Seminar exercises Product-mix decisions

Corporation 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.

Resource-product matrix 1 i n R a 11 1i 1n b 2 21 2i 2n i1 i i i n m m1 m i mn Product types Capacities Resources Resource utilization coefficients

Contribution margin per unit (CM) Rnvironmental matrix P1 … Pi Pn MIN MAX Price (p) Contribution margin per unit (CM)

Contribution margin (Unit Price) – (Variable Costs Per Unit) = (Contribution Margin Per Unit) (Contribution Margin Per Unit) x (Units Sold) = (Product’s Contribution to Corporate Profit) (Contributions to Profit From All Products) – (Firm’s Fix Costs) = (Total Profit)

Resource-Product Relation types Non-convertible relations R1 a11 R2 a22 R3 a32 Partially convertible relations R4 a43 a44 a45 R5 a56 a57 R6 a66 a67

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*P1+0,5*P2 < 50 e2: 0,5*P1+1*P2 < 50 e3: 0,1*P2 < 10 p1, p2: 10 < P1 < 100 p3, p4: 10 < P2 < 100 ofF: 200 P1+200P2=MAX 200 200

variables (amount of produced goods) Objective function refers to choosing the best element from some set of available alternatives. X*P1 + Y*P2 = max weights (depends on what we want to maximize: price, contribution margin) variables (amount of produced goods)

Solution with linear programming 33 jugs and 33 plaits a per week Contribution margin: 13 200 HUF / week P1 e1 100 ofF e3 e1: 1*P1+0,5*P2 < 50 e2: 0,5*P1+1*P2 < 50 e3: 0,1*P2 < 10 p1,p2: 10 < P1 < 100 p3, p4: 10 < P2 < 100 ofF: 200 P1+200P2=MAX 33,3 e2 100 P2 33,3

What is the product-mix, that maximizes the revenues and the contribution to profit!   P1 P2 P3 P4 P5 P6 b (hrs/y) R1 4 2 000 R2 2 1 3 000 R3 1 000 R4 3 6 000 R5 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

Solution P1: P2-P3: Which one is the better product? Resource constraint 2000/4 = 500 > market constraint 400 P2-P3: Which one is the better product? Rev. max.: 270/2 < 200/1 thus P3 P3=(3000-200*2)/1=2600>1000 P2=200+1600/2=1000<1100 Contr. max.: 110/2 > 50/1 thus P2 P2=(3000-200*1)/2=1400>1100 P3=200+600/1=800<1000

P5-P6: linear programming P4: does it worth? Revenue max.: 1000/1 > 500 Contribution max.: 200 P5-P6: linear programming e1: 2*P5 + 3*P6 ≤ 6000 e2: 2*P5 + 2*P6 ≤ 5000 p1, p2: 50 ≤ P5 ≤ 1500 p3, p4: 100 ≤ P6 ≤ 2000 cfÁ: 50*P5 + 150*P6 = max cfF: 30*P5 + 20*P6 = max

P5 e1 Profit max: P5=1500, P6=1000 TR max: P5=50, P6=1966 3000 e2 2500 cfF cfÁ 2000 2500 P6