Consider the transportation problem and its initial solution in the table below. What is the improvement index for the empty cell Source 1 - Destination 3? $\begin{array}{|l|r|r|r|r|r|}
\hline \text { COSTS } & \text { Dest. 1 } & \text { Dest. 2 } & \text { Dest. 3 } & \text { Dest . 4 } & \text { Supply } \
\hline \text { Source 1 } & 12 & 18 & 9 & 11 & 45 \
\hline \text { Source 2 } & 19 & 7 & 30 & 15 & 145 \
\hline \text { Source 3 } & 8 & 10 & 14 & 16 & 50 \
\hline \text { Demand } & 80 & 30 & 70 & 60 & 240 /240 \
\hline
\end{array}$

$\begin{array}{|lrrrrr|}
\hline \text { Shipments } & \text { Dest. 1 } & \text { Dest. 2 } & \text { Dest. 3 } & \text { Dest 4 } & \text { Row Total } \
\text { Source 1 } & 45 & 0 & 0 & 0 & 45 \
\text { Source 2 } & 35 & 30 & 70 & 10 & 145 \
\text { Source 3 } & 0 & 0 & 0 & 50 & 50 \
\text { Col. Total } & 80 & 30 & 70 & 60 & 240/240 \
\hline
\end{array}$
A) 0
B) -14
C) -23
D) -70
E) 115

Question

Consider the transportation problem and its initial solution in the table below. What is the improvement index for the empty cell Source 1 - Destination 3? $\begin{array}{|l|r|r|r|r|r|}
\hline \text { COSTS } & \text { Dest. 1 } & \text { Dest. 2 } & \text { Dest. 3 } & \text { Dest . 4 } & \text { Supply } \
\hline \text { Source 1 } & 12 & 18 & 9 & 11 & 45 \
\hline \text { Source 2 } & 19 & 7 & 30 & 15 & 145 \
\hline \text { Source 3 } & 8 & 10 & 14 & 16 & 50 \
\hline \text { Demand } & 80 & 30 & 70 & 60 & 240 /240 \
\hline
\end{array}$

$\begin{array}{|lrrrrr|}
\hline \text { Shipments } & \text { Dest. 1 } & \text { Dest. 2 } & \text { Dest. 3 } & \text { Dest 4 } & \text { Row Total } \
\text { Source 1 } & 45 & 0 & 0 & 0 & 45 \
\text { Source 2 } & 35 & 30 & 70 & 10 & 145 \
\text { Source 3 } & 0 & 0 & 0 & 50 & 50 \
\text { Col. Total } & 80 & 30 & 70 & 60 & 240/240 \
\hline
\end{array}$
A) 0
B) -14
C) -23
D) -70
E) 115

Quizplus · Accepted Answer

The improvement index for an empty cell in a transportation problem is calculated by adding the costs of the occupied cells in the smallest closed path that includes the empty cell and subtracting the costs of the occupied cells that would be removed if the empty cell were filled. For Source 1 - Destination 3, the smallest closed path involves Source 1 - Destination 1, Source 2 - Destination 1, and Source 2 - Destination 3. The costs are 12 (Source 1 - Destination 1) + 30 (Source 2 - Destination 3) - 19 (Source 2 - Destination 1) = 23. The improvement index is the cost of the empty cell (9) minus this sum, which is 9 - 23 = -14.

Consider the Transportation Problem and Its Initial Solution in the Table