Problem Name Variables X1 X2 X3 Sign Rhs Slack Objective

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The problem requires determining the optimal number of different game projects—specifically console, handheld, and PC games—that can be developed within the given man-hour constraints to maximize profit. The constraints include the availability of skilled personnel (programmers, artists, designers, managers) and their respective man-hours, with specific resource requirements for each game type. The goal is to find the optimal combination of projects that maximizes profit while respecting the resource limitations, and to identify which resource pools are underutilized or over capacity.

Paper For Above instruction

In the rapidly evolving landscape of the gaming industry, effective resource allocation is paramount to maximizing profit and maintaining competitiveness. With the recent legislative changes limiting overtime, companies must now strategize within stricter manpower and time constraints. This paper explores the application of linear programming techniques to optimize project selection—specifically, the development of console, handheld, and PC games—based on available personnel and their respective man-hours allocated to different tasks.

Initially, the company’s staffing levels include 238 programmers, 225 artists, 180 designers, and 57 production managers. Each category's total available man-hours over a quarter—13 weeks—are calculated by multiplying the number of personnel by 520 hours, resulting in 123,760, 117,000, 93,600, and 29,640 hours, respectively. These resources are allocated to various tasks: development, art, design, and management, with specific hourly requirements for each game type. The profit margins differ: $1.8 million for console games and $1 million for handheld games.

Using linear programming, the primary objective is to maximize profit by determining the optimal number of console and handheld games produced. The constraints ensure that the total hours allocated to each task do not exceed the available staff hours and that production quantities are non-negative. In mathematical terms, decision variables x1 and x2 can represent the number of console and handheld games produced, respectively. The objective function is to maximize total profit: Z = 1.8 million x1 + 1 million x2.

Constraints include resource limitations such as:

• Development hours: 10,920 per console game, 7,280 per handheld.
• Art hours: 13,000 per console, 2,600 per handheld.
• Design hours: 3,120 per console, 9,360 per handheld.
• Management hours: 2,080 per console, 2,600 per handheld.
• Total hours allocated per resource type must not surpass the company's capacity, e.g., development hours cannot exceed 123,760 hours, and similarly for other pools.

Solving these constraints with a linear programming approach, typically via simplex method or software tools, yields the optimal production quantities for console and handheld games. The analysis reveals the number of each game that maximizes profit while respecting human resource constraints.

Furthermore, the resource utilization is analyzed to identify underutilized pools—that is, pools with remaining capacity—and overextended pools that require expansion. For example, if the total hours used in development are less than the available 123,760 hours, this indicates underutilization, suggesting potential for increased production or resource reallocation.

In subsequent analysis, the company considers expanding into PC game development. With the addition of 44 programmers, 58 artists, and 2 managers, available man-hours across all resource pools increase—development hours to 171,600 (adding 44 * 520 hours), art hours to 124,480, and management hours to 36,880. The resource constraints are updated accordingly, and the linear program is rerun to find optimal mixes for console, handheld, and PC projects.

The introduction of PC projects with their specific resource requirements and profit margins leads to recalculation of optimal project combinations. The analysis indicates whether building a new PC department is profitable within the resource constraints and helps identify any pools requiring further expansion. For example, if the addition of PC projects exceeds resource capacities, the company must evaluate whether additional hiring or reallocations are feasible to meet the increased demand.

In conclusion, applying linear programming to project selection and resource allocation in the gaming industry enables strategic decision-making that maximizes profit under operational constraints. The approach helps identify not only the optimal number of projects of each type but also offers insight into resource management, underutilized capacities, and areas requiring expansion. Such analytical frameworks are essential for businesses operating in dynamic markets where resource constraints and profit maximization are critical to success.

References

• Winston, W. L. (2004). Operations Research: Applications and Algorithms. Thomson-Brooks/Cole.