One of the specialities at Dixie\'s Cafe is pulled pork, which simmers over a lo

Question

One of the specialities at Dixie's Cafe is pulled pork, which simmers over a low flame all day. Since the cooking time is so long, Dixie must decide in the morning how many servings of pork to cook for that night's dinner service. Moreover, the pork cooked orn a given day cannot be served the next day; it must be thrown away. Pulled pork is the highest-profit item on the menu at Dixie's Cafe. It earns Dixie a profit of $8 per serving, whereas a other items earn a profit of $4. Customers who want pulled pork pork costs the Cafe $2.5. (a) First suppose the demand for pulled pork on a given evening is normally dis- tributed with a mean of 18 and a variance of 16. How many servings of pulled pork should Dixie prepare in the morning? (Fractional servings are OK.) What is the expected cost of the optimal solution? (b) Now, suppose the demand is distributed as an exponential random variable with mean 18. How many servings should Dixie prepare?

Explanation / Answer

(a) Underage cost, Cu = profit of pulled pork - profit of other items = 8 - 4 = $ 4

Overage cost, Co = cost of pulled pork = $ 2.5

Critical ratio F(z) = Cu/(Cu+Co) = 4/(4+2.5) = 0.6154

z = NORMSINV(0.6154) = 0.2934

Average demand, µ = 18

Standard deviation of demand, ? = SQRT(16) = 4

Number of servings of pulled pork that Dixie should prepare in the morning =µ + z*? = 18 + 0.2934*4 = 19.17

Expected cost of the optimal solution = 19.17*2.5 = $ 47.93

(b) Mean, ? = 1/18 = 0.056

Exponential distribution = 1-e-?x = 1-e-0.056x = 0.6154 (critical ratio)

Solving the above equation for x, we get,

e-0.056x = 1-0.6154 = 0.3846

x = -ln(0.3846)/0.056 = 17.06

Number of Servings Dixie should prepare = 17

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