Chapter $8,$ Assignment $1$

$1.(10$ pts$.)$ Consider the transportation problem having the following parameter table:

Destination

$12345$ Supply

$1246574$

$2763$ M $46$

Source $3875256$

$4000004$

Demand $44255$

Use each of the following methods to obtain an initial basic feasible solution.

a$.$ Northwest Corner Rule

b$.$ Vogel's Approximation method

c$.$ Russell's Approximation method

From one of the initial BF solutions $($your choice$),$ conduct the optimality test, and show iterations to determine the optimal solutions.

$2.(4$ pts$.)$ Two reservoirs are available to supply the water needs of three cities. Each reservoir can supply up to $50$ million gallons of water per day. Each city would like to receive $40$ million gallons per day. For each million gallons per day of unmet demand, there is a penalty. At city $1,$ the penalty is $$20$; at city $2,$ the penalty is $$22$; and at city $3,$ the penalty is $$23.$ The cost of transporting $1$ million gallons of water from each reservoir to each city are shown in the following table. Formulate a balanced transportation problem $($by introducing dummy demands and$/$or dummy supplies$)$ that can be used to minimize the sum of shortage and transport costs. Do not work iterations, just set up the new table.

To

From City $1$ City $2$ City $3$

Reservoir $1$ $$7$ $$8$ $$10$

Reservoir $2$ $$9$ $$7$ $$8$

$3.(6$ pts$.)$ A company supplies goods to three customers, who each require $30$ units. The company has two warehouses. Warehouse $1$ has $40$ units available and warehouse $2$ has $30$ units available. The costs of shipping $1$ unit from warehouse to customer are shown in the following table. There is a penalty for each unmet customer unit of demand; with customer $1$ a penalty cost of $$90$ is incurred; with customer $2,$ $$80$; and with customer $3,$ $$110.$ Formulate a balanced transportation problem to minimize the sum of shortage and shipping costs.

To

From Customer $1$ Customer $2$ Customer $3$

Warehouse $1$ $$15$ $$35$ $$25$

Warehouse $2$ $$10$ $$50$ $$40$

Use Vogel's approximation followed by iterations to optimize this transportation problem.