seminar exercises the product-mix problem agnes kotsis
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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