Random Variable and Highest Expected Profit
I. Introduction Arrowmark Vending has the contract to supply pizza at football games for a university. The operations manager, Tom Kealey, faces the challenge of determining how many pizzas to make available at the games. We have been provided with demand distributions for pizza based on past experience and know that Tom will only supply plain cheese and pepperoni and cheese combo pizzas. We also know that there is a fixed cost of $1,000 allocated equally between the two types of pizzas, and that the costs to make plain cheese pizza and pepperoni and cheese pizza are $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 achieve the highest expected profit from pizza sales at the game. II. Analysis In order to determine at which production level Tom will achieve the highest expected profit, it is first necessary to determine the potential 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 200 plain cheese pizzas are produced and 200 are demanded, the potential profit is $400. This profit consists of $1800 in sales revenue minus $1400 in costs ($900+$500 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 $500. Using these distributions, we are then able calculate the distribution’s mean, which is the expected value of the profits at each level of production.
The expected profits 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 Profits at each Production Level| | | 200| 300| 400| 500| 600| 700| 800| 900| Plain Cheese 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| $175| $220| $265| 220| 175| | 800| $20| $43| $65| $88| $110| $133| 155| 132. 5| | 900| $20| $43| $65| $88| $110| $133| 155| 177. 5| | Total| $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| $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 expected 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.
