Stochastic Fractional Programming Approach to a Mean and Variance Model of a Transportation Problem

Abstract

In this paper, we propose a stochastic programming model, which considers a ratio of two nonlinear functions and probabilistic constraints. In the former, only expected model has been proposed without caring variability in the model. On the other hand, in the variance model, the variability played a vital role without concerning its counterpart, namely, the expected model. Further, the expected model optimizes the ratio of two linear cost functions where as variance model optimize the ratio of two non-linear functions, that is, the stochastic nature in the denominator and numerator and considering expectation and variability as well leads to a non-linear fractional program. In this paper, a transportation model with stochastic fractional programming (SFP) problem approach is proposed, which strikes the balance between previous models available in the literature.

1. Introduction

The transportation engineering problem is one of the most primitive applications of linear programming problems. The basic transportation problem was initially developed by Hitchcock [1] and has grown to the stage wherein supply chain management uses it significantly. Even one can say that supply chain’s success is closely linked to the appropriate use of transportation. Linear fractional transportation problem was first discussed by Swarup [2] and since then it did not receive much attention. This paper deals with a fractional transportation model in which parameters involved in the model are probabilistic in nature.

When the market demands for a commodity are stochastic in nature, the problem of scheduling shipments to a number of demand points from several supply points is a stochastic transportation problem [3]. Jörnsten et al. [4, 5] studied stochastic transportation model for petroleum transport and proposed a cross-decomposition algorithm to solve the said problem. The stochastic transportation problem can be formulated as a convex transportation problem with nonlinear objective function and linear constraints. Holmberg [6] compared different methods based on decomposition techniques and linearization techniques for this problem; Holmberg tried to find the most efficient method or combination of methods. Holmberg also discussed and tested a separable programming approach, the Frank-Wolfe method with and without modifications, the new technique of mean value cross-decomposition, and the more well-known Lagrangian relaxation with subgradient optimization, as well as combinations of these approaches.

Ratio optimization problems are commonly called fractional programs. One of the earliest fractional programs is an equilibrium model for an expanding economy introduced by Von Neumann in 1937, at a time when linear programming hardly existed. The linear and nonlinear models of fractional programming problems have been studied by Charnes and Cooper [7] and Dinkelbach [8]. The fractional programming problems have been studied extensively by many researchers. Mjelde [9] maximized the ratio of the return and the cost in resource allocation problems; Kydland [10], on the other hand, maximized the profit per unit time in a cargo-loading problem. Arora and Ahuja [11] discussed a fractional bulk transportation problem in which the numerator is quadratic in nature and the denominator is linear.

Stochastic fractional programming (SFP) offers a way to deal with planning in situations where the problem data is not known with certainty. Such situations arise where technological aspects of the system under study may be highly complicated or incapable of being observed completely. Stochastic Programming and Fractional Programming constitute two of the more vibrant areas of research in optimization. Both areas have blossomed into fields that have solid mathematical foundations, reliable algorithms and software, and a plethora of applications that continue to challenge current state-of-art computing resources. For various reasons, these areas have matured independently. Many of the existing procedures that are of practical importance for solving stochastic programming and fractional programming problems rely mostly on simplified assumptions. Wide range of applications of stochastic and fractional programming can be seen in [12–17].

A constrained linear stochastic fractional programming (LSFP) problem involves optimizing the ratio of two linear functions subject to some constraints in which at least one of the problem data is random in nature with nonnegative constraints on the variables. In addition, some of the constraints may be deterministic.

The LSFP framework attempts to model uncertainty in the data by assuming that the input or a part thereof is specified by a probability distribution, rather than being deterministic. Gupta [18] described a model on capacitated stochastic transportation problem, which maximizes profitability. LSFP has been extensively studied by Gupta et al. [19, 20] and Charles et al. [14–17, 21–31], the concepts of LSFP are available in [21, 22], various algorithms to solve LSFP have been discussed in [23, 26, 28, 29], financial derivative applications of nonlinear SFP are studied in [25, 27], and multiobjective LSFP problems are discussed in [24, 30]. Charles and Dutta [30] discusses an application to assembled printed circuit board of multi-objective LSFP, and an algorithm to identify redundant fractional objective function in multi-objective SFP is clearly discussed in [31, 32].

In this paper, a special class of transportation problems has been considered wherein the stochastic fractional programming (SFP) is the handy technique to optimize the transportation problem. The said special class of uncapacitated transportation problems has two distinct cost matrices in which costs involved in the problem are random in nature and are assumed to follow normal distribution, and the demand vector under study is also random wherein the demand vector is assumed to follow probability distributions like normal and uniform. The proposed mean-variance model attempts to optimize the profit over shipping cost under uncertain environment, subject to regular supply constraints along with stochastic demand constraints.

