A manufacturer of hospital supplies has a uniform annual demand for 320,000 boxes of bandages. It costs $10 to store one box of bandages for one year and $ 160 to set up the plant for production. How many times a year should the company produce boxes of bandages in order to minimize the total storage and setup costs?

Question

A manufacturer of hospital supplies has a uniform annual demand for 320,000 boxes of bandages. It costs $10 to store one box of bandages for one year and $ 160 to set up the plant for production. How many times a year should the company produce boxes of bandages in order to minimize the total storage and setup costs?

Answer 1

To minimize the total storage and setup costs, we can use the Economic Order Quantity (EOQ) model. The EOQ formula is given as:

EOQ = √((2DS)/H),

where
D = annual demand (320,000 boxes),
S = setup cost ($160), and
H = holding cost per unit per year ($10).

Let's calculate the EOQ:

EOQ = √((2 * 320,000 * 160) / 10)
= √(10,240,000 / 10)
= √1,024,000
≈ 1,013.79

Rounding this to a whole number, we get an EOQ of 1,014 boxes.

The optimal number of times to produce bandage boxes in a year can be calculated using the formula:

Number of production batches = D / EOQ,

where D = annual demand (320,000 boxes).

Number of production batches = 320,000 / 1,014
≈ 315.81

Rounding to the nearest whole number, the company should produce bandage boxes around 316 times in a year to minimize the total storage and setup costs.

Answer 2

To determine the optimal production quantity that minimizes the total storage and setup costs, we can use the Economic Order Quantity (EOQ) model.

The EOQ formula is given by:
EOQ = √((2DS)/H)

Where:
D = Annual demand (320,000 boxes)
S = Setup cost per production run ($160)
H = Holding cost per unit per year ($10)

First, we calculate the EOQ:
EOQ = √((2 * 320,000 * 160) / 10)

Now, we can calculate the number of production runs per year:
Number of runs = D / EOQ
Number of runs = 320,000 / EOQ

To find the optimal number of times the company should produce boxes of bandages in a year, we need to determine the EOQ value and then calculate the number of runs.

Let's calculate it step by step:

Step 1: Calculate EOQ
EOQ = √((2 * 320,000 * 160) / 10)
EOQ = √(10,240,000 / 10)
EOQ = √1,024,000
EOQ ≈ 1,012

Step 2: Calculate the number of runs
Number of runs = 320,000 / 1,012
Number of runs ≈ 316

Therefore, the company should produce boxes of bandages approximately 316 times a year to minimize the total storage and setup costs.

Answer 3

Assume "store one year" means 365 box-days and not "a year or part thereof".

k=320000 boxes/year
= 320000/365 boxes/day
= 876.7 boxes/day

If supplies are produced every n days, then it must produce nk boxes at each production, and storage requirements are
P=(0+1+2+3+...n-1)k box-days for each production
=> P=(n/2)*(0+n-1)k=n(n-1)k/2 box-days/production

Over the whole year, there are 365/n productions, so total cost is
C(n)=160(365/n)+10(P/365)*(365/n)
=160*(365/n)+10(n-1)k/2

To minimize cost, C'(n)=0 =>
C'(n)=-58400/n²+5k=0 => n=3.65,
so to minimize cost the manufacturer must produce bandages
N=365/n=365/3.65=100 times a year, or every 3.65 days.
Check:
C(3)=28233.7899543379
C(3.65)=27616.43835616438
C(4)=27750.68493150684

So if production is every 4 days, it will cost $134 more than the optimum at 3.65 days which could be awkward to manage (overtime, shifts, etc.)