Maximizing Profits using Linear Programming
Directions:
Use linear programming to maximize profit at a custom printing shop that allows for overtime in a single department. All submitted work should be properly formatted and include:
- A typeset of the linear program.
- An analysis in narrative form (paragraph structure please).
- Incorporation of the solution and sensitivity analysis in simplified terms. Scientific notation must be interpreted for a layman’s audience. Tables and charts should be properly formatted and not run over multiple pages.
Framework:
Custom printing is a contracting industry for which jobs must be selected efficiently. One such firm is trying to determine the best mix of jobs it should seek, given current capacity constraints in four main departments: typesetting, camera, pressroom, and bindery. It has classified its commercial work into three classes: Albany, Berkshire, and Chatham, each requiring different amounts of time in the four major departments.
The production requirements in hours per unit of product are as follows:
A|B|C
Typesetting 0|2|3
Camera 3|1|3
Pressroom 3|6|2
Bindery 5|4|0
Assuming these units of work are produced using regular time, the contribution to overhead and profit is $200 for each unit of Albany work, $300 for each unit of Berkshire work, and $100 for each unit of Chatham work.
The firm currently has the following regular-time capacity available in each department for the next time period: typesetting, 40 hours; camera, 160 hours; pressroom, 200 hours; bindery, 160 hours. In addition to this regular time, the firm could utilize an overtime shift in typesetting, which would make available an additional 35 hours in that department. The premium for this overtime (i.e., incremental costs in addition to regular time) would be $4/hour.
The firm has also made the decision that in order to meet current demand, that it will produce at least 10 units of each type of product.
- Analysis should include addressing the following information:
- Define 5 decision variables that quantify the amount of work produced based on the type of work done.
- Create an objective function that maximizes profit.
- Create a system of linear constraints to describe the linear programming model.
- Find the optimal production mix that maximizes profit.
- Is there a unique mix?
- Perform a sensitivity analysis on the objective function. What changes to marginal profit would cause you to rerun the system for new production mix.
- If overtime were to increase by $2 per hour, would you need to rerun the program?
- Perform the sensitivity analysis for the constraints and comment on the following questions.
- Which constraints have no slack available and are binding?
- Which constraints have slack and are nonbinding?
- Explain why there is a difference in marginal shadow price for regular typesetting and overtime typesetting.
- How much would profit increase if the overtime typesetting hours were increased by 1?
- How many hours will the camera department remain idle in the optimal state?
- How much would profit increase if the bindry department were to have 5 hours added?