The organization of this paper is as follows. Section 2 discusses the uncapacitated transportation problem of SFP along with some basic assumptions. A deterministic equivalent of probabilistic demand constraints are described in Section 3 along with explanation for some of the preliminary properties of transportation problem of SFP and expectation, and also variance and mean-variance models for the uncapacitated transportation problem of SFP are established. In this Section 4 provides an algorithm to solve this problem. Discussion on this paper with a summary and recommendations for future research is in Section 5.

2. The UnCapacitated Transportation Problem of LSFP

This section deals with the uncapacitated TP of LSFP for the distribution of a single homogenous commodity from 𝑚 sources to 𝑛 destinations, where the demand for the commodity at each of the 𝑛 destinations is a random variable. An uncapacitated TP of LSFP in a criterion space is defined as follows: maximize𝑅(𝑋)=𝑁(𝑋)+𝛼𝐷(𝑋)+𝛽=∑^𝑚𝑖=1∑^𝑛𝑗=1𝑝_𝑖𝑗𝑥_𝑖𝑗+𝛼∑^𝑚𝑖=1∑^𝑛𝑗=1𝑐_𝑖𝑗𝑥_𝑖𝑗+𝛽,(2.1) subject to ^𝑛_𝑗=1𝑥_𝑖𝑗≤𝑎_𝑖,𝑖=1,2,…,𝑚,(2.2)_𝑚_𝑖=1𝑥_𝑖𝑗=𝑟_𝑗,𝑗=1,2,…,𝑛,(2.3) where 0≤X_𝑚×𝑛=‖𝑥_𝑖𝑗‖∈𝑅^𝑚×𝑛 is a feasible set, 𝑆={𝑋∣(2.2)-(2.3),𝑋≥0,𝑋∈𝑅^𝑚×𝑛} is nonempty, convex, and compact set in 𝑅^𝑚×𝑛, 𝑥_𝑖𝑗 is an unknown quantity of the good shipped from supply point 𝑖 to demand point 𝑗, profit matrix 𝑁_𝑚×𝑛=‖𝑝_𝑖𝑗‖ which determines the profit 𝑝_𝑖𝑗∼𝑁(𝑢_𝑝𝑖𝑗,𝑠^2𝑝_𝑖𝑗) gained from shipment from 𝑖 to 𝑗, cost matrix 𝐷_𝑚×𝑛=‖𝑐_𝑖𝑗‖ which determines the cost 𝑐_𝑖𝑗∼𝑁(𝑢_𝑐𝑖𝑗,𝑠^2𝑐_𝑖𝑗) per unit of shipment from 𝑖 to 𝑗, the denominator function 𝐷(𝑋)+𝛽 is assumed to be positive throughout the constraint set, scalars 𝛼,𝛽, which determines some constant profit and cost, respectively, supply point 𝑖 must have 𝑎_𝑖 units available, stochastic demand point 𝑗 must obtain 1−𝑙_𝑗 level of 𝑟_𝑗 units, and 1−𝑙_𝑗(0<𝑙_𝑗<1) is the least probability with which 𝑗th stochastic demand constraint is satisfied.

Stochastic equation (2.3) can be rewritten as follows:Pr^𝑚_𝑖=1𝑥_𝑖𝑗≥𝑟_𝑗≥1−𝑙^𝑗Pr_𝑚,𝑗=1,2,…,𝑛,(2.4)_𝑖=1𝑥_𝑖𝑗≤𝑟_𝑗≥1−𝑙^𝑗,𝑗=1,2,…,𝑛.(2.5)

Assumption 2.1. (a) The values of every point of supply and demand are positive.
(b) Total supply is not less than total demand.
(c) Noninteger solutions are acceptable.

