A renowned chocolatier, Francesco Schr
ö
eder
,
makes three kinds of chocolate confectionery: artisanal truffles, handcrafted chocolate nuggets, and premium gourmet chocolate bars. He uses the highe

Question

A renowned chocolatier, Francesco Schr
ö
eder
,
 ﻿makes three kinds of chocolate confectionery: artisanal truffles, handcrafted chocolate nuggets, and premium gourmet chocolate bars. He uses the highest quality of cacao butter, dairy cream, and honey as the main ingredients. Francesco makes his chocolates each morning, and they are usually sold out by the early afternoon.
For a pound of artisanal truffles, Francesco uses 
1
 ﻿cup of cacao butter, 
1
 ﻿cup of honey, and 
1
/
2
 ﻿cup of cream. The handcrafted nuggets are milk chocolate and take 
1
/
2
 ﻿cup of cacao, 
2
/
3
 ﻿cup of honey, and 
2
/
3
 ﻿cup of cream for each pound. Each pound of the chocolate bars uses 
1
 ﻿cup of cacao butter, 
1
/
2
 ﻿cup of honey, and 
1
/
2
 ﻿cup of cream. One pound of truffles, nuggets, and chocolate bars can be purchased for $
3
5
,
 ﻿$
2
5
,
 ﻿and $
2
0
,
 ﻿respectively. A local store places a daily order of 
1
0
 ﻿pounds of chocolate nuggets, which means that Francesco needs to make at least 
1
0
 ﻿pounds of the chocolate nuggets each day. Before sunrise each day, Francesco receives a delivery of 
5
0
 ﻿cups of cacao butter, 
5
0
 ﻿cups of honey, and 
3
0
 ﻿cups of dairy cream.
1
.
 ﻿Formulate

VIPSolutions · Accepted Answer

# Francesco's Chocolate Production LP Problem

## 1. LP Model Formulation and Solution

### **Decision Variables**
- Let $ x_1 $: Pounds of **artisanal truffles** made per day  
- Let $ x_2 $: Pounds of **handcrafted nuggets** made per day  
 Let $ x3 $: of **gourmet bars** per day

**Objective Function**
Maximize **revenue**:
$$
\text{Maximize } Z = 35x_1 + 25x_2 + 20x_3
$$

### **Constraints**

#### **Ingredient Constraints**
- **Cacao butter**:  
  $$
  1x_1 + .5x_2 + 1x_3 \leq 50
  $$
- **Honey**:  
  $$
  1x_1 + \frac{2}{3}x_2 + .5x_3 \leq 50
  $$
- **Cream**:  
  $$
  .5x_1 + \frac{2}{3}x_2 + .5x_3 \leq 30
  $$

#### **Nugget Order Constraint**
- Local store order:
  $$
  x_2 \geq 10
  $$

#### **Non-negativity**
- $ x_1, x_2, x_3 \geq  $

---

### **Step 1: Set Up Constraints in Standard Form**
$$
\begin{align*}
\text{Maximize } & Z = 35x_1 + 25x_2 + 20x_3 \
\text{Subject to:} \
& x_1 + .5x_2 + x_3 \leq 50 \quad \text{(Cacao butter)} \
& x_1 + .6667x_2 + .5x_3 \leq 50 \quad \text{(Honey)} \
& .5x_1 + .6667x_2 + .5x_3 \leq 30 \quad \text{(Cream)} \
& x_2 \geq 10 \
& x_1, x_2, x_3 \geq 
\end{align*}
$$

---

### **Step 2: Solve the LP**

#### **Plug in the minimum for $ x_2 $ (10) and check slack for other resources**

Set $ x_2 = 10 $:

- Cacao butter: $ x_1 + .5 \times 10 + x_3 \leq 50 \Rightarrow x_1 + x_3 \leq 45 $
- Honey: $ x_1 + .6667 \times 10 + .5x_3 \leq 50 \Rightarrow x_1 + 6.667 + .5x_3 \leq 50 \Rightarrow x_1 + .5x_3 \leq 43.333 $
- Cream: $ .5x_1 + .6667 \times 10 + .5x_3 \leq 30 \Rightarrow .5x_1 + 6.667 + .5x_3 \leq 30 \Rightarrow .5x_1 + .5x_3 \leq 23.333 $

Let’s let $ x_3 = y $ and $ x_1 = x $:

From the cream constraint:
$$
.5x + .5y \leq 23.333 \implies x + y \leq 46.666
$$

From cacao butter:
$$
x + y \leq 45
$$

From honey:
$$
x + .5y \leq 43.333
$$

The most restrictive is honey.

