1. **Problem Statement:** A company produces two types of tables, T1 and T2. Each requires different hours for producing parts, assembling, and polishing. Available hours per month are limited. The goal is to find how many units of T1 and T2 to produce to maximize total monthly gain. 2. **Define Variables:** Let $x$ = number of T1 tables produced per month. Let $y$ = number of T2 tables produced per month. 3. **Constraints:** - Producing parts: $2x + 3y \leq 7000$ - Assembling: $1x + 2.5y \leq 4000$ - Polishing: $2x + 1.5y \leq 5500$ - Non-negativity: $x \geq 0$, $y \geq 0$ 4. **Objective Function:** Maximize total gain: $$Z = 12000x + 14000y$$ 5. **Solve the system:** We use the constraints to find feasible region and corner points. 6. **Find intersection points of constraints:** - Intersection of producing parts and assembling: $$\begin{cases} 2x + 3y = 7000 \\ x + 2.5y = 4000 \end{cases}$$ Multiply second equation by 2: $$2x + 5y = 8000$$ Subtract first from this: $$2x + 5y - (2x + 3y) = 8000 - 7000 \Rightarrow 2y = 1000 \Rightarrow y = 500$$ Substitute $y=500$ into $x + 2.5y = 4000$: $$x + 2.5(500) = 4000 \Rightarrow x + 1250 = 4000 \Rightarrow x = 2750$$ - Intersection of assembling and polishing: $$\begin{cases} x + 2.5y = 4000 \\ 2x + 1.5y = 5500 \end{cases}$$ Multiply first by 2: $$2x + 5y = 8000$$ Subtract second: $$2x + 5y - (2x + 1.5y) = 8000 - 5500 \Rightarrow 3.5y = 2500 \Rightarrow y = \frac{2500}{3.5} = 714.29$$ Substitute $y$ into $x + 2.5y = 4000$: $$x + 2.5(714.29) = 4000 \Rightarrow x + 1785.71 = 4000 \Rightarrow x = 2214.29$$ - Intersection of producing parts and polishing: $$\begin{cases} 2x + 3y = 7000 \\ 2x + 1.5y = 5500 \end{cases}$$ Subtract second from first: $$2x + 3y - (2x + 1.5y) = 7000 - 5500 \Rightarrow 1.5y = 1500 \Rightarrow y = 1000$$ Substitute $y=1000$ into $2x + 1.5(1000) = 5500$: $$2x + 1500 = 5500 \Rightarrow 2x = 4000 \Rightarrow x = 2000$$ 7. **Evaluate objective function $Z$ at corner points:** - At $(0,0)$: $Z=0$ - At $(0, 2333.33)$ from $2x + 3y = 7000$ when $x=0$: $Z=14000 \times 2333.33 = 32666620$ - At $(2750, 500)$: $Z=12000 \times 2750 + 14000 \times 500 = 33000000 + 7000000 = 40000000$ - At $(2214.29, 714.29)$: $Z=12000 \times 2214.29 + 14000 \times 714.29 = 26571480 + 10000060 = 36571540$ - At $(2000, 1000)$: $Z=12000 \times 2000 + 14000 \times 1000 = 24000000 + 14000000 = 38000000$ 8. **Maximum gain:** The maximum $Z$ is $40000000$ at $x=2750$, $y=500$. 9. **Limiting resource and shadow price:** The limiting resource is the constraint that is binding at the optimal solution. Check which constraints are equalities at $(2750, 500)$: - Producing parts: $2(2750) + 3(500) = 5500 + 1500 = 7000$ (binding) - Assembling: $1(2750) + 2.5(500) = 2750 + 1250 = 4000$ (binding) - Polishing: $2(2750) + 1.5(500) = 5500 + 750 = 6250 > 5500$ (not feasible, so polishing is not binding here, so re-check) Since polishing constraint is violated, the point $(2750, 500)$ is not feasible. We must check feasibility carefully. Check polishing at $(2750, 500)$: $$2(2750) + 1.5(500) = 5500 + 750 = 6250 > 5500$$ Not feasible. Check $(2214.29, 714.29)$ polishing: $$2(2214.29) + 1.5(714.29) = 4428.58 + 1071.43 = 5500.01 \approx 5500$$ Feasible. Check $(2000, 1000)$ polishing: $$2(2000) + 1.5(1000) = 4000 + 1500 = 5500$$ Feasible. Check $(0, 2333.33)$ polishing: $$2(0) + 1.5(2333.33) = 0 + 3500 = 3500 < 5500$$ Feasible. Recalculate $Z$ for feasible points: - $(2214.29, 714.29)$: $36571540$ - $(2000, 1000)$: $38000000$ - $(0, 2333.33)$: $32666620$ Maximum feasible gain is $38000000$ at $(2000, 1000)$. 10. **Iso-Contribution Curve:** An Iso-Contribution Curve shows combinations of $x$ and $y$ that yield the same total gain $Z$. Equation: $$12000x + 14000y = Z$$ For this firm, for a fixed $Z$, the curve is: $$y = \frac{Z - 12000x}{14000}$$ This line helps visualize how changing production mix affects total gain. 11. **Limitations of the technique:** - Assumes linearity in costs and gains which may not hold in reality. - Ignores market demand and other practical constraints.