3. Deterministic Equivalents of Probabilistic Demand Constraints and E-Model

Let 𝑟_𝑗 be a random variable in constraint (2.4) that follows 𝑁(𝑢_𝑟𝑗,𝑠^2𝑟_𝑗),𝑗=1,2,…,𝑛, where 𝑢_𝑟𝑗 is the 𝑗th mean and 𝑠^2𝑟_𝑗 is the 𝑗th variance. The 𝑗th deterministic demand constraint (2.4) is obtained from Charles and Dutta [21] and is given as follows:Pr^𝑚_𝑖=1𝑥_𝑖𝑗≥𝑟_𝑗≥1−𝑙^𝑗(or)Pr𝑟_𝑗≤^𝑚_𝑖=1𝑥_𝑖𝑗≥1−𝑙^𝑗(or)Pr𝑍_𝑗≤𝑧_𝑗≥1−𝑙_𝑗,(3.1) where 𝑍_𝑗=(𝑟_𝑗−𝑢_𝑟𝑗)/𝑠_𝑟𝑗 follows standard normal distribution and 𝑧_𝑗=(∑^𝑚𝑖=1𝑥_𝑖𝑗−𝑢_𝑟𝑗)/𝑠_𝑟𝑗. Thus, 𝜙(𝑧_𝑗)≥𝜙(𝐾_1−𝑙𝑗), where 1−𝑙_𝑗=𝜙(𝐾_1−𝑙𝑗), is the cumulative distribution function of standard normal distribution. Clearly, 𝜙(⋅) is a nondecreasing continuous function, hence 𝑧_𝑗≥𝐾_1−𝑙𝑗. The 𝑗th deterministic demand constraint (2.4) is as follows: ^𝑚_𝑖=1𝑥_𝑖𝑗≥𝑢_𝑟𝑗+𝐾_1−𝑙𝑗𝑠_𝑟𝑗.(3.2) Similar to constraint (3.2), one can obtain the constraint given below from (2.5): ^𝑚_𝑖=1𝑥_𝑖𝑗≤𝑢_𝑟𝑗+𝐾_𝑙𝑗𝑠_𝑟𝑗.(3.3) Inequalities (3.2) and (3.3) can be combined as follows:𝑢_𝑟𝑗+𝐾_1−𝑙𝑗𝑠_𝑟𝑗≤^𝑚_𝑖=1𝑥_𝑖𝑗≤𝑢_𝑟𝑗+𝐾_𝑙𝑗𝑠_𝑟𝑗.(3.4) Let 𝑟_𝑗 be the uniform random variable which ranges from 𝑢^low_𝑗 to 𝑢^up_𝑗, that is, 𝑟_𝑗∼𝑈(𝑢^low_𝑗,𝑢^up_𝑗), the probabilistic demand constraint in system (2.1) is equivalent to∑^𝑚𝑖=1𝑥_𝑖𝑗≥𝜏_𝑗, where 𝑙^𝑗=1−𝑙_𝑗, and ∫^𝑢up𝑗_𝜏𝑗(𝑑𝑥/(𝑢^up_𝑗−𝑢^low_𝑗))=𝑙^𝑗, that is, 𝜏_𝑗=𝑙_𝑗𝑢^up_𝑗+𝑙^𝑗𝑢^low_𝑗. Hence, the deterministic equivalent of the 𝑗𝑡h probabilistic demand constraint (2.4) is^𝑚_𝑖=1𝑥_𝑖𝑗≥𝑙_𝑗𝑢^up_𝑗+𝑙^𝑗𝑢^low_𝑗.(3.5) Similar to (3.5) one can obtain the constraint given below from (2.5): ^𝑚_𝑖=1𝑥_𝑖𝑗≤𝑙_𝑗𝑢^low_𝑗+𝑙^𝑗𝑢^up_𝑗.(3.6) Constraints (3.5) and (3.6) can be combined as follows:𝑙_𝑗𝑢^up_𝑗+𝑙^𝑗𝑢^low_𝑗≤^𝑚_𝑖=1𝑥_𝑖𝑗≤𝑙_𝑗𝑢^low_𝑗+𝑙^𝑗𝑢^up_𝑗.(3.7)

Definition 3.1. If the total supply lies in the interval of total deterministic demand, the transportation problem of SFP has feasible solutions. Case 1. The normally distributed demand is ∑^𝑛𝑗=1(𝑢_𝑟𝑗+𝐾_1−𝑙𝑗𝑠_𝑟𝑗)≤∑^𝑚𝑖=1𝑎_𝑖≤∑^𝑛𝑗=1(𝑢_𝑟𝑗+𝐾_𝑙𝑗𝑠_𝑟𝑗).Case 2. Uniformly distributed demand the ∑^𝑛𝑗=1(𝑙_𝑗𝑢^up_𝑗+𝑙^𝑗𝑢^low_𝑗)≤∑^𝑚𝑖=1𝑎_𝑖≤∑^𝑛𝑗=1(𝑙_𝑗𝑢^low_𝑗+𝑙^𝑗𝑢^up_𝑗).

Lemma 3.2. The transportation problem of SFP always has a feasible solution, that is, feasible set 𝑆 is nonempty.

Lemma 3.3. The set of feasible solutions is bounded.

Lemma 3.4. The transportation problem of SFP is solvable.

