Random Variable and Highest Expected Profit

I. Introduction Arrowmark Vending has the contract to hang on pizza at football game games for a university. The operations manager, Tom Kealey, faces the challenge of determining how galore(postnominal) pizzas to pay off available at the games. We have been provided with demand distributions for pizza based on past experience and know that Tom allow only supply plain cease and pepperoni and cheese combo pizzas. We also know that thither is a fixed cost of $1,000 allocated equally between the two types of pizzas, and that the cost to make plain cheese pizza and pepperoni and cheese pizza ar $4. 50 and $5. 0 respectively. Both pizzas sell for $9. 00 and unsold pizzas have no value. The purpose of this report is to provide Tom with some information regarding how many of each type of pizza he should produce if he wants to fall upon the highest expected avail from pizza sales at the game. II. Analysis In order to determine at which production train Tom will achieve the highest expected avail, it is first necessary to determine the potence profit or loss associated with producing at each demand level. To do this, a discrete probability distribution is composed for each potential level of production.For example, if two hundred plain cheese pizzas be produced and 200 are demanded, the potential profit is $400. This profit consists of $one hundred eighty0 in sales revenue minus $1400 in costs ($900+$500 fixed). This profit will result regardless of whether much than 200 are demanded. Accordingly, if 400 cheese pizzas are produced and only 200 demanded, there is a potential loss of $500. Using these distributions, we are then able calculate the distributions mean, which is the expected value of the meshing at each level of production.The expected profits in this fiber are the weighted average of the potential profit values, in which the weights are the probabilities. The expected profits associated with each type of pizza are provided in the tables belo w Expected Profits at each Production take aim 200 three hundred 400 500 600 700 800 900 Plain lay off admit 200 $40 -$5 -$50 -$95 -$140 -$185 -230 -275 300 $60 $128 $60 -$8 -$75 -$143 -210 -277. 5 400 $60 $128 $195 $128 $60 -$8 -75 -142. 500 $80 $170 $260 $350 $260 $170 80 -10 600 $80 $170 $260 $350 $440 $350 260 170 700 $40 $85 $cxxx $clxxv $220 $265 220 175 800 $20 $43 $65 $88 $110 $133 one hundred fifty-five 132. 5 900 $20 $43 $65 $88 $110 $133 one hundred fifty-five 177. 5 measure $400 $760 $985 $1,075 $985 $715 $355 $(50) 300 400 500 600 700 800 Pepperoni and Cheese Demand 300 $70 $20 -$30 -$80 -$130 -$180 400 $140 $220 $120 $20 -$80 -$180 500 $175 $275 $375 $250 $125 $0 600 $175 $275 $375 $475 $350 $225 700 $105 $ one hundred sixty-five $225 $285 $345 $270 800 $35 $55 $75 $95 $115 $135 Total $700 $1,010 $1,140 $1,045 $725 $270 III. Recommendation If Kealey wants to achieve the highest expected profit from pizza sales at the game, he shoul d produce 500 cheese pizzas and 500 pepperoni and cheese pizzas. looking at at the tables, we know this is the best option because we see the highest expected profit of $1,075 associated with this production level for cheese pizza and $1,140 in profit for pepperoni and cheese pizza. This number takes into account the probabilities at each demand level, so we offer be reasonably assured that this is an accurate recommendation.