$(7)$ Derive the EOQ model, optimal $Q^{**},$ and optimal $Y(Q^{**})$ for the case that the assumption of nstantaneous production is relaxed. Show all steps including derivation of $Y(Q)$ and steps for determining optimal $Q^{**}$ and $Y(Q^{**})-$ Assume that production is linear.

$(3)$ Discuss and show how the model and the process for determining $Q^{**}$ and $Y(Q^{**})$ would change if backorders were allowed.