The proof of the above said properties when demand follows normal distribution are as follows: Let 𝑥^∗𝑖𝑗 be defined as 𝑎_𝑖𝑢^𝑟_𝑗+𝐾_1−𝑙𝑗𝑠_𝑟𝑗𝑇₁≤𝑥^∗𝑖𝑗≤𝑎^𝑖𝑢^𝑟_𝑗+𝐾_𝑙𝑗𝑠_𝑟𝑗𝑇₂,𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛,(3.8) where 𝑇₁=∑^𝑛𝑗=1(𝑢_𝑟𝑗+𝐾_1−𝑙𝑗𝑠_𝑟𝑗),𝑇₂=∑^𝑛𝑗=1(𝑢_𝑟𝑗+𝐾_𝑙𝑗𝑠_𝑟𝑗) are positive.

Substituting 𝑥^∗𝑖𝑗 for the supply and demand constraints, that is, from constraints (2.2) and (2.4), the following can be obtained:^𝑛_𝑗=1𝑥^∗𝑖𝑗≤^𝑛_𝑗=1𝑎_𝑖𝑢^𝑟_𝑗+𝐾_1−𝑙𝑗𝑠_𝑟𝑗𝑇₁=𝑎^𝑖𝑇_1𝑛𝑗=1𝑢^𝑟_𝑗+𝐾_1−𝑙𝑗𝑠_𝑟𝑗=𝑎^𝑖,𝑖=1,2,…,𝑚,_𝑛_𝑗=1𝑥^∗𝑖𝑗≥^𝑛_𝑗=1𝑎_𝑖𝑢^𝑟_𝑗+𝐾_𝑙𝑗𝑠_𝑟𝑗𝑇₂=𝑎^𝑖𝑇_2𝑛𝑗=1𝑢^𝑟_𝑗+𝐾_𝑙𝑗𝑠_𝑟𝑗=𝑎^𝑖,𝑖=1,2,…,𝑚,(3.9) and hence ∑^𝑛𝑗=1𝑥^∗𝑖𝑗=𝑎_𝑖.From (3.8), one can obtain^𝑚_𝑖=1𝑎_𝑖𝑢^𝑟_𝑗+𝐾_1−𝑙𝑗𝑠_𝑟𝑗𝑇₁≤^𝑚_𝑖=1𝑥^∗𝑖𝑗≤^𝑚_𝑖=1𝑎_𝑖𝑢^𝑟_𝑗+𝐾_𝑙𝑗𝑠_𝑟𝑗𝑇₂𝑢_𝑟𝑗,+𝐾_1−𝑙𝑗𝑠_𝑟𝑗𝑇_1𝑚𝑖=1𝑎_𝑖≤^𝑚_𝑖=1𝑥^∗𝑖𝑗≤𝑢^𝑟𝑗+𝐾_𝑙𝑗𝑠_𝑟𝑗𝑇_2𝑚𝑖=1𝑢_𝑟𝑗𝑎_𝑖,+𝐾_1−𝑙𝑗𝑠_𝑟𝑗≤𝑢^𝑟𝑗+𝐾_1−𝑙𝑗𝑠_𝑟𝑗𝑇_1𝑚𝑖=1𝑎_𝑖≤^𝑚_𝑖=1𝑥^∗𝑖𝑗≤𝑢^𝑟𝑗+𝐾_𝑙𝑗𝑠_𝑟𝑗𝑇_2𝑚𝑖=1𝑢_𝑟𝑗𝑎_𝑖≤𝑢_𝑟𝑗+𝐾_𝑙𝑗𝑠_𝑟𝑗,+𝐾_1−𝑙𝑗𝑠_𝑟𝑗≤^𝑚_𝑖=1𝑥^∗𝑖𝑗≤𝑢_𝑟𝑗+𝐾_𝑙𝑗𝑠_𝑟𝑗,𝑗=1,2,…,𝑚.(3.10)

Hence, constraints (2.2) and (2.4) are satisfied by 𝑥^∗𝑖𝑗. Since from Assumption 2.1(a)-(b) the constraint (3.2) it follows that 𝑥^∗𝑖𝑗>0,𝑖=1,2,…,𝑚, 𝑗=1,2,…,𝑛, it becomes obvious that 𝑥^∗=(𝑥^∗𝑖𝑗) is a feasible solution of the transportation problem of stochastic fractional programming. Thus it has been clearly shown that the feasible set 𝑆 is not empty.

Further, from (2.2), (3.4), and (3.7) along with nonnegativity constraints, it is clear that 0≤𝑥^∗𝑖𝑗≤𝑎_𝑖,𝑖=1,2,…,𝑚;𝑗=1,2,…,𝑛.

