1. **Problem statement:** Rehema has 900 tonnes of bricks at Mtakuja and 600 tonnes at Tupendane. She wants to transport bricks to sites A, B, and C requiring 500, 600, and 400 tonnes respectively. Let: - $x$ = bricks transported from Mtakuja to A - $y$ = bricks transported from Mtakuja to B Define: - From Mtakuja to C: $900 - (x + y)$ (since total from Mtakuja is 900) - From Tupendane to A: $500 - x$ - From Tupendane to B: $600 - y$ - From Tupendane to C: $400 - [900 - (x + y)] = x + y - 500$ 2. **(a) Formulate inequalities and objective function:** - Constraints for supply at Mtakuja: $$x + y + \text{(Mtakuja to C)} = 900 \implies x + y \leq 900$$ - Constraints for supply at Tupendane: $$\text{(Tupendane to A)} + \text{(Tupendane to B)} + \text{(Tupendane to C)} = 600$$ Using variables: $$(500 - x) + (600 - y) + (x + y - 500) = 600$$ This simplifies to $600 = 600$, confirming balance. - Demand constraints for sites: From site A: $$x + (500 - x) = 500$$ From site B: $$y + (600 - y) = 600$$ From site C: $$(900 - (x + y)) + (x + y - 500) = 400$$ Checks out. - Non-negativity: $$x \geq 0, y \geq 0, 900 - (x + y) \geq 0 \Rightarrow x + y \leq 900$$ $$500 - x \geq 0 \Rightarrow x \leq 500$$ $$600 - y \geq 0 \Rightarrow y \leq 600$$ $$x + y - 500 \geq 0 \Rightarrow x + y \geq 500$$ 3. **Objective function (minimize total transport cost):** Distances/cost per tonne: - Mtakuja to A: 600 - Mtakuja to B: 300 - Mtakuja to C: 400 - Tupendane to A: 400 - Tupendane to B: 200 - Tupendane to C: 600 Total cost: $$Z = 600x + 300y + 400(900 - x - y) + 400(500 - x) + 200(600 - y) + 600(x + y - 500)$$ Simplify: $$Z = 600x + 300y + 360000 - 400x - 400y + 200000 - 400x + 120000 - 200y + 600x + 600y - 300000$$ Collect terms: $$Z = (600x - 400x - 400x + 600x) + (300y - 400y - 200y + 600y) + (360000 + 200000 + 120000 - 300000)$$ $$= (400x) + (300y) + 380000$$ 4. **(b) Find values for $x$ and $y$ to minimize $Z$ under constraints:** Constraints summary: $$x + y \leq 900$$ $$x + y \geq 500$$ $$x \leq 500$$ $$y \leq 600$$ $$x, y \geq 0$$ Since $Z = 400x + 300y + 380000$, to minimize $Z$, minimize $x$ and $y$ subject to constraints. Try $x=0$, $y=500$ (since $x+y=500$ is min sum) Check $y \leq 600$ ok. Compute $Z$ at $(0,500)$: $$Z = 400*0 + 300*500 + 380000 = 150000 + 380000 = 530000$$ Try $x=500$, $y=0$: $$Z = 400*500 + 300*0 + 380000 = 200000 + 380000 = 580000$$ Try $x=100$, $y=400$ ($x+y=500$): $$Z = 400*100 + 300*400 + 380000 = 40000 + 120000 + 380000 = 540000$$ Minimum is at $(0,500)$. 5. **Check bricks distribution:** - Mtakuja to A: $x=0$ - Mtakuja to B: $y=500$ - Mtakuja to C: $900 - 0 - 500 = 400$ - Tupendane to A: $500 - 0 = 500$ - Tupendane to B: $600 - 500 = 100$ - Tupendane to C: $0 + 500 - 500 = 0$ All non-negative and satisfy demands. **Final answer:** - From Mtakuja: A=0, B=500, C=400 - From Tupendane: A=500, B=100, C=0 Minimum transport cost is $530000$ units.