A specialty chemicals facility manufactures two products and B in barrels. Products A and B utilize the same raw material; A uses 120 kg/bbl, while B requires 100 kg/bbl. There is an upper limit on the raw material supply of 9000 kg/day. Another constraint is warehouse storage space (40 m^2 total;

Question

A specialty chemicals facility manufactures two products [latex]A[/latex] and [latex]B[/latex] in barrels. Products [latex]A[/latex] and [latex]B[/latex] utilize the same raw material; A uses [latex]120 kg / bbl[/latex], while B requires [latex]100 kg / bbl[/latex]. There is an upper limit on the raw material supply of [latex]9000 kg /[/latex] day. Another constraint is warehouse storage space [latex]\left(40 m ^{2}\right.[/latex] total; both [latex]A[/latex] and [latex]B[/latex] require [latex]\left.0.5 m ^{2} / bbl \right) .[/latex] In addition, production time is limited to [latex]7 h[/latex] per day. A and B can be produced at [latex]20 bbl / h[/latex] and [latex]10 bbl / h[/latex], respectively. If the profit per bbl is [latex]\$ 10[/latex] for A and [latex]\$ 14[/latex] for B, find the production levels that maximize profit.

Accepted Answer

Let [latex]x_{A}[/latex] be bbl/day of A produced [latex]x_{B}[/latex] be bbl/day of B produced

Objective is to maximize profit

[latex]\max P=10 x_{A}+14 x_{B}[/latex] (1)

Subject to

Raw material constraint: [latex]\quad 120 x_{A}+100 x_{B} \leq 9,000[/latex] (2)

Warehouse space constraint: [latex]0.5 x_{A}+0.5 x_{B} \leq 40[/latex] (3)

Production time constraint: [latex](1 / 20) x_{A}+(1 / 10) x_{B} \leq 7[/latex] (4)

	[latex]X_{A}[/latex]	[latex]X_{B}[/latex]
Initial values	1	1
Final values	20	60
max P =	1040
Constraints
[latex]120X_{A} + 100X_{B}[/latex]	8400
[latex]0.5X_{A} + 0.5X_{B}[/latex]	40
[latex](1/20)X_{A} + (1/10)X_{B}[/latex]	7

Table S19.11. Excel solution

Thus the optimum point is [latex]x_{A}=20[/latex] and [latex]x_{B}=60[/latex]

The maximum profit [latex]=\$ 1040 /[/latex] day

Question 19.11: A specialty chemicals facility manufactures two products and...

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