Expectation and variance of the profit and cost function of the probabilistic fractional objective function are defined as follows:𝐸(𝑁(𝑋))=^𝑚_𝑛𝑖=1_𝑗=1𝐸𝑝_𝑖𝑗𝑥_𝑖𝑗+𝛼=^𝑚_𝑛𝑖=1_𝑗=1𝐸(𝐷(𝑋))=_𝑚𝑢_𝑝𝑖𝑗𝑥_𝑖𝑗+𝛼,_𝑛𝑖=1_𝑗=1𝐸𝑐_𝑖𝑗𝑥_𝑖𝑗+𝛽=^𝑚_𝑛𝑖=1_𝑗=1𝑉(𝑁(𝑋))=_𝑚𝑢_𝑐𝑖𝑗𝑥_𝑖𝑗+𝛽,_𝑛𝑖=1_𝑗=1𝑉𝑝_𝑖𝑗𝑥_𝑖𝑗=^𝑚_𝑛𝑖=1_𝑗=1𝑉(𝐷(𝑋))=_𝑚𝑆^{2𝑝𝑖𝑗}𝑥^2𝑖𝑗,_𝑛𝑖=1_𝑗=1𝑉𝑐_𝑖𝑗𝑥_𝑖𝑗=^𝑚_𝑛𝑖=1_𝑗=1𝑆^{2𝑐𝑖𝑗}𝑥^2𝑖𝑗.(3.11) Hence the deterministic fractional objective function is as follows:𝑅^𝐸𝑉(𝑋)=𝑤¹∑^𝑚𝑖=1∑^𝑛𝑗=1𝑢_𝑝𝑖𝑗𝑥_𝑖𝑗+𝛼+𝑤₂∑^𝑚𝑖=1∑^𝑛𝑗=1𝑆^{2𝑝𝑖𝑗}𝑥^2𝑖𝑗𝑤₁∑^𝑚𝑖=1∑^𝑛𝑗=1𝑢_𝑐𝑖𝑗𝑥_𝑖𝑗+𝛽+𝑤₂∑^𝑚𝑖=1∑^𝑛𝑗=1𝑆^{2𝑐𝑖𝑗}𝑥^2𝑖𝑗,(3.12) where 𝑤₁ and 𝑤₂ are preselected nonnegative numbers indicating the relative importance for optimization of the mean and the square root of the variance covariance matrix. The special cases corresponding to 𝑤₂=0 and 𝑤₁=0 are, respectively, known as the 𝐸-model and the 𝑉-model. The objective function (3.12) is very a well-known mean-variance model.

Since the numerator and denominator functions of the fractional objective function (3.12) are in Kataoka’s [32] form and the denominator is assumed to be positive over the bounded feasible set 𝑆, it means that fractional objective function 𝑅^𝐸𝑉(𝑋) is also bounded over the same feasible set 𝑆, and hence it can be concluded that transportation problem of SFP is solvable.

The 𝐸-model for the uncapacitated TP of LSFP when demand follows normal distribution is as follows: maximize𝑅^𝐸(𝑋)=∑^𝑚𝑖=1∑^𝑛𝑗=1𝑢_𝑝𝑖𝑗𝑥_𝑖𝑗+𝛼subjectto_𝑛∑^𝑚𝑖=1∑^𝑛𝑗=1𝑢_𝑐𝑖𝑗𝑥_𝑖𝑗+𝛽,_𝑗=1𝑢𝑥_𝑖𝑗≤𝑎_𝑖,𝑖=1,2,…,𝑚,_𝑟𝑗+𝐾_1−𝑙𝑗𝑠_𝑟𝑗≤^𝑚_𝑖=1𝑥_𝑖𝑗≤𝑢_𝑟𝑗+𝐾_𝑙𝑗𝑠_𝑟𝑗,𝑗=1,2,…,𝑛,(3.13) where 0≤𝑋_𝑚×𝑛=‖𝑥_𝑖𝑗‖∈𝑅^𝑚×𝑛 is a feasible set, 𝑆={𝑋|(2.2)and(3.4),𝑋≥0,𝑋∈𝑅^𝑚×𝑛} is nonempty, convex and compact set in 𝑅^𝑚×𝑛,𝑥_𝑖𝑗 is an unknown quantity of the good shipped from supply point 𝑖 to demand point 𝑗, 𝑅^𝐸(𝑋) is the fractional objective function defined as ratio of the profit function over the cost function, the profit and cost function is assumed to be positive throughout the constraint set, supply point 𝑖 must have at most 𝑎_𝑖 units, deterministic demand point 𝑗 must obtain at least 𝑢_𝑟𝑗+𝐾_1−𝑙𝑗𝑠_𝑟𝑗 units and at most 𝑢_𝑟𝑗+𝐾_𝑙𝑗𝑠_𝑟𝑗 units. Similarly one can define 𝐸-model of system (2.1), when demand follows uniform distribution or/and normal distribution.

