Bakery Production Planning Exercise

Bakery Production Planning Exercise

A food company produces three types of bread: country bread, cornbread, and rye bread.
All three types of bread must pass through three processing stages:

1. Kneading
2. Resting / Dough handling
3. Baking

The following table shows the required processing hours per ton of bread for each type at each stage:

Week 1 availability:

• Kneading: 90 hours
• Resting: 60 hours
• Baking: 94 hours

Questions

(a)

How many tons of each bread should be produced in the first week of December to exactly use all available processing hours?

(b)

Now assume that the available hours differ across weeks. Specifically:

• Week 2 availability:
  • Kneading: 70 hours
  • Resting: 47 hours
  • Baking: 93 hours
• Week 3 availability:
  • Kneading: 79 hours
  • Resting: 53 hours
  • Baking: 82 hours

How many tons of each bread should be produced in both the second and third weeks of December to fully use the available processing hours each week?

Solutions

Define Variables

Let:

• $x$ = tons of country bread
• $y$ = tons of cornbread
• $z$ = tons of rye bread

(a) System of equations for Week 1

Based on the processing times:

1. Kneading:

$$2x + 4y + 3z = 90$$

2. Resting:

$$x + 2y + 3z = 60$$

3. Baking:

$$3x + 2y + 4z = 94$$

Now solve this system!

Step 1: Simplify

Let’s use substitution or elimination (I’ll solve it efficiently for you):

From Resting equation, express $x$:

$$x = 60 – 2y – 3z$$

Substitute into Kneading equation:

$$2(60 – 2y – 3z) + 4y + 3z = 90$$ $$120 – 4y – 6z + 4y + 3z = 90$$ $$120 – 6z + 3z = 90$$ $$-3z = -30$$ $$z = 10$$

Now substitute $z = 10$ into $x = 60 – 2y – 3z$:

$$x = 60 – 2y – 30 = 30 – 2y$$

Substitute $x$ and $z$ into Baking equation:

$$3(30 – 2y) + 2y + 4(10) = 94$$ $$90 – 6y + 2y + 40 = 94$$ $$130 – 4y = 94$$ $$-4y = -36$$ $$y = 9$$

Finally, compute $x$:

$$x = 30 – 2 \times 9 = 12$$

✅ Answer (a):

• Country bread: 12 tons
• Cornbread: 9 tons
• Rye bread: 10 tons

(b) Week 2 and Week 3

Week 2 equations:

1. Kneading:

$$2x + 4y + 3z = 70$$

2. Resting:

$$x + 2y + 3z = 47$$

3. Baking:

$$3x + 2y + 4z = 93$$

Step 1: Express $x$ from Resting:

$$x = 47 – 2y – 3z$$

Substitute into Kneading:

$$2(47 – 2y – 3z) + 4y + 3z = 70$$ $$94 – 4y – 6z + 4y + 3z = 70$$ $$94 – 6z + 3z = 70$$ $$-3z = -24$$ $$z = 8$$

Substitute $z = 8$ into $x$:

$$x = 47 – 2y – 3 \times 8 = 47 – 2y – 24 = 23 – 2y$$

Substitute into Baking:

$$3(23 – 2y) + 2y + 4(8) = 93$$ $$69 – 6y + 2y + 32 = 93$$ $$101 – 4y = 93$$ $$-4y = -8$$ $$y = 2$$

Finally, compute $x$:

$$x = 23 – 2 \times 2 = 19$$

✅ Week 2 answer:

• Country bread: 19 tons
• Cornbread: 2 tons
• Rye bread: 8 tons

Week 3 equations:

1. Kneading:

$$2x + 4y + 3z = 79$$

2. Resting:

$$x + 2y + 3z = 53$$

3. Baking:

$$3x + 2y + 4z = 82$$

Step 1: Express $x$:

$$x = 53 – 2y – 3z$$

Substitute into Kneading:

$$2(53 – 2y – 3z) + 4y + 3z = 79$$ $$106 – 4y – 6z + 4y + 3z = 79$$ $$106 – 6z + 3z = 79$$ $$-3z = -27$$ $$z = 9$$

Substitute $z = 9$ into $x$:

$$x = 53 – 2y – 3 \times 9 = 53 – 2y – 27 = 26 – 2y$$

Substitute into Baking:

$$3(26 – 2y) + 2y + 4 \times 9 = 82$$ $$78 – 6y + 2y + 36 = 82$$ $$114 – 4y = 82$$ $$-4y = -32$$ $$y = 8$$

Finally, compute $x$:

$$x = 26 – 2 \times 8 = 26 – 16 = 10$$

✅ Week 3 answer:

• Country bread: 10 tons
• Cornbread: 8 tons
• Rye bread: 9 tons

Final Summary