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Sunrise company supplies milkshakes with three flavors including matcha, earl grey, and strawberry. The demand for milkshake rapidly increased due to the high temperature and summer holiday started. Unfortunately, the inventory levels of ingredients of milk, cream, and sugar are a bit lower to meet the demand and their supplier is unable to replenish immediately. Shop manager, Tony, is going to maximize the profit by planning the certain amount of each flavor of milkshakes to produce before replenishment.

The profit generated by matcha, earl grey, and strawberry milkshakes are $2.7, $2.6, and $2.5 per liter sold.  Tony just checked the inventory left of milk, cream, and sugar are 65 liters (L), 45 liters (L), and 32 kilograms (KG) respectively.  Moreover, each cup consumes 0.3 L milk for the two tea flavors and 0.27 L for strawberry.  It consumes 0.20 L, 0.19 L, and 0.23 L for cream, as well as 0.1 KG, 0.08 KG, 0.15 KG for sugars.
Tony found that 1L of cream has a bit unpleasant odor and so discarded then. Will the optimal solution change and what can be said about the effect on total profit?

This scenario involves a linear programming sensitivity analysis where Tony is trying to maximize profit given the constraints of limited milk, cream, and sugar inventories. The report you've provided seems to represent an optimization output where the optimal production quantities, shadow prices, and sensitivity ranges (allowable increases/decreases) are detailed.

From the report:

1. **Optimal solution without changes**: 
   - **Matcha milkshake**: 186.6667 liters (L) produced, with profit contribution $2.7/L.
   - **Earl Grey milkshake**: 0 liters produced, with profit contribution $2.6/L.
   - **Strawberry milkshake**: 33.3333 liters produced, with profit contribution $2.5/L.
   
   The shadow price tells us how much the objective (profit) would increase if 1 additional unit of a constrained resource were available. For instance, the shadow price of milk is 8.0667, meaning an additional liter of milk would increase profit by $8.07.

2. **Effect of cream shortage**:
   The cream shortage and discarding of 1 liter of cream reduce the total available cream to 44 liters. Checking the **allowable decrease** for cream in the report (1.66667 liters), reducing cream by 1 liter will stay within the allowable range. Therefore, **the current solution remains optimal** even with the discard, meaning the production plan will not change. The shadow price for cream is -3, meaning this shortage actually reduces the overall profit.

3. **Profit impact**:
   Given the shadow price of cream is negative (-3), this implies that losing 1 liter of cream will decrease the total profit by $3. Therefore, the total profit will decrease slightly due to the reduced cream inventory.

### Summary:
- The optimal solution will **not change** with the discarding of 1 liter of cream, as the reduction is within the allowable decrease.
- The total profit will **decrease by $3** due to the shortage in cream.

Let me know if you want further clarification or any additional details!

Here are some related questions you might consider:
1. How would an increase in sugar availability affect the production quantities?
2. What happens if the milk constraint is relaxed beyond the allowable increase?
3. Could Tony increase total profit by producing any Earl Grey milkshakes?
4. What if the cream's allowable decrease was exceeded by more than 1 liter?
5. How would the total profit change if the price per liter of matcha milkshake increased?

**Tip**: Shadow prices help in understanding how sensitive your solution is to changes in resource availability. A positive shadow price indicates a potential profit increase when additional resources are added.

Sunrise company supplies milkshakes with three flavors: matcha, earl grey, and strawberry. Due to high demand, Tony, the shop manager, is maximizing profit by planning the production quantities for each flavor with limited ingredients of milk, cream, and sugar. What is the optimal solution and how does discarding 1 liter of cream affect it?

Learn how to maximize profit for milkshakes using linear programming with limited milk, cream, and sugar. Includes detailed sensitivity analysis and the effect of discarding 1 liter of cream on the optimal solution and profit. Suitable for undergraduate students in operations research.

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