Lemma 3.5. This lemma is proposed with the 𝑅^𝐸(⋅) being defined in the earlier section as the fractional objective function: (1.1)𝑅^𝐸(𝜆) is a convex function for 𝜆∈𝑅. (1.2)𝑅^𝐸(𝜆) is strictly decreasing function on 𝑅. (1.3)𝑅^𝐸(𝜆) is continuous function on 𝑅. (1.4) The equation 𝑅^𝐸(𝜆)=0 has unique solution, say 𝜆^∗. (1.5)𝑅^𝐸(𝜆)≥0 for all 𝑥∈𝑆.

Theorem 3.6. A necessary and sufficient condition for 𝜆^∗=∑^𝑚𝑖=1∑^𝑛𝑗=1𝑢_𝑝𝑖𝑗𝑥^∗𝑖𝑗+𝛼∑^𝑚𝑖=1∑^𝑛𝑗=1𝑢_𝑐𝑖𝑗𝑥^∗𝑖𝑗+𝛽=maximize^𝑥∈𝑆∑^𝑚𝑖=1∑^𝑛𝑗=1𝑢_𝑝𝑖𝑗𝑥_𝑖𝑗+𝛼∑^𝑚𝑖=1∑^𝑛𝑗=1𝑢_𝑐𝑖𝑗𝑥_𝑖𝑗+𝛽(3.14) is R^E𝜆^∗=R^E𝑥^∗,𝜆^∗=maximize_𝑥∈𝑆^𝑚_𝑛𝑖=1_𝑗=1𝑢_𝑝𝑖𝑗𝑥_𝑖𝑗+𝛼−𝜆^∗^𝑚_𝑛𝑖=1_𝑗=1𝑢_𝑐𝑖𝑗𝑥_𝑖𝑗+𝛽=0.(3.15)

Note 3. It may be noted that optimal solution 𝑥^∗may not be unique for the extremes (i.e., max/min). The 𝑉-model for the uncapacitated TP of SFP when demand follows normal distribution is as follows: maximizeR^V(𝑋)=∑^𝑚𝑖=1∑^𝑛𝑗=1𝑆^{2𝑝𝑖𝑗}𝑥^2𝑖𝑗∑^𝑚𝑖=1∑^𝑛𝑗=1𝑆^{2𝑐𝑖𝑗}𝑥^2𝑖𝑗(3.16) subject to ∑^𝑛𝑗=1𝑥_𝑖𝑗≤𝑎_𝑖,𝑖=1,2,…,𝑚,𝑢_𝑟𝑗+𝐾_1−𝑙𝑗𝑠_𝑟𝑗≤∑^𝑚𝑖=1𝑥_𝑖𝑗≤𝑢_𝑟𝑗+𝐾_𝑙𝑗𝑠_𝑟𝑗𝑗=1,2,…,𝑛, where 0≤𝑋_𝑚×𝑛=||𝑥_𝑖𝑗||∈𝑅^𝑚×𝑛 is a feasible set, 𝑆={𝑋|(2.2)and(3.4),𝑋≥0,𝑋∈𝑅^𝑚×𝑛} is nonempty, convex, and compact set in 𝑅^𝑚×𝑛, 𝑥_𝑖𝑗 is an unknown quantity of the good shipped from supply point 𝑖 to demand point 𝑗, 𝑅^𝑉(𝑋) is the fractional objective function defined as ratio of standard deviation of the profit function over standard deviation of the cost function, the profit and cost function is assumed to be positive throughout the constraint set, supply point 𝑖 must have at most 𝑎_𝑖 units, deterministic demand point 𝑗 must obtain at least 𝑢_𝑟𝑗+𝐾_1−𝑙𝑗𝑠_𝑟𝑗 units and at most 𝑢_𝑟𝑗+𝐾_𝑙𝑗𝑠_𝑟𝑗 units. Similarly one can define 𝑉-model of system (2.1) when demand follows uniform distribution or/and normal distribution.

Lemma 3.7 .. The following results are true. (2.1)𝑅^𝑉2(𝜆) is a convex function for λ ∈𝑅.(2.2)𝑅^𝑉2(𝜆) is strictly decreasing function on 𝑅.(2.3)𝑅^𝑉2(𝜆) is continuous function on 𝑅.(2.4)The equation 𝑅^𝑉2(𝜆) = 0 has unique solution, say 𝜆^∗.(2.5)𝑅^𝑉2(𝜆)≥0 for all 𝑥∈𝑆.

