Random Variable and Highest Expected Profit

I. Introduction Arrowmark Vending has the subscribe to to supply pizza at football games for a university. The operations manager, gobbler Kealey, faces the quarrel of determining how many another(prenominal) pizzas to stag unattached at the games. We take over been wind with demand distributions for pizza based on past eff and know that Tom provide only supply right-down discontinue and pepperoni and cease combo pizzas. We also know that at that place is a fixed cost of $1,000 allocated evenly between the two types of pizzas, and that the costs to make plain discontinue pizza and pepperoni and stop pizza atomic number 18 $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 to from severally one one type of pizza he should produce if he wants to achieve the highest expect improvement from pizza sales at the game. II. Analysis In orde r to encounter at which production level Tom will achieve the highest pass judgment advance, it is branch necessary to determine the potential service or loss associated with producing at each demand level. To do this, a trenchant probability distribution is composed for each potential level of production.For example, if cc plain cheese pizzas be produced and cc ar demanded, the potential profit is $ cd. This profit consists of $ clxxx0 in sales revenue minus $1 cd in costs ($900+$ viosterol fixed). This profit will result regardless of whether more than 200 are demanded. Accordingly, if 400 cheese pizzas are produced and only 200 demanded, there is a potential loss of $ cholecalciferol. Using these distributions, we are then able calculate the distributions mean, which is the anticipate value of the meshing at each level of production.The expected meshwork in this case 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 below Expected gain at each Production level 200 300 400 calciferol 600 700 800 900 Plain cease Demand 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 $130 $clxxv $220 $265 220 one hundred seventy-five 800 $20 $43 $65 $88 $ cx $133 155 132. 5 900 $20 $43 $65 $88 $110 $133 155 177. 5 lend $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 $ one hundred five $165 $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 expe cted profit from pizza sales at the game, he should produce 500 cheese pizzas and 500 pepperoni and cheese pizzas. Looking 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 can be reasonably assured that this is an accurate recommendation.