#### **Find intersection of the most restrictive constraints**

From above:
- $ x + .5y \leq 43.333 $
- $ x + y \leq 45 $

Let’s try to maximize revenue. Since truffles ($ x_1 $) pay more (\$35) than bars (\$20), prioritize $ x_1 $.

Let’s try $ x_3 =  $:
- $ x + .5() \leq 43.333 \implies x \leq 43.333 $
- $ x +  \leq 45 \implies x \leq 45 $
- $ x + y \leq 46.666 \implies x \leq 46.666 $

So, $ x_1 = 43.333 $ and $ x_3 =  $ is allowed.

Check ingredient usage:
- Cacao butter: $ 43.333 + .5*10 +  = 43.333 + 5 = 48.333 \leq 50 $ ✔️
- Honey: $ 43.333 + .6667*10 + .5* = 43.333 + 6.667 = 50 \leq 50 $ ✔️ (tight)
- Cream: $ .5*43.333 + .6667*10 + .5* = 21.6665 + 6.667 = 28.3335 \leq 30 $ ✔️

Thus, optimal:

$$
x_1 = 43.33,\quad x_2 = 10,\quad x_3 = 
$$

#### **Max Revenue Calculation**
$$
Z = 35 \times 43.33 + 25 \times 10 + 20 \times  = 1,516.55 + 250 +  = \$1,766.55
$$

---

## **Summary Table (rounded to two decimals):**

| Product        | Pounds to Make | Revenue per lb | Total Revenue |
|----------------|---------------|---------------|--------------|
| Artisanal Truffles ($ x_1 $) | 43.33         | \$35          | \$1,516.55    |
| Nuggets ($ x_2 $)            | 10.00         | \$25          | \$250.00      |
| Gourmet Bars ($ x_3 $)       | .00          | \$20          | \$.00        |
| **Total**      |               |               | \$1,766.55    |

---

## 2. **Shadow Prices and Range of Feasibility**

### **Binding Constraints**
- **Honey constraint** is binding (used up exactly).
- **Nuggets minimum** is binding ($ x_2 = 10 $).
- **Cacao butter** and **cream** have slack.

#### **Shadow Price Calculation**
- **Honey constraint**: The shadow price is the increase in objective value per additional unit of honey.
    - If honey increases by 1 cup, we can make 1 more lb of truffles (since each uses 1 cup), increasing revenue by \$35.
    - But need to check if other resources allow this.
    - **Cacao butter**: making one more truffle needs 1 cup (currently using 48.33, so 1.67 cups left).
    - **Cream**: needs .5 cup/truffle (currently using 28.3335, so 1.6665 cups left).

So, one more cup of honey allows us to make 1 more truffle, increasing revenue by \$35.

**Shadow price for honey constraint: \$35.00**

#### **Range of Feasibility for Honey**

- If honey increases, next constraint to bind is cacao butter.
- Cacao butter used: $ 43.33 + .5 \times 10 = 48.33 $, so $ 50 - 48.33 = 1.67 $ cups left.
- Each extra pound of truffle uses 1 cup cacao butter, so max increase in honey is 1.67 cups.
- New honey limit: 50 + 1.67 = 51.67 cups.
- **Range: $50 \leq \text{Honey} \leq 51.67$ cups**

#### **Nuggets Minimum (Constraint)**
- If the requirement increases, it binds more, but if it decreases, it becomes non-binding.
- **Shadow price:**  (does not improve revenue if required to make more nuggets).

#### **Cacao Butter and Cream**
- Both are non-binding (have slack), so shadow price is .

### **Summary Table**

| Constraint         | Status   | Shadow Price | Range of Feasibility |
|--------------------|----------|--------------|-----

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