Theorem 3.8 .. A necessary and sufficient condition for 𝜆^∗=∑^𝑚𝑖=1∑^𝑛𝑗=1𝑆^{2𝑝𝑖𝑗}𝑥^2∗𝑖𝑗∑^𝑚𝑖=1∑^𝑛𝑗=1𝑆^{2𝑐𝑖𝑗}𝑥^2∗𝑖𝑗=maximize_𝑥∈𝑆∑^𝑚𝑖=1∑^𝑛𝑗=1𝑆^{2𝑝𝑖𝑗}𝑥^2𝑖𝑗∑^𝑚𝑖=1∑^𝑛𝑗=1𝑆^{2𝑐𝑖𝑗}𝑥^2𝑖𝑗(3.17) is R^V2𝜆^∗=R^V2𝑥^∗,𝜆^∗=optimize_𝑥∈𝑆^𝑚_𝑛𝑖=1_𝑗=1𝑆^{2𝑝𝑖𝑗}𝑥^2𝑖𝑗−𝜆^∗𝑚_𝑛𝑖=1_𝑗=1𝑆^{2𝑐𝑖𝑗}𝑥^2𝑖𝑗=0.(3.18)

Note 3. It may be noted that optimal solution 𝑥^∗ may not be unique for the extremes (i.e., max/ min). The mean-variance model for the uncapacitated TP of SFP when demand follows normal distribution is as follows: maximizeR^EV(X)=𝑤¹∑^𝑚𝑖=1∑^𝑛𝑗=1𝑢_𝑝𝑖𝑗𝑥_𝑖𝑗+𝛼+𝑤₂∑^𝑚𝑖=1∑^𝑛𝑗=1𝑆^{2𝑝𝑖𝑗}𝑥^2𝑖𝑗𝑤₁∑^𝑚𝑖=1∑^𝑛𝑗=1𝑢_𝑐𝑖𝑗𝑥_𝑖𝑗+𝛽+𝑤₂∑^𝑚𝑖=1∑^𝑛𝑗=1𝑆^{2𝑐𝑖𝑗}𝑥^2𝑖𝑗,(3.19) subject to ∑^𝑛𝑗=1𝑥_𝑖𝑗≤𝑎_𝑖,𝑖=1,2,…,𝑚,𝑢_𝑟𝑗+𝐾_1−𝑙𝑗𝑠_𝑟𝑗≤∑^𝑚𝑗=1𝑥_𝑖𝑗≤𝑢_𝑟𝑗+𝐾_𝑙𝑗𝑠_𝑟𝑗,𝑗=1,2,…,𝑛, where 0≤𝑋_𝑚×𝑛=‖𝑥_𝑖𝑗‖∈𝑅^𝑚×𝑛 is a feasible set, 𝑆={𝑋|(2.2)and(3.4),𝑋≥0,𝑋∈𝑅^𝑚×𝑛} is nonempty, convex, and compact set in 𝑅^𝑚×𝑛, and 𝑥_𝑖𝑗 is an unknown quantity of the good shipped from supply point 𝑖 to demand point 𝑗. Similarly one can define mean-variance model of system (2.1) when demand follows uniform distribution or/and normal distribution.

Theorem 3.9. A necessary and sufficient condition for 𝜆^∗=𝑤¹∑^𝑚𝑖=1∑^𝑛𝑗=1𝑢_𝑝𝑖𝑗𝑥^∗𝑖𝑗+𝛼+𝑤₂∑^𝑚𝑖=1∑^𝑛𝑗=1𝑆^{2𝑝𝑖𝑗}𝑥^2∗𝑖𝑗𝑤₁∑^𝑚𝑖=1∑^𝑛𝑗=1𝑢_𝑐𝑖𝑗𝑥^∗𝑖𝑗+𝛽+𝑤₂∑^𝑚𝑖=1∑^𝑛𝑗=1𝑆^{2𝑐𝑖𝑗}𝑥^2∗𝑖𝑗=maximize_𝑥∈𝑆𝑤₁∑^𝑚𝑖=1∑^𝑛𝑗=1𝑢_𝑝𝑖𝑗𝑥_𝑖𝑗+𝛼+𝑤₂∑^𝑚𝑖=1∑^𝑛𝑗=1𝑆^{2𝑝𝑖𝑗}𝑥^2𝑖𝑗𝑤₁∑^𝑚𝑖=1∑^𝑛𝑗=1𝑢_𝑐𝑖𝑗𝑥_𝑖𝑗+𝛽+𝑤₂∑^𝑚𝑖=1∑^𝑛𝑗=1𝑆^{2𝑐𝑖𝑗}𝑥^2𝑖𝑗(3.20) is R^EV𝜆^∗==mRa^EVxim𝑥i^∗z,e_𝑥∈𝑆𝜆^∗⎡⎢⎢⎣𝑤₁^𝑚_𝑛𝑖=1_𝑗=1𝑢_𝑝𝑖𝑗𝑥_𝑖𝑗+𝛼+𝑤₂⎷_𝑚_𝑛𝑖=1_𝑗=1𝑆^{2𝑝𝑖𝑗}𝑥^2𝑖𝑗−𝜆^∗⎛⎜⎜⎝𝑤₁^𝑚_𝑛𝑖=1_𝑗=1𝑢_𝑐𝑖𝑗𝑥_𝑖𝑗+𝛽+𝑤₂⎷_𝑚_𝑛𝑖=1_𝑗=1𝑆^{2𝑐𝑖𝑗}𝑥^2𝑖𝑗⎞⎟⎟⎠⎤⎥⎥⎦=0.(3.21)

