The Fish House (TFH) in Norfolk, Virginia, sells fresh fish and seafood. TFH rec

Question

The Fish House (TFH) in Norfolk, Virginia, sells fresh fish and seafood. TFH receives daily shipments of farm-raised trout from a nearby supplier. Each trout costs $2.45 and is sold for $3.95. To maintain its reputation for freshness, at the end of the day, TFH sells any leftover trout to a local pet food manufacturer for $1.25 each. The owner of TFH wants to determine how many trout to order each day. Historically, the daily demand for trout is:

Demand 10 11 12 13 14 15 16 17 18 19 20   

Probability 0.02 0.06 0.09 0.11 0.13 0.15 0.18 0.11 0.07 0.05 0.03

Construct a payoff matrix for this problem.

Hint: The matrix should have 11 states of nature and 11 alternatives. USE A FORMULA TO CALCULATE THE 121 PAYOFF AMOUNTS (**YOU WILL NEED THE MIN AND MAX FUNCTIONS**). The number sold to customers paying $3.95 is the minimum of (1) the quantity ordered and (2) the demand. THe number sold to the pet-food manufacturer is the maximum of (1) the quantity odered minus the demand and (2) zero.

It is the formulaas in excel that I am having dificulty with.

Explanation / Answer

The following table shows the payoffs. Note that I have shown the entire calculation, but you can actually solve it incrementally (note the final formula column). When you order an additional trout, it will cost you an additional 2.45

For all trout sold demanded equal to this additional number or higher, you earn an extra 3.95. The number demanded equal to this amount or higher = 1 - CDF(trout - 1)

On the other hand, if the demand is less than the number ordered, we will earn only 1.25. The demand less than the number is CDF(trout - 1).

Thus, the incremental profit is (1 - CDF(trout-1))* 3.95 + CDF(trout-1)* 1.25 - 2.45.

The optimal number to sell is 15. The profit is 19.908 there

The formula in the formula column for the increment was 1.5*(1-f(y-1))-1.2*f(y-1), as for each additional trout that you buy, you make an additional 3.95-2.45 when you can sell the trout, and you lose 1.2 on each one you can't sell

Trout P(trout) CDF(trout) Cost Sold at 3.95 Sold at 1.25 Total Revenue Profit Increment Formula 10 0.02 0.02 24.5 39.5 39.5 15 11 0.06 0.08 26.95 43.371 0.025 43.396 16.446 1.446 1.446 12 0.09 0.17 29.4 47.005 0.125 47.13 17.73 1.284 1.284 13 0.11 0.28 31.85 50.2835 0.3375 50.621 18.771 1.041 1.041 14 0.13 0.41 34.3 53.1275 0.6875 53.815 19.515 0.744 0.744 15 0.15 0.56 36.75 55.458 1.2 56.658 19.908 0.393 0.393 16 0.18 0.74 39.2 57.196 1.9 59.096 19.896 -0.012 -0.012 17 0.11 0.85 41.65 58.223 2.825 61.048 19.398 -0.498 -0.498 18 0.07 0.92 44.1 58.8155 3.8875 62.703 18.603 -0.795 -0.795 19 0.05 0.97 46.55 59.1315 5.0375 64.169 17.619 -0.984 -0.984 20 0.03 1 49 59.25 6.25 65.5 16.5 -1.119 -1.119

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