Transportation Problems

            Transportation plays an important role in our economy as well as in managerial decision making. Rather than to cover entire field of transportations we will focus on one particular type of transportation problem. This type of transportation problem usually involves the physical movement of goods and services from various supply origins to multiple demand destinations within the given constraints of supply and demand in such a way that the total transportation cost is minimized. A transportation problem is a special type of linear programming (LP) problem that may be solved using the simplex method. But even small transportation problems will involve large number of variables and linear constraints and a direct application of the simplex method may be prohibitive even for electronic computers. However a transportation problem has a special mathematical structure which permits it to be solved by a fairly efficient method known as transportation method.

            The basic transportation problem was originally developed by F.L. Hitchcock (1941) in his study entitled “the distribution of a product from several sources to numerous locations.” In 1947, T.C. Koopmans independently published a study on “optimum utilization of the transportation system.” Subsequently the linear programming formulation and the associated systematic procedure for solution were given by George B. Dantzig (1951) several extensions of its model and methods have subsequently been developed.

Transportation Models

            Transportation models deals with the transportation of a product manufactured at different plants or factories (supply origins) to a number of manufactured at different warehouses (demand destinations). The objective is to satisfy the destination requirements within the plants capacity constraints at the minimum transportation cost. Transportation models thus typically arise in situations involving physical movement of goods from plants to warehouses, warehouses to wholesalers wholesalers to retailers and retailers to customers. Solution of the transportation models requires the determination of how many units should be transported from each supply origin to each demands destination in order to satisfy all the destination demands white minim sing the total associated cost of transportation.

            The easiest way to recognize a transportation problem is to consider a typical situation as shown in the following. Assume that a manufacturer has three plants P₁ p₂   producing the same product. From these plants the product is transported to three warehouses w₁ w₂ and w₃. Each plant has a limited supply (capacity) and each warehouse has specific demand. Each plant can transport to each warehouse but the transportation costs vary for different combinations. The problem is to determine the quantity each plant should transport to each warehouse in order to minimize total transportation costs.

            It was stated earlier that a transportation problem is a special type of LP problem. To illustrate this, let us see how this problem can formulated as a LP problem. Let x_ij represent the quantity transported from plant p_i to warehouse W_j. similarly C_ij is the per unit transportation cost from plant p_i to warehouse w_j the objective is the minimize total transportation costs. The LP objective function is:

Minimize Z= c₁₁ x₁₁ + c₁₂x₁₂ + c₁₃x₁₃ + c₂₁x₂₁ + c₂₂ x₂₂

                                              + c₂₃ x₂₃ + c₃₁ x₃₁ + c₃₂ x₃₂ + c₃₃x₃₃      

The supply constraints are :

                                       X₁₁ + x₁₂ + x₁₃ = s₁

                                       X₂₁ + x₂₂ + x₂₃ = s₂

                                       X₃₁ + x₃₂ +x₃₃ = s₃

The demand constraints are:

                                       X₁₁ + x₂₁ + x₃₁ = d₁

                                       X₂₁ + x₂₂ + x₃₂ = d₂

                                       X₃₁ + x₃₂ +x₃₃ = d₃

And x_ij > 0 for I = 1,2,3; j = 1,2,3. Additionally it is also assumed the total supply available at the plants will exactly satisfy the demand required at the destinations, i. e., s₁ +s₂ +s₃= d₁+d₂ +d₃

Remarks 1. The problem where total supply is equal to total demand is called the balance transportation problem. If the total supply is not equal to total demand then such types of problem are referred to as unbalanced transportation problems and will be discussed later in this chapter.

2. a total of six constraints are given one for demand and supply. It can be established theoretically that for the above transportation problem only five (rather than six) constraints are needed to get the feasible solution. since total supply is equal to total demand any solution satisfying five of  the six constraints will also satisfy the remaining constraint. Therefore in general, if we have m rows (supply) any n columns (demand) in a given transportation problem, then the problem can be solved completely if we have exactly (m+n-1) basic variables. Thus a basic feasible solution to a balance transportation problem would be represented in the transportation table as having exactly (m+n-1) positive X_ij  ‘s (allocations). These allocations are referred to as occupied cells and others as unoccupied (empty) cells. If the numbers of occupied cells are less than (m+n-1) allocations then it becomes a case of degeneracy which will be discussed in a later section of the chapter.

3. The general transportation problem with m- factories (capacity centres or source of supply) and n- warehouses (requirement centres).

            The problem of the company is to distribute the available product to different warehouses in such a way so as to minimize the total transportation cost for all possible factory-warehouse shipping patterns.

