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# 数学代考|运筹学代写Operations Research代考|RENR535 ASSUMPTIONS OF TRANSPORTATION PROBLEMS

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## 数学代考|运筹学代写Operations Research代考|ASSUMPTIONS OF TRANSPORTATION PROBLEMS

• Requirement assumption: Each supplier has a fixed number of units to be supplied, i.e. it has a fixed output that has to be supplied to various destinations. Each demand centre has a fixed demand, which is to be met by the number of units supplied from various and limited suppliers.
• Feasibility: A transportation problem is considered to be feasible if total output from various supply sources equals total demand of various destinations. From Table 5.1, this can be illustrated as:
$$\begin{gathered} \mathrm{b}{1}+\mathrm{b}{2}+\mathrm{b}{3}=\mathrm{c}{1}+\mathrm{c}{2}+\mathrm{c}{3}+\mathrm{c}{4} \ \text { i.e. } \sum \mathrm{b}{\mathrm{i}}=\sum \mathrm{c}_{\mathrm{j}} \end{gathered}$$
In some cases, this assumption gets violated as practically each source indicates its maximum capacity to produce and each demand centre indicates its maximum demand. However, such a situation can be resolved, which will be discussed later in detail.
• Cost assumption: the cost of transporting units from one source to a particular destination is directly proportional to the number of units to be transported. For example, if the transportation cost per unit from one source to a destination is $\$ 5$, then transporting ten units would be$\$50$.
• Integer values: A transportation problem would have every demand and supply value as an integer value. The number of units allocated to each destination and cost of transporting are also integer values.

## 数学代考|运筹学代写Operations Research代考|Formulation of model

The above discussion shows that a transportation problem can be solved through its formulation by using a linear programming technique. Such an LPP model can further be reached to a final solution by solving it through the simplex method. However, the above example involving only a few resources in terms of supply and demand resulted in the creation of quite a tedious LPP. It would become more complex if the problem consisted of many more supply and distribution centres, which happen in many industrial problems. Such an issue can be resolved by applying the transportation version of the simplex method. Solving of such a version is detailed below.

The first step is to check the feasibility of transportation problem. Feasibility is tested by comparing total demand with total supply. The transportation model demands them to be equal. In this case:
Total supply $=55,000+40,000+25,000=120,000$
Total demand $=65,000+45,000+10,000=120,000$
As both are equal, we can proceed with further steps. Thus, the model would be as shown in Table 5.3.

The simplex form of the above model shows that the problem of minimization of transportation cost is limited by six constraints – three of which indicate supply constraints and three demand constraints. All the constraints are equality constraints and so solving it by the simplex method would involve the introduction of six artificial variables. These artificial variables in the form of basic variables would result in an initial feasible solution satisfying non-negativity criteria. However, a final feasible solution would be achieved through numerous iterations, making the entire procedure cumbersome. The transportation model solves the same problem by using a simpler procedure where directly a basic feasible model is constructed in the form of the transportation initial table shown in Table 5.3.

## 数学代考|运筹学代写Operations Research代考|ASSUMPTIONS OF TRANSPORTATION PROBLEMS

$$\mathrm{b} 1+\mathrm{b} 2+\mathrm{b} 3=\mathrm{c} 1+\mathrm{c} 2+\mathrm{c} 3+\mathrm{c} 4 \text { i.e. } \sum \mathrm{bi}=\sum \mathrm{c}_{\mathrm{j}}$$

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