1. **State the problem:** Mossaic Tiles, Ltd. wants to maximize profit by deciding how many batches of large tiles ($x$) and small tiles ($y$) to produce weekly, subject to constraints on molding time, baking time, glazing time, clay availability, and non-negativity. 2. **Define variables:** Let $x$ = number of batches of 100 large tiles produced per week. Let $y$ = number of batches of 100 small tiles produced per week. 3. **Formulate the objective function:** Maximize profit: $$Z = 190x + 240y$$ 4. **Formulate constraints:** - Molding time (in minutes): $$18x + 15y \leq 60 \text{ hours} = 3600 \text{ minutes}$$ - Baking time (in hours): $$0.27x + 0.58y \leq 105$$ - Glazing time (in hours): $$0.16x + 0.20y \leq 40$$ - Clay availability (in pounds): $$32.8x + 20y \leq 6000$$ - Non-negativity: $$x \geq 0, \quad y \geq 0$$ 5. **Standard form:** Convert inequalities to equalities by adding slack variables $s_1, s_2, s_3, s_4 \geq 0$: $$18x + 15y + s_1 = 3600$$ $$0.27x + 0.58y + s_2 = 105$$ $$0.16x + 0.20y + s_3 = 40$$ $$32.8x + 20y + s_4 = 6000$$ 6. **Graphical solution:** Plot constraints and find feasible region vertices: - Intersection points of constraints give candidate solutions. - Evaluate $Z$ at each vertex to find maximum. 7. **Calculate intersection points:** For example, solve $18x + 15y = 3600$ and $0.27x + 0.58y = 105$ simultaneously. 8. **Determine unused resources at optimal point:** Calculate slack variables $s_i$ by substituting optimal $x,y$ into constraints. 9. **Sensitivity analysis:** - Objective coefficients: find ranges of $190$ and $240$ where optimal basis remains. - Right-hand side values: find allowable increases/decreases for resource limits. 10. **Profit threshold for only small tiles:** Set $x=0$, solve for $y$ maximizing profit: $$Z = 240y$$ Find minimum $240$ such that producing only small tiles is optimal. 11. **Effect of molding time reduction:** Update molding constraint: $$16x + 12y \leq 3600$$ Re-solve to find new optimal solution. 12. **Effect of additional 100 pounds clay:** New clay constraint: $$32.8x + 20y \leq 6100$$ Check if this increases profit. 13. **Investment in kiln glazing hours:** Additional 20 hours glazing at cost 90000. Check if increased profit from extra hours exceeds 90000. 14. **Effect of kiln shutdown reducing glazing hours to 37:** Update glazing constraint: $$0.16x + 0.20y \leq 37$$ Recalculate optimal solution and profit. **Final answer:** The linear programming model is: $$\max Z = 190x + 240y$$ subject to $$18x + 15y \leq 3600$$ $$0.27x + 0.58y \leq 105$$ $$0.16x + 0.20y \leq 40$$ $$32.8x + 20y \leq 6000$$ $$x,y \geq 0$$ Solving graphically or by computer yields the optimal production mix maximizing profit under constraints.