I= index for origins (factory); I = 1,2,…,m

J= index for destination (warehouse); j=1,2,…,n

x_ij = number of units shipped from origin I to destination j.

c_ij = cost per unit of shipping from origin I to destination j.

s_i = supply or capacity in units at origin i.

d_j = demand in units at destination j.

(i)                 The objective is to determination x_ij that would minimize the total transportation cost:

Z= x₁₁ c₁₁ + x₁₂c₁₂ +…..+ x_1nc_1n

   + x₂₁c₂₁ + x₂₂c₂₂ +….+ x_2n c_2n

             :           :          :               :

           .           .          .               .

   + x_m1 c_m1 + x_m2c_m2+….+ x_mn c_mn

Subject to the linear constraints:

(ii)               Total supply from the ith origin to all the destinations is equal to total quantity produced at the ith origin, i.e.,

X₁₁ + x₁₂ +…. + x_in = si; i= 1,2,…,m

(iii)             Total quantity transported at the jth destination from various origin s should be equal to the quantity required at the jth destination, i.e.,

X_ij + x_2j + .. … + x_mj = d_j; j= 1,2,..n

(iv)                                          X_ij > 0, for all I and j.

The general mathematical model may be given as follows:

Minimize                         Z = m   n

                                               ?    ?       c_ij x_ij

                                               I=1 j=1

Subject to the constraints

                         n

                         ? x_ij = s_i; i=1,2,………….,m   (capacity constraint)

                          J=1

                         M

                         ? x_ij = d_j; j=1,2,…………..,n     (requirement constraint)

                         I= 1

                                    Xij > 0, for all i.j.,

For a feasible solution to exist, it is necessary that total capacity equals total requirement i.e.,

                                                  m             n

                                                  ?    s_i=    ?

                                                  I=1         j=1

4. Feasible Solution. A set of non-negative values x_ij i=1,2,…….., m; j=1,2,……, n that satisfies (ii), (iii) and (iv) is called a feasible solution to the transportation problem.

5. Basic Feasible. An initial feasible solution with an allocation of (m+n-1) number of variables, x_ij; i=1,2,……..m; j =1,2,…,n is called a basic feasible solution.

6. Optimum Solution. A feasible solution (not necessarily basic) is said to be optimum feasible solution.

7. Balanced or Unbalanced Transportation Problems. A transportation problem can be balanced or unbalanced. It is said to be balanced if the total demand of all the warehouses equals the amount produced in all the factories. If in reality capacity is greater than requirement then a dummy warehouse may be used to create desired equality if capacity is less than requirement then be dummy factory may be introduced. The transportation cost in both the dummy cases is assumed to be zero. The concept of unbalanced transportation problem will be discussed later in this chapter.

8. Dual. This general transportation problem with (m+n) constraints is expressed in the form of equalities instead of inequalities. We convert each constraint to an equivalent pair of inequalities before we can formulate the dual. The equivalent can be expressed as follows:

Minimize      Z = ?    ?     c_ij x_ij

                             I=1/j=1

Subject to the constraints

                                 N

                                 ?     x_ij > s_i ;                                          I = 1,2,…..,m

                                 I=1

                                 N

                                 ?     (-x_ij) > - s_i;                                     I = 1,2,…….,m

                                 M

                                 ?      x_ij > d_j;                                          j = 1,2,….....,n

                                 M

                                 ?      (-x_ij) > -d_j                                      j = 1,2,….....,n

                                 I=1

                                 x_ij > 0, for all I j

Let u_i + and u_j- be the decision variables corresponding to capacity constraint i (i=1,2,…,m). Similarly v_j+ and v_j for the requirement constraint j (j=1,2,….,n). The dual model of the above problem is given as follows:

Maximize Z* =  ?  (u_i⁺-u_i^-) s_i + ?  (v_j⁺+v_j^-) d_j

                             I=1

Subject to the constraints

                             (u_i⁺-u_i^-) + (v_j⁺-v_j^-) < c_ij, for all i.j

                                     U_i⁺, u_i^-,v_i⁺,v_j^- > 0, for all i.j

                                     U_i = u_i⁺-u_i^- and v_j= v_j⁺-v_j^-

The revised dual model can now be written as

Maximize      Z* = ?   u_is_i + ?   v_jd_j

                              I=1         j=1

Subject to the constraints

                             U_i + v_j < c_ij [for all ij]

And                      u_j, v_j unrestricted in sing for all i.j

The dual solution of the transportation problem implicitly measures the comparative locational advantage of the sites of factories (source of capacities) and the delivered price of the product. 

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