Seminar exercises The Product-mix Problem Agnes Kotsis

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.

Resource-product matrix Product types Capacities Resources Resource utilization coefficients

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

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

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

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: 0,1*T2 < 10 p1, p2: 10 < T1 < 100 p3, p4: 10 < T2 < 100 ofF: 200 T1+200T2=MAX 200 200

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)

Solution with linear programming 33 jugs and 33 plaits a per week Contribution margin: 13 200 HUF / week T1 e1 100 ofF e3 e1: 1*T1+0,5*T2 < 50 e2: 0,5*T1+1*T2 < 50 e3: 0,1*T2 < 10 p1,p2: 10 < T1 < 100 p3, p4: 10 < T2 < 100 ofF: 200 T1+200T2=MAX 33,3 e2 100 T2 33,3

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

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=(3000-200*2)/1=2600>1000 T2=200+1600/2=1000<1100 Contr. max.: 110/2 > 50/1 thus T2 T2=(3000-200*1)/2=1400>1100 T3=200+600/1=800<1000

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*T5 + 150*T6 = max cfF: 30*T5 + 20*T6 = max

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