4. Algorithm: Sequential Linear Programming for TP of SFP

(1)Start with an initial point 𝑋⁽⁰⁾ and set the iteration number 𝑡=0 (there are many ways to get the initial guess 𝑋⁽⁰⁾, one among is to solve maximize_𝑥∈𝑆∑^𝑚𝑖=1∑^𝑛𝑗=1𝑢_𝑝𝑖𝑗𝑥_𝑖𝑗). (2)Decide the importance of mean and variance by means of assigning values to 𝑤₁ and𝑤₂.(3)Obtain 𝜆⁽⁰⁾=𝑤¹∑^𝑚𝑖=1∑^𝑛𝑗=1𝑢_𝑝𝑖𝑗𝑥_𝑖𝑗+𝛼+𝑤₂∑^𝑚𝑖=1∑^𝑛𝑗=1𝑆^{2𝑝𝑖𝑗}𝑥^2𝑖𝑗𝑤₁∑^𝑚𝑖=1∑^𝑛𝑗=1𝑢_𝑐𝑖𝑗𝑥_𝑖𝑗+𝛽+𝑤₂∑^𝑚𝑖=1∑^𝑛𝑗=1𝑆^{2𝑐𝑖𝑗}𝑥^2𝑖𝑗.(4.1)(4)Linearize the constraint form of objective function about the points (𝑋^(𝑡),𝜆^(𝑡)) as 𝑅^𝐸𝑉(𝑋,𝜆)≈𝑅^𝐸𝑉(𝑋^(𝑡),𝜆^(𝑡))+∇𝑅^𝐸𝑉(𝑋^(𝑡),𝜆^(𝑡))^𝑇(𝑋−𝑋^(𝑡),𝜆−𝜆^(𝑡))^𝑇.(5)Formulate the approximate TP of LSFP as maximize_𝑥∈𝑆𝜆subjectto𝑅^𝐸𝑉𝑋^(𝑡),𝜆^(𝑡)+∇𝑅^𝐸𝑉𝑋^(𝑡),𝜆^(𝑡)^𝑇𝑋−𝑋^(𝑡),𝜆−𝜆^(𝑡)^𝑇=0.(4.2)(6)Solve the approximating TP of SFP to obtain the solution vector 𝑋^(t+1) and scalar 𝜆^(𝑡+1).(7)Find 𝑅^𝐸𝑉(𝑋^(𝑡+1),𝜆^(𝑡+1)).(8)If|𝑅^𝐸𝑉(𝑋^(𝑡+1),𝜆^(𝑡+1))|≤𝜀, where𝜀 is a prescribed small positive tolerance, all the demand and supply constraints can be assumed to have been satisfied. Hence stop the procedure by considering optimal 𝑋 is approximately equal to 𝑋^(𝑡+1), that is,𝑋^opt=𝑋^(𝑡+1).(9)Else, once again linearize the constraint form of objective function about the points (𝑋^(𝑡+1),𝜆^(𝑡+1)) as 𝑅^𝐸𝑉(𝑋,𝜆)≈𝑅^𝐸𝑉(𝑋^(𝑡+1),𝜆^(𝑡+1))+∇𝑅^𝐸𝑉(𝑋^(𝑡+1),𝜆^(𝑡+1))^𝑇(𝑋−𝑋^(𝑡+1),𝜆−𝜆^(𝑡+1))^𝑇 and add this as an additional constraint to TP of SFP defined in step (4).(10) Increment the iteration number by 1 and go to step (4).

5. Discussion and Future Research

A transportation model with stochastic programming approach is considered, and an algorithm to this effect has been presented. The reason to use SFP was to deal with planning in situations where the problem data is known only in the stochastic environment. Such situations arise in high technological complex systems. This proposed model would provide useful solution under those circumstances when the company likes to optimize the ratio of profit over the cost per unit of shipment in a way to meet the stochastic demands with a clear account for variation. This paper can be extended to an integer solution using branch and bound technique. Mixed model for TP of SFP and stochastic fractional recourse programming may be the interest of future research.

Acknowledgments

The authors are grateful to the Editors, anonymous referees for their valuable comments and suggestions. The author V. Charles is thankful to Carmen Mazzerini at the CENTRUM investigocion for her assistance. Thanks are due to NRF (South Africa) for their financial assistance to the author V. S. S. Yadavalli.