1. **Problem 1: Minimizing Transportation Cost for Beer Distribution** We have 4 depots (Chilanga Freedom, Chelston Green, George Compound, Jack Compound) and 4 beer brands (Chat Beer, Lusaka Beer, Shake Shake, Nkhosi Beer) with given supply and demand. The goal is to minimize total transportation cost. 2. **Set up the transportation problem:** Supply vector: $[1100, 1400, 600, 1500]$ (hectoliters for each beer brand) Demand vector: $[900, 1500, 1200, 1000]$ (hectoliters for each depot) Cost matrix (rows = beer brands, columns = depots): $$\begin{bmatrix} 10 & 30 & 25 & 15 \\ 20 & 15 & 20 & 10 \\ 10 & 30 & 20 & 20 \\ 30 & 40 & 35 & 45 \end{bmatrix}$$ 3. **Find initial basic feasible solution using Vogel's Approximation Method (VAM):** - Calculate penalties for rows and columns. - Allocate as much as possible to the lowest cost cell with highest penalty. - Adjust supply and demand and repeat until all are allocated. 4. **Apply stepping stone or MODI method to test for optimality:** - Calculate opportunity costs for unused routes. - If all opportunity costs are non-negative, current solution is optimal. - Otherwise, adjust allocations along a closed path to reduce total cost. 5. **Optimal solution (allocations in hectoliters):** - Chat Beer: 900 to Chilanga Freedom, 200 to Jack Compound - Lusaka Beer: 1400 to Chelston Green - Shake Shake: 600 to George Compound - Nkhosi Beer: 600 to Chelston Green, 1000 to Jack Compound 6. **Calculate total minimum transportation cost:** $$\text{Cost} = 10\times900 + 15\times1400 + 20\times600 + 30\times600 + 15\times1000 + 15\times200 = 9000 + 21000 + 12000 + 18000 + 15000 + 3000 = 78000$$ --- 7. **Problem 2: Assigning Contracts to Suppliers to Maximize Revenue** Given bid sums matrix (in thousands): $$\begin{bmatrix} 2 & 4 & 3 & 5 & 4 \\ 7 & 4 & 6 & 8 & 4 \\ 2 & 9 & 8 & 10 & 4 \\ 8 & 6 & 12 & 7 & 4 \\ 2 & 8 & 5 & 8 & 8 \end{bmatrix}$$ Suppliers: A, B, C, D, E Contracts: C1, C2, C3, C4, C5 8. **Use the Hungarian Algorithm to maximize total revenue:** - Convert maximization to minimization by subtracting each element from the maximum element (12). - Apply Hungarian method to find optimal assignment. 9. **Optimal assignment:** - Supplier A: Contract C4 (5) - Supplier B: Contract C1 (7) - Supplier C: Contract C2 (9) - Supplier D: Contract C3 (12) - Supplier E: Contract C5 (8) 10. **Maximum total revenue:** $$7 + 9 + 12 + 5 + 8 = 41$$ (thousands) --- **Final answers:** - Minimum transportation cost: 78000 Kwacha - Maximum revenue from contract assignment: 41000 (thousands)