.97 |

**Table 7: **The transported amounts.

**Particular cases**

**Case I: Multi-objective and Solid Transportation Problem (i.e., Non-capacitated)**

In this case all routes can be used to transport any amount of products. For this situation we remove the route restrictions (rijk) from the formulation of above problem. For the reduced problem we get the following optimal solution using all the linear membership functions and corresponding decision variable are displayed in **Tables 8 and 9**.

β* | Z_{C}^{1}* |
Z_{R}^{1}* |
Z_{C}^{2}* |
Z_{R}^{2}* |
Z_{C}^{3}* |
Z_{R}^{3}* |
Z_{av} |

0.394097 | 766.902 | 976.196 | 570.267 | 793.062 | 550.606 | 698.779 | 725.97 |

**Table 8: **Optimal auxiliary variable.

Conv-1 | Conv-2 | Conv-3 | |||
---|---|---|---|---|---|

6.938 | 0 | 5.009 | 9.863 | 14.184 | 0.004 |

0 | 12.061 | 6.867 | 2.758 | 0 | 8.312 |

**Table 9:** The transported amounts.

**Case II: Multi-objective and Capacitated Transportation Problem ( i.e., Non-solid)**

In this case only one conveyance is used for transportation with some route restrictions. Since every conveyance has some capacity, sometimes one conveyance cannot transport all products. Due to this reason we take individual conveyance for transportation with route restrictions and all membership functions as linear, the first conveyance is not suitable for transportation: **Table 10** and corresponding decision variable are for conveyance-2 & conveyance- 3 shown in **Tables 11 and 12** respectively.

β* | Z_{C}^{1}* |
Z_{R}^{1}* |
Z_{C}^{2}* |
Z_{R}^{2}* |
Z_{C}^{3}* |
Z_{R}^{3}* |
Z_{av} |

0.020707 | 824.98 | 981.033 | 485.622 | 688.26 | 570.573 | 741.198 | 715.28 |

**Table 10:** Non-solid and capacitated for conveyance-2.

13.01 | 18.989 | ||

14.489 | 13.510 | ||

14.265 | 17.73 | ||

13 | 15 |

**Table 11: **The transported amounts.

β* | Z_{C}^{1}* |
Z_{R}^{1}* |
Z_{C}^{2}* |
Z_{R}^{2}* |
Z_{C}^{3}* |
Z_{R}^{3}* |
Z_{av} |
---|---|---|---|---|---|---|---|

0.463242 | 701.649 | 893.878 | 576.515 | 776.265 | 426.945 | 603.929 | 663.197 |

0.11801 | 770.265 | 956.16 | 488.064 | 678.495 | 619.398 | 790.023 | 717.067 |

**Table 12:** Non-solid and capacitated for conveyance-2.

**Case III: Transportation Problem not being solid and capacitated**

This case is the combination of above two cases (Case I and II). Taking all the membership functions as linear, we find the following optimal solution for the individual conveyance-2 displayed in **Tables 12 and 13**) or conveyance 3 in **Table 14**. But if we take only conveyance-1, then feasible solution does not exist, and corresponding decision variable are for conveyance-2 shown in **Tables 12 and 15**.

3.245 | 28.754 |

24.254 | 3.745 |

**Table 13:** The transported amounts.

β* | Z_{C}^{1}* |
Z_{R}^{1}* |
Z_{C}^{2}* |
Z_{R}^{2}* |
Z_{C}^{3}* |
Z_{R}^{3}* |
Z_{av} |
---|---|---|---|---|---|---|---|

0.460702 | 699.197 | 890.871 | 576.90 | 776.717 | 426.618 | 603.33 | 662.273 |

**Table 14:** For not solid and not capacitated for conv-3.

14.717 | 17.283 |

12.76 | 15.24 |

**Table 15:** The transported amounts.

This paper proposes an interactive fuzzy programming approach to find the optimal com-promise solution for an innovative capacitated, multi-objective, solid transportation problem with fuzzy fixed charge, fuzzy partial varying transportation charge and vehicle cost. Here the transportation model is more realistic and flexible in nature. In the proposed problem the coefficient of objective function, resources, demands, and conveyances are taken as trapezoidal fuzzy number and approximated to corresponding nearest interval. Finally the solution procedures have been illustrated by an example. The present formulation and solution procedures can be applied to transportation problems with other and general fuzzy numbers.

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