Consider the following question from the slides:

"The Cozy Tea Shop offers two types of tea mixes to customers: “Rainy Day” and “Too Cold Outside”. A kg of Rainy Day contains ½ kg of black tea and ½ kg of cinnamon/orange tea. A kg of Too Cold Outside contains ¼ kg of black tea and ¾ kg of cinnamon/orange tea. The Cozy Tea Shop makes a profit of $3.50 per kg of Rainy Day tea and $4.00 per kg profit on the Too Cold Outside tea.

If the Cozy Tea Shop received 200 kg of black tea and 250 kg of cinnamon/orange tea today, how much of each tea mix the Cozy Tea Shop must prepare to maximize profits today?"

We found that The Cozy Tea Shop should make 350 and 100 lbs of Rainy Day and Too Cold Outside tea mixes. What is the marginal value of additional cinnamon/orange tea? (i.e., if the tea shop got 1 kg more of cinnamon/orange tea, how much would the profit increase or decrease by?)

Question

Consider the following question from the slides:

"The Cozy Tea Shop offers two types of tea mixes to customers: “Rainy Day” and “Too Cold Outside”. A kg of Rainy Day contains ½ kg of black tea and ½ kg of cinnamon/orange tea. A kg of Too Cold Outside contains ¼ kg of black tea and ¾ kg of cinnamon/orange tea. The Cozy Tea Shop makes a profit of $3.50 per kg of Rainy Day tea and $4.00 per kg profit on the Too Cold Outside tea.

If the Cozy Tea Shop received 200 kg of black tea and 250 kg of cinnamon/orange tea today, how much of each tea mix the Cozy Tea Shop must prepare to maximize profits today?"

We found that The Cozy Tea Shop should make 350 and 100 lbs of Rainy Day and Too Cold Outside tea mixes. What is the marginal value of additional cinnamon/orange tea? (i.e., if the tea shop got 1 kg more of cinnamon/orange tea, how much would the profit increase or decrease by?)

BlackTom AI · Accepted Answer

First, I restate the problem to ensure understanding: the shop uses two tea mixes, Rainy Day (RD) and Too Cold Outside (TCO). RD uses 0.5 kg black tea and 0.5 kg cinnamon/orange tea per kg with a profit of $3.50 per kg. TCO uses 0.25 kg black tea and 0.75 kg cinnamon/orange tea per kg with a profit of $4.00 per kg. Today we have 200 kg of black tea and 250 kg of cinnamon/orange tea. The claimed optimal production is 350 kg RD and 100 kg TCO. We’re asked for the marginal value (shadow price) of additional cinnamon/orange tea, i.e., if we add 1 kg of cinnamon/orange tea, by how much would profit change? Step 1: Verify the stated production is feasible and uses all resources. With RD = 350 kg and TCO = 100 kg: - Black tea usage = 0.5*350 + 0.25*100 = 175 + 25 = 200 kg (fits exactly the 200 kg limit). - Cinnamon/orange tea usage = 0.5*350 + 0.75*100 = 175 + 75 = 250 kg (fits exactly the 250 kg limit). Thus, both resources are binding at the stated solution, which is typical when a solution is optimal under a tight resource constraint. Step 2: Set up the linear program and its dual to obtain the shadow price of cinnamon/orange tea. The primal is: maximize 3.50 x + 4.00 y subject to 0.5 x + 0.25 y <= 200 (black tea) 0.5 x + 0.75 y <= 250 (cinnamon/orange tea) x, y >= 0 The dual (for a maximization with <= constraints) is: minimize 200 u + 250 v subject to 0.5 u + 0.5 v >= 3.50 (for RD, coefficient of x) 0.25 u + 0.75 v >= 4.00 (for TCO, coefficient of y) u, v >= 0. Step 3: Use complementary slackness or solve the dual with the given optimal primal to find the dual values. Since both primal constraints are binding and both x and y are positive, both dual variables u and v are positive, and the corresponding equalities hold: 0.5 u + 0.5 v = 3.50 => u + v = 7.00 0.25 u + 0.75 v = 4.00 Solving these two equations: From u = 7 - v, substitute into the second: 0.25(7 - v) + 0.75 v = 4 => 1.75 - 0.25 v + 0.75 v = 4 => 1.75 + 0.50 v = 4 => 0.50 v = 2.25 => v = 4.50. Then u = 7 - 4.50 = 2.50. Step 4: Interpret the dual value. The dual variable v corresponds to the cinnamon/orange tea resource. Its value (shadow price) is 4.50 per additional kilogram of cinnamon/orange tea. Conclusion: If the shop could obtain 1 more kilogram of cinnamon/orange tea (holding other resources constant and recomputing the optimal mix), the maximum profit would increase by 4.50 dollars. The value 3.50 would correspond to the profit per kilogram of RD, which is not the resource’s marginal value; the marginal value depends on the binding resource in the optimal solution, which in this case is cinnamon/orange tea with a shadow price of 4.50.

BU.520.601.T2.FA25 Final Exam

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