Question

A company must allocate oil produced from their two plants to meet all the demands of two markets. The cost per litre of allocation from plant \( i \in \{1,2\} \) to market \( j \in \{1,2\} \) is denoted by \( C_{ij} \). The market demand \( D_j \), plant production capacity \( K_i \) and \( C_{ij} \) values are given in the table.

	Market 1 (j = 1)	Market 2 (j = 2)	Capacity \( K_i \) (litres)
Plant 1 (i = 1)	\( C_{11} = 250 \)	\( C_{12} = 280 \)	\( K_1 = 500 \)
Plant 2 (i = 2)	\( C_{21} = 150 \)	\( C_{22} = 180 \)	\( K_2 = 600 \)
Demand \( D_j \) (litres)	\( D_1 = 300 \)	\( D_2 = 400 \)

The company hired an intern to formulate an optimization model to decide on the quantity (\( X_{ij} \)) to be allocated from plant \( i \) to market \( j \), and the formulation is given below:

Minimize \( Z = \sum_{i=1}^{2} \sum_{j=1}^{2} C_{ij} X_{ij} \)

Subject to:
\( \sum_{j=1}^{2} X_{ij} \le K_i \quad \forall i \in \{1,2\} \)
\( \sum_{i=1}^{2} X_{ij} \le D_j \quad \forall j \in \{1,2\} \)
\( X_{ij} \ge 0 \)

The optimal value of the objective function of the linear programming problem formulated by the intern is ______ (in integer).

Hint:

Check the direction of the demand constraint carefully. A "less than or equal to" sign on demand does not force any oil to actually be delivered.

Answer 1

Correct Answer: 0

Step 1: Understanding the Concept:
This question is not really about solving a transportation problem, it is about reading the mathematical model exactly as written.
We must find the optimal value of the LP as the intern formulated it, even if the formulation is a poor model of the real allocation problem.

Step 2: Key Formula or Approach:
Look at the direction of every constraint.
The capacity constraints \( \sum_j X_{ij} \le K_i \) are correct for a "cannot exceed capacity" condition.
But the demand constraints are also written as \( \sum_i X_{ij} \le D_j \), an upper bound, not a requirement to meet demand.
Since \( X_{ij} \ge 0 \) is also allowed, the point \( X_{ij} = 0 \) for all i, j satisfies every constraint in the model.

Step 3: Detailed Explanation:
Check feasibility of the all-zero solution:
Capacity: \( 0 \le 500 \) and \( 0 \le 600 \), both true.
Demand: \( 0 \le 300 \) and \( 0 \le 400 \), both true.
Non-negativity: \( 0 \ge 0 \), true.
So \( X_{11}=X_{12}=X_{21}=X_{22}=0 \) is a feasible point.
All cost coefficients \( C_{ij} \) are strictly positive (250, 280, 150, 180), so any positive shipment only increases the objective \( Z \).
Since the model minimizes \( Z \) and nothing forces any shipment to be positive, the minimum is achieved by shipping nothing at all:
\[ Z_{min} = 250(0) + 280(0) + 150(0) + 180(0) = 0 \]

Final Answer:
The flawed formulation allows zero shipment to be optimal, so the optimal objective value is 0.
\[ \boxed{Z_{min} = 0} \]