### Optimal Manufacturing Strategy for Three Products

## by Andrea Arnstein

Submitted : Fall 2013

The objective of this problem was to successfully maximize a company’s annual profit by optimizing the production of three independent products A, B, and C while dealing with storage and production constraints. To optimize these outcomes, a cost equation, annual profit equation, and constraint equation had to be formulated and utilized. Products A, B, and C had to be converted to N_{A, }N_{B, }and_{ }N_{C} in order to account for the full years production (600 shifts / year). In order to account for the cost and annual profit equations, the cost of the material, the cost to make each product, and the given cost of the equipment to construct each product yearly became the cost equation and that was subtracted from the price that each product would sell for. The final annual profit equation after conversion and manipulation was = 1200N_{A} - 200N_{A}^{1.2} + 1800N_{B} - 400N_{B}^{1.2}+ 2400N_{C} -600N_{C}^{1.2}.

To account for the other constraint, I took the companies storage constraint per shift along with the size of each product and solved for the variable N_{C}, doing this N_{C} became 1250 – ½ N_{A }– ¾ N_{B}. Substituting N_{C} into the final annual profit equation as listed above, I then solved for the partial derivative of N_{A} and N_{B}.

The partial derivatives being:

d/dNA = 360N_{C}^{0.2}-240 N_{A}^{0.2}, d/dNB = 540N_{C}^{0.2}-480 N_{A}^{0.2}

I then set each equation equal to zero and solved for an equation of N_{C}. I then solved for N_{A} and N_{B }by comparing the equations to one another and solving for a common variable, and then I solved for N_{C}. The final numbers for N_{A}, N_{B}, and N_{C} was 1541.88, 367.361, 203.885 (units / shift) respectively.

In reference to the annual profit, I took the numbers found for N_{A}, N_{B}, and N_{C} and inserted them into the final annual profit equation and found that the annual profit for the company would be $828,953.005. For the floor usage each value of N_{A}, N_{B}, and N_{C }multiplied by the products production space and was compared against the whole storage area (5000ft^{2}) and the percentages were found. The floor usage of N_{A }was 61.65%, N_{B} was 22.04%, and N_{C} was 16.311%.

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