1. Seminar exercises The Product-mix Problem
Agnes Kotsis

2. Corporate system-matrix
1.) Resource-product matrix Describes the connections between the company’s resources and products as linear and deterministic relations via coefficients of resource utilization and resource capacities.
2.) Environmental matrix (or market-matrix): Describes the minimum that we must, and maximum that we can sell on the market from each product. It also desribes the conditions.

3. Resource-product matrix
Product types
Capacities
Resources
Resource utilization coefficients

4. Contribution margin per unit (f)
Environmental matrix T1 … Ti Tn
MIN
MAX
Price (p)
Contribution margin per unit (f)

5. Contribution margin
Unit Price - Variable Costs Per Unit = Contribution Margin Per Unit
Contribution Margin Per Unit x Units Sold = Product’s Contribution to Profit
Contributions to Profit From All Products – Firm’s Fixed Costs = Total Firm Profit

6. Resource-Product Relation types
Non-convertible relations E1
a11 E2
a22 E3
a32
Partially convertible relations E4
a43
a44
a45 E5
a56
a57 E6
a66
a67

7. Product-mix in a pottery – corporate system matrix
Jug
Plate
Capacity
50 kg/week
100 HUF/kg
50 hrs/week
800 HUF/hr
10 kg/week
Clay (kg/pcs)
1,0
0,5
Weel time (hrs/pcs)
0,5
1,0
Paint (kg/pcs)
0,1
Minimum (pcs/week) 10
Maximum (pcs/week)
100
Price (HUF/pcs)
700
1060
Contribution margin (HUF/pcs)
e1: 1*T1+0,5*T2 < 50
e2: 0,5*T1+1*T2 < 50
e3: ,1*T2 < 10
p1, p2: < T1 < 100
p3, p4: < T2 < 100
ofF: T1+200T2=MAX
200
200

8. variables (amount of produced goods)
Objective function
refers to choosing the best element from some set of available alternatives.
X*T1 + Y*T2 = max
weights (depends on what we want to maximize: price, contribution margin)
variables (amount of produced goods)

9. Solution with linear programming
33 jugs and 33 plaits a per week
Contribution margin: HUF / week T1 e1
100
ofF e3
e1: 1*T1+0,5*T2 < 50
e2: 0,5*T1+1*T2 < 50
e3: ,1*T2 < 10
p1,p2: < T1 < 100
p3, p4: < T2 < 100
ofF: T1+200T2=MAX
33,3 e2
100 T2
33,3

10. What is the product-mix, that maximizes the revenues and the contribution to profit!T1 T2 T3 T4 T5 T6
b (hrs/y) E1 4
2 000 E2 2 1
3 000 E3
1 000 E4 3
6 000 E5
5 000
MIN (pcs/y)
100
200 50
MAX (pcs/y)
400
1100
1 000
500
1 500
2000
p (HUF/pcs)
270 30
150
f (HUF/pcs)
110
-10 20

11. Solution T1: T2-T3: Which one is the better product?
Resource constraint 2000/4 = 500 > market constraint 400
T2-T3: Which one is the better product?
Rev. max.: 270/2 < 200/1 thus T3
T3=( *2)/1=2600>1000
T2= /2=1000<1100
Contr. max.: 110/2 > 50/1 thus T2
T2=( *1)/2=1400>1100
T3= /1=800<1000

12. T5-T6: linear programming
T4: does it worth?
Revenue max.: 1000/1 > 500
Contribution max.: 200
T5-T6: linear programming
e1: 2*T5 + 3*T6 ≤ 6000
e2: 2*T5 + 2*T6 ≤ 5000
p1, p2: 50 ≤ T5 ≤ 1500
p3, p4: 100 ≤ T6 ≤ 2000
cfÁ: 50*T *T6 = max
cfF: 30*T5 + 20*T6 = max

13. T5 e1
Contr. max: T5=1500, T6=1000
Rev. max: T5=50, T6=1966
3000 e2
2500
cfF
cfÁ
2000
2500 T6