Question

Consider a transportation problem setting where suppliers must make deliveries to satisfy the customer demands. Let m be the number of supply points, n the number of demand points, s i bet the supply available at point i for i∈{1,…,m},d j be the demand at point j for j∈{1,…,n}, and c ij be the unit transportation cost from supply point i for i∈{1,…,m} to demand point j for j∈{1,…,n}. a-) Assume that the problem is balanced, i.e, ∑ i=1 m s i =∑ j=1 n d j , unmet demand is not allowed and total supply can satisfy the total demand. Model this problem as a linear integer program. b-) Now, assume that the total demand exceeds fie total supply, i. e, ∑ i=1 m​ δ i​ <∑ j=1n​ d j . As it is not possible to satisfy the demand of all customers, we would like to ship the products in such a way that the transportation cost does not exceed a given budiget and the total penalty cost for unmet demand is minimized. Let B the given budget, and p j for j∈{1,…,n} be the penalty of unmet demand at demand point j c-) If we slowly increase the budget. B, what would be the trend in the the objective function value for part b-), i.e., does it increase or decrease with increasing budget? Is this change indefinite?

Answer 1

Linear programming is used to model the transportation problem, with an integer program set to minimize transportation costs in the first scenario, and to minimize penalty costs subject to a budget in the second scenario. As the budget increases in scenario b), the total penalty cost will typically decrease, but not indefinitely, since once all demand is met, additional budget increases no longer affect the outcome.

Step-by-step explanation:

The transportation problem in both cases can be modeled using linear programming, a method used in operations research and economics to find the best outcome in a mathematical model with some requirements.

In the first scenario (a), where the problem is balanced and unmet demand is not allowed, you would set up a linear integer program with the objective to minimize the total transportation cost. The decision variables are the number of units to ship from supply point i to demand point j, which are bounded by the supply at point i and the demand at point j, and the total supply equals total demand. The constraints ensure that the supply from each supply point does not exceed its available supply, and that the demand at each demand point is fully satisfied.

In the second scenario (b), where total demand exceeds total supply, you'd modify the objective to minimize the total penalty cost for unmet demand while ensuring that the transportation cost does not exceed the given budget B. This introduces an additional constraint reflecting the budget limit and penalty costs pj which are applied for unmet demand at demand point j.

Lastly, for part c, as we slowly increase the budget (B), the objective function value, which is the total penalty cost, would generally decrease. This is because you have more room within your budget to transport goods and therefore can meet more demand, reducing penalties. However, this change is not indefinite - once all demand is met, any further increase in the budget will not affect the objective function.