Profit Optimization Problem

Cheatsheet Content

Problem Overview Company supplies bearings from warehouses to customers. Goal: Maximize profit by optimizing supply allocation. Scenario: Evaluate replacing warehouse C with warehouse D to increase profit. Initial Data: Warehouses A, B, C Warehouse Profit per piece to P (Rs.) Profit per piece to Q (Rs.) Supply Capacity A 3 4 4000 B 2 3 4000 C 5 4 3000 Customer Demands: P = 6000, Q = 6000 Total Supply: $4000 (A) + 4000 (B) + 3000 (C) = 11000$ Total Demand: $6000 (P) + 6000 (Q) = 12000$ Company cannot meet full demand ($11000 Optimization Strategy (Current Scenario) To maximize profit, prioritize supplying from warehouses that offer higher profit margins. For each warehouse, determine which customer (P or Q) yields a higher profit per piece. Warehouse A: Profit P = Rs. 3, Profit Q = Rs. 4. Prioritize Q. Warehouse B: Profit P = Rs. 2, Profit Q = Rs. 3. Prioritize Q. Warehouse C: Profit P = Rs. 5, Profit Q = Rs. 4. Prioritize P. Profit Calculation (Current Scenario) Allocation from Warehouse A (Capacity: 4000) Supply to Q: 4000 units (Profit: $4000 \times 4 = 16000$) Remaining demand for Q: $6000 - 4000 = 2000$ Allocation from Warehouse B (Capacity: 4000) Supply to Q: 2000 units (Profit: $2000 \times 3 = 6000$) Supply to P: 2000 units (Remaining capacity $4000 - 2000 = 2000$; Profit: $2000 \times 2 = 4000$) Remaining demand for P: $6000 - 2000 = 4000$ Allocation from Warehouse C (Capacity: 3000) Supply to P: 3000 units (Profit: $3000 \times 5 = 15000$) Remaining demand for P: $4000 - 3000 = 1000$ Total Profit (Current): $16000 + 6000 + 4000 + 15000 = 41000$ Rs. New Data: Warehouse D (replacing C) Supply Capacity: 5000 Profit per piece to P: Rs. 6 Profit per piece to Q: Rs. 5 Rental Charges for D: Rs. 5000 per year New Total Supply: $4000 (A) + 4000 (B) + 5000 (D) = 13000$ Now, total supply ($13000$) exceeds total demand ($12000$), so all demand can be met. Optimization Strategy (New Scenario with D) Prioritize supplying from warehouses offering higher profit margins. Warehouse A: Profit P = Rs. 3, Profit Q = Rs. 4. Prioritize Q. Warehouse B: Profit P = Rs. 2, Profit Q = Rs. 3. Prioritize Q. Warehouse D: Profit P = Rs. 6, Profit Q = Rs. 5. Prioritize P. Order of priority by profit: Warehouse D to P (Rs. 6) Warehouse D to Q (Rs. 5) Warehouse A to Q (Rs. 4) Warehouse C (now D) to Q (Rs. 5) (This is less than D to P) Warehouse A to P (Rs. 3) Warehouse B to Q (Rs. 3) Warehouse B to P (Rs. 2) Revised priority (highest profit first): D to P (Rs. 6) D to Q (Rs. 5) A to Q (Rs. 4) C (not D) to Q (Rs. 4) A to P (Rs. 3) B to Q (Rs. 3) B to P (Rs. 2) Let's re-evaluate the allocation based on the highest profit per unit. Profit per unit for each customer from each warehouse: A to P: 3, A to Q: 4 B to P: 2, B to Q: 3 D to P: 6, D to Q: 5 Sorted profits (descending): D-P (6), D-Q (5), A-Q (4), C-Q (4), A-P (3), B-Q (3), B-P (2) Profit Calculation (New Scenario with D) Allocation from Warehouse D (Capacity: 5000) Supply to P: 5000 units (Profit: $5000 \times 6 = 30000$) Remaining demand for P: $6000 - 5000 = 1000$ Allocation from Warehouse A (Capacity: 4000) Supply to Q: 4000 units (Profit: $4000 \times 4 = 16000$) Remaining demand for Q: $6000 - 4000 = 2000$ Allocation from Warehouse B (Capacity: 4000) Supply to P: 1000 units (Remaining demand for P; Profit: $1000 \times 2 = 2000$) Supply to Q: 2000 units (Remaining demand for Q; Profit: $2000 \times 3 = 6000$) Remaining capacity for B: $4000 - 1000 - 2000 = 1000$. This capacity is unused as all demand is met. Total Profit (New): $30000 + 16000 + 2000 + 6000 = 54000$ Rs. Subtract rental charges for D: $54000 - 5000 = 49000$ Rs. Increase in Profit New Profit: Rs. 49000 Current Profit: Rs. 41000 Increase: $49000 - 41000 = 8000$ Rs. The increase in the profit of the company after replacing warehouse C by D is Rs. 8000.