Q$2.$ To calculate the project on$-$hand inventories, planned production, and available$-$to$-$promise inventory, we need to consider the current inventory, customer orders, and the forecasted demand for June and July.

Given information:

Forecasted demand for June: $120$ pumps $(30$ pumps per week for $4$ weeks$)$

Forecasted demand for July: $160$ pumps $(40$ pumps per week for $4$ weeks$)$

Current inventory: $64$ pumps

Let's break down the calculations for each of the three components:

$1.$ Project On$-$Hand Inventories:

This represents the total inventory available at the beginning of each month. To calculate the project on$-$hand inventories for June and July, we need to consider the current inventory and subtract the customer orders that have been committed $($booked$)$ and must be filled.

June:

Beginning Inventory: $64$ pumps

Customer Orders: None mentioned.

Project On$-$Hand Inventory for June $=$ Beginning Inventory $-$ Customer Orders

$=64$ pumps $-0$ pumps

$=64$ pumps

July:

Beginning Inventory: Project On$-$Hand Inventory for June

Customer Orders: None mentioned.

Project On$-$Hand Inventory for July $=$ Project On$-$Hand Inventory for June $-$ Customer Orders

$=64$ pumps $-0$ pumps

$=64$ pumps

Explanation:

Therefore, the project on$-$hand inventories for both June and July are $64$ pumps.

Planned Production

This represents the production needed to meet the forecasted demand for each month. We can calculate the planned production by subtracting the project's on$-$hand inventory from the forecasted demand for each month.

June:

Forecasted Demand: $120$ pumps.

Project On$-$Hand Inventory: $64$ pumps

Planned Production for June $=$ Forecasted Demand $-$ Project On$-$Hand Inventory $=120$ pumps $-64$ pumps

$=56$ pumps

July:

Forecasted Demand: $160$ pumps.

Project On$-$Hand Inventory: $64$ pumps

Planned Production for July $=$ Forecasted Demand $-$ Project On$-$Hand Inventory

$=160$ pumps $-64$ pumps

$=96$ pumps

Explanation:

Therefore, the planned production for June is $56$ pumps, and for July is $96$ pumps.

Q$3.$ Explanation:

To formulate the objective function and constraints for the construction firm's decision$-$making problem, we need to maximize the total profit while considering the available resources. Let's define the decision variables and establish the constraints:

Decision Variables:

Let x be the number of model A houses to build.

Let y be the number of model B houses to build.

Objective Function:

The objective is to maximize the total profit.

Maximize:

Profit $=10000$x $+20000$y

Constraints:

Labor hours constraint:

$4000$x $+10000$y $<=400000$

Stone constraint:

$2$x $+3$y $<=150$

Lumber constraint:

$2000$x $+2000$y $<=200000$

Non$-$negativity constraint:

x $>=0$

y $>=0$

Explanation:

These constraints ensure that the firm does not exceed the available labor hours, stone, and lumber while maintaining non$-$negativity for the number of houses built.