Context:

During the next two months an automobile manufacturer must meet $($on time$)$ the following demands for trucks and cars: month $1,400$ trucks and $800$ cars; month $2,300$ trucks and $300$ cars. During each month at most $1000$ total vehicles can be produced.

Each truck uses two tons of steel, and each car uses one ton of steel. During month $1,$ steel costs $$700$ per ton; but during month $2,$ the cost of steel is projected to rise to $$1,000$ per ton. At most $2500$ tons of steel can be purchased each month. $($Steel can be used only during the month in which it is purchased.$)$

At the beginning of month $1,100$ trucks and $200$ cars are in the inventory. Some of the cars and trucks produced in month $1$ can be held in inventory and sold in month $2.$ At the end of each month, a holding cost of $$200$ per vehicle is assessed.

The objective is to minimize the total cost of the steel plus the cost of holding any excess inventory from month $1$ to month $2$ while meeting the demand requirements in each month.

Hints:

$$ You will need decision variables for the numbers of cars and of trucks produced in each month.

$$ You may also need decision variables for any additional cars and trucks produced in month $1$ that will be carried over in inventory until month $2.$

$$ Assume that you will not have any excess inventory of cars or trucks at the end of month $2.$

$$ You will need two constraints for the demands for cars in each month and two constraints for the demands for trucks in each month. You will also need two constraints for the total capacities of $1000$ vehicles in each month and two constraints for availability of steel in each month, for a total of eight constraints.

Questions:

What is the total cost $($the value of the objective function$)$ of the optimal solution?

In the optimal solution, how many cars are produced in month $1?$

In the optimal solution, how many trucks are produced in month $1?$

In the optimal solution, how many cars are produced for inventory in month $1?$

In the optimal solution, how many trucks are produced for inventory in month $1?$

In the optimal solution, are the constraints on the amount of steel that can be purchased each month binding constraints?

In the optimal solution, are the shadow prices for cars or trucks larger?

If you could increase production capacity by $100$ vehicles in only one of the two months, would it be better to increase capacity in month $1$ or month $2?$Data

Begin Car Inventory

Begin Truck Inventory

Inventory Holding Cost

cars

$ per vehicle per month

Decision Variables

Cars Produced in Month $1$

Cars Produced in Month $2$

Trucks Produced in Month $1$

Trucks Produced in Month $2$

Cars Produced for Inventory in Month $1$

Trucks Produced for Inventory in Month $1$

Objective Function

Minimize Total Cost

Constraints

Demand

Production