Seminar exercisesSeminar exercisesThe The Product-mix Product-mix

ProblemProblemAgnes KotsisAgnes Kotsis

Corporate system-matrixCorporate system-matrix1.) Resource-product matrix 1.) Resource-product matrix

Describes theDescribes the connections between the connections between the company’s resources and products as company’s resources and products as linear and deterministic relations via linear and deterministic relations via coefficients of resource utilizationcoefficients of resource utilization and and resource capacities.resource capacities.

22.) Environmental matrix (or market.) Environmental matrix (or market--matrix): matrix): Describes the minimum that we must, and Describes the minimum that we must, and maximum that we can sell on the market maximum that we can sell on the market from each product. It also desribes the from each product. It also desribes the conditions.conditions.

Resource-product matrixResource-product matrix

Produktumok

Erőforrások

T1 Ti Tn Erőforrások nagysága

(kapacitás) óra/időszak

E1 a11 a1i a1n b1 E2 a21 a2i a2n b2 Ei a i1 a i i a i n b i Em am1 am i amn bm erőforrás felhasználási

koeficiensek

Product types

Resources Capacities

Resource utilization

coefficients

Environmental matrixEnvironmental matrix

T1 … Ti … Tn

MIN

MAX

Price (p)Contribution margin per

unit (f)

Contribution marginContribution margin

Unit Unit Price - Variable Costs Per Unit Price - Variable Costs Per Unit = Contribution Margin Per Unit = Contribution Margin Per Unit

Contribution Margin Per Unit x Contribution Margin Per Unit x Units Sold = Product’s Contribution Units Sold = Product’s Contribution to Profit to Profit

Contributions to Profit From All Contributions to Profit From All Products – Firm’s Fixed Costs = Products – Firm’s Fixed Costs = Total Firm Profit Total Firm Profit

Resource-Product Resource-Product Relation typesRelation types

T1 T2 T3 T4 T5 T6 T7

E1 a11

E2 a22

E3 a32

E4 a43 a44 a45

E5 a56 a57

E6 a66 a67

Non-convertible relations Partially convertible relations

Product-mix in a potterProduct-mix in a pottery y – corporate system – corporate system

matrixmatrixJug Plate

Clay (kg/pcs) 1,0 0,5

Weel time (hrs/pcs)

0,5 1,0

Paint (kg/pcs) 0 0,1

Capacity

50 kg/week 100 HUF/kg

50 hrs/week 800 HUF/hr

10 kg/week 100 HUF/kg

Minimum (pcs/week)

10 10

Maximum (pcs/week)

100 100

Price (HUF/pcs) 700 1060

Contribution margin (HUF/pcs)

e1: 1*T1+0,5*T2 < 50e2: 0,5*T1+1*T2 < 50e3: 0,1*T2 < 10p1, p2: 10 < T1 < 100p3, p4: 10 < T2 < 100ofF: 200 T1+200T2=MAX

200 200

Objective functionObjective function

refers to choosing the best element refers to choosing the best element from some set of available from some set of available alternatives.alternatives.

X*X*TT11 + Y* + Y*TT22 = max = max

variables (amount of produced

goods)

weights(depends on what we want to maximize:

price, contribution margin)

Solution with linear Solution with linear programmingprogramming

T1

T2

33,3

33,3

33 jugs and 33 plaits a per week

Contribution margin: 13 200 HUF / week

e1: 1*T1+0,5*T2 < 50e2: 0,5*T1+1*T2 < 50e3: 0,1*T2 < 10p1,p2: 10 < T1 < 100p3, p4: 10 < T2 < 100ofF: 200 T1+200T2=MAX

e1

e2

e3ofF

100

100

What is the productWhat is the product--mix, that mix, that maximizes the revenues and the maximizes the revenues and the

contributioncontribution to profit!to profit!

  T1 T2 T3 T4 T5 T6 b (hrs/y)

E1 4           2 000

E2   2 1       3 000

E3       1     1 000

E4         2 3 6 000

E5         2 2 5 000

MIN (pcs/y) 100 200 200 200 50 100

MAX (pcs/y) 400 1100 1 000 500 1 500 2000

p (HUF/pcs) 200 270 200 30 50 150

f (HUF/pcs) 100 110 50 -10 30 20

SolutionSolution TT11: :

Resource constraint 2000/4 = 500 Resource constraint 2000/4 = 500 > market constraint > market constraint 400400

TT22-T-T33: Which one is the better product?: Which one is the better product?Rev. max.Rev. max.: : 270/2 < 200/1270/2 < 200/1 thus Tthus T33

TT33=(3000-200*2)/1=2600>=(3000-200*2)/1=2600>10001000

TT22=200+1600/2==200+1600/2=10001000<1100<1100

Contr. max.Contr. max.: : 110/2 > 50/1110/2 > 50/1 thus T thus T22

TT22=(3000-200*1)/2=1400>=(3000-200*1)/2=1400>11001100

TT33=200+600/1==200+600/1=800800<1000<1000

TT44: : does it worth?does it worth?Revenue max.: 1000/1 > Revenue max.: 1000/1 > 500500Contribution max.: Contribution max.: 200200

TT55-T-T66: : linear programminglinear programming ee11: : 2*T2*T55 + 3*T + 3*T66 ≤ 6000≤ 6000

ee22:: 2*T2*T55 + 2*T + 2*T66 ≤ 5000 ≤ 5000

pp11, p, p22:: 50 ≤ 50 ≤ TT55 ≤ 1500≤ 1500

pp33, p, p44:: 100 ≤ 100 ≤ TT66 ≤ 2000≤ 2000

cfcfÁÁ:: 5050*T*T55 + 150*T + 150*T66 = max = max

cfcfFF:: 30*T30*T55 + 20*T + 20*T66 = max = max

e2

e1

cfF

cfÁ

Contr. max: T5=1500, T6=1000Rev. max: T5=50, T6=1966

T5

T62000

3000

2500

2500