Chapter 4, Problem 21RP

Explanation of Solution

Formulating an LP:

Suppose that,

$\begin{array}{l}{w}_t=Numberofshoemakersintheworkforceduringmontht.\hfill \\ h_t=Numberofshoemakershiredatthebeginningofthemontht.\hfill \\ f_t=Numberofshoemakersfiredatthebeginningofthemontht.\hfill \\ r_t=Pairofshoesproduceduringregulartimeinmontht.\hfill \\ o_t=Pairofshoesproduceduringovertimeinmontht.\hfill \\ i_t=Pairofshoesintheinventoryattheendofmontht.\hfill \\ z=Costofthecompany.\hfill \end{array}$

The regular salary of a shoemaker is $2,000 per month and it costs $1,500 to hire a worker and $1,900 to a fire a worker.

The overtime labor cost is as follows:

$50\left(\frac{15}{60}\right)=\$12.5$

The material cost of a pair of shoes is $20.

The labor cost is:

$\frac{2000}{150}\left(\frac{15}{60}\right)=\$3.33$

Therefore, the total production cost of a pair of shoes is:

$20+3.33=\$23.33$

Therefore, the monthly cost of holding shoes in inventory is:

$23.33(\frac{3}{100})=\$0.7$

Therefore, the cost of the company is as follows:

$z=2000{\displaystyle \sum _{t=1}^6{w}_t}+1500{\displaystyle \sum _{t=1}^6{h}_t+1900{\displaystyle \sum _{t=1}^6{f}_t+0}}$