Suppose we relax the first assumption and allow for multiple items that are independent except for a restriction on the amount of space available to store the products. The following model describes this situation:

Let Dj $=$annual demand for item jCj $=$unit cost of item jSj $=$cost per order placed for item ji $=$inventory carrying charge as a percentage of the cost per unitW $=$the maximum amount of space available for all goodswj $=$space required for item j

The decision variables are Qj$,$ the amount of item j to order. The model is:

In the objective function, the first term is the annual cost of goods, the second is the annual ordering cost $($D$($j$)/($Q$)$j is the number of orders$),$ and the last term is the annual inventory holding cost $($Q$($j$)/(2)$ is the average amount of inventory$).$

Set up a spreadsheet model for the following data:

Item $1$Item $2$Item $3$Annual Demand$3,0003,0002,000$Item Cost $($$$)905070$Order Cost $($$$)150135125$Space Required $($sq$.$ feet$)502540$

W $=5,000$

i $=0\mathrm{.}3$

Solve the problem using Excel Solver. Hint: You will need to start with decision variable values that are greater than $0$ for Solver to find a solution.

If required, round your answers to two decimal places.

Optimal Solution:

Q$1=$

Q$2=$

Q$3=$

If required, round your answer to the nearest dollar. Do not round intermediate calculations.

Total cost $=$

Let S represent the amount of steel produced $($in tons$).$ Steel production is related to the amount of labor used $($L$)$C S$=20$L$^(0\mathrm{.}30)$C$^(0\mathrm{.}70)$

In this formula L represents the units of labor input and C the units of capital input. Each unit of labor costs $$50,$ and each unit of capital costs $$100.$

$($a$)$ Formulate an optimization problem that will determine how much labor and capital are needed in order to produce $50,000$ tons of steel at minimum cost. If

your answer is zero, enter $"0".$

$($b$)$ Solve the optimization problem you formulated in part a$.$ Hint: When using Excel Solver, start with an initial L$>0$ and C$>0.$

Do not round intermediate calculations. Round your answers to the nearest whole numbe Let $\backslash ($ S $\backslash )$ represent the amount of steel produced $($in tons$).$ Steel production is related to the amount of labor used $(\backslash ($ L $\backslash ))$ and the amount of capital used $(\backslash ($ C $\backslash ))$ by the following function:

$\backslash [$

S$=20$ L$^\{0\mathrm{.}30\}$ C$^\{0\mathrm{.}70\}$

$\backslash ]$

In this formula $\backslash ($ L $\backslash )$ represents the units of labor input and $\backslash ($ C $\backslash )$ the units of capital input. Each unit of labor costs $\backslash (\backslash$$ $50\backslash ),$ and each unit of capital costs $\backslash (\backslash$$ $100\backslash ).$

$($a$)$ Formulate an optimization problem that will determine how much labor and capital are needed in order to produce $50,000$ tons of steel at minimum cost. If your answer is zero, enter $"0".$

$($b$)$ Solve the optimization problem you formulated in part a$.$ Hint: When using Excel Solver, start with an initial $\backslash ($ L$>0\backslash )$ and $\backslash ($ C$>0\backslash ).$

Do not round intermediate calculations. Round your answers to the nearest whole number.

$\backslash ($ L$=\backslash )$

$\backslash ($ C$=\backslash )$

Cost $\backslash (=\backslash$$ $\backslash )$