Brian Q’ Rourke, owner of The Horse & Hound Pub in Dublin, is preparing for St. Patricks Day, amd he must determine how much cider to stock. Brian sells three brands of cider on tap – Murphy’s Blackton, and red Rye. The cost per gallon (to the pub owner) of each brand is as follows:
Brand Cost/Unit
Murphy’s $1.50
Blackton 0.90
Red Rye 0.50
The pub budjet of $2,000 cider for St. Patricks Day. Brian sells Murphy’s at a rate $3.00 per gallon, Blackton at $2.50 per gallon and Red Rye at $1.75 per gallon. Based on previous St. Patrick’s Day celebrations, Brian has determined the maximum customer demand to be 400 gallons of Murphy’s, 500 gallons of Blackton, and 300 gallons of Red Rye. The pub has the capacity to stock 1,000 gallons of each brand of cider; Brian wants to stock completely. Brian wants to determine the number of gallons of each brands of cider to order in order to maximize profit.
a. Formulate a linear programming model for this problem.
b. Solve the model by using the computer.
a. To formulate a linear programming model for this problem, we need to determine the decision variables, objective function, and constraints.
Decision Variables:
Let M represent the number of gallons of Murphy's cider to order.
Let B represent the number of gallons of Blackton cider to order.
Let R represent the number of gallons of Red Rye cider to order.
Objective Function:
We want to maximize the profit, which can be represented by the total revenue minus the total cost. The total revenue is the sum of the revenue from selling Murphy's, Blackton, and Red Rye cider. The total cost is the sum of the cost per gallon multiplied by the number of gallons ordered for each brand. Therefore, the objective function is:
Maximize: 3M + 2.5B + 1.75R - (1.5M + 0.9B + 0.5R)
Constraints:
1. Budget constraint: The total cost of the cider order cannot exceed the $2,000 budget. This can be represented as:
1.5M + 0.9B + 0.5R <= 2000
2. Demand constraint: The number of gallons ordered for each brand cannot exceed the maximum customer demand. This can be represented as:
M <= 400
B <= 500
R <= 300
3. Stock constraint: The number of gallons ordered for each brand cannot exceed the available stock capacity. This can be represented as:
M <= 1000
B <= 1000
R <= 1000
b. To solve the model using a computer, we can use a linear programming solver tool or software. These tools can take the formulated model and provide the optimal solution. The solver will find the values of M, B, and R that maximize the objective function while satisfying the constraints.