Linear Optimization and Decision Making Problems

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Linear Optimization and Decision Making Problems

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Answer 1 Part a
Let X units of standard desks and Y unit of deluxe desks will be produced in next week.
Then the objective is to maximize the profit which will be yield by selling these desks in next
week (Vanderbei, 2014).
Hence, the objective function is Max Z= 150 * X+ 320 *Y
The constraints are of less than equal to type. There will be three constraints, one for labour
hours, then for pine requirement and finally for oak requirement.
The three constraints are,
LABOUR (Hours) 10*X+16*Y ≤ 400
PINE (Square feet) 80*X+60*Y ≤ 5000
OAK (Square feet) 18*Y ≤ 750
The non-negative constraints are X ≥ 0 and Y ≥ 0.
Answer 1 Part b
1. The shadow price for labour hours is non zero. Therefore labour hour constraint is the
only Binding constraint (Munier, Hontoria, & Jiménez-Sáez, 2019).
2. Maximize profit will be Max Z= 150 * X (=0) + 320 *Y (= 25) = $ 8000
3. Slack values for
Pine = 5000- 1500 = 3500,
Oak = 750 -450 = 300, and
Labor = 400 – 400 = 0 (Binding constraint).

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4. Considering that the shadow price for labour hours = 20, an increase in labor hours by 50
will increase the objective profit by = 20 * 50 = $ 1000. The increased objective profit
will be = $ 8000 + $ 1000 = $ 9000.
5. According to the solution of the LPP, labor hours is the only Binding constraint. Hence,
increasing labor hours can be increased maximum by 266.667 hours. Now, the optimal
production units for standard desks stay as an original optimum solution given in the
information of Solver. The only increase in production is possible for deluxe desks,
which will increase the optimal profit. If labor hours is increased by 266.67, then from
the first constraint we get 16*Y ≤ 666.667 => Y ≤ 41.667 units. From the second
constraint we get 60*Y ≤ 5000, which is satisfied by Y = 41.667. From the third
constraint we get 18*Y ≤ 750 => Y ≤ 41.667. Therefore, increase in labor hours by
2666.667 will not alter the optimality of the solution. The increased value for production
of deluxe desks will be Y = 41.67 units.
6. Pine and Oak are two non-Binding constraints with shadow price of zero. Hence,
increase in availability by one unit for both of these constraints will not change the
objective profit.
7. Final value of Oak is at its minimum value of 450 square feet in the present optimal
solution. Hence, further decrease in Oak will affect the optimality of the solution. So, no
further reduction in percentage of Oak is possible.
8. Final value of Pine is at its minimum value of 1500 square feet in the present optimal
solution. Hence, further decrease in Pine will affect the optimality of the solution. So, no
further reduction in percentage of Pine is possible.
9. Reduced cost of standard desks is -50, which implies that production of one unit of
standard desk will result in loss of $ 50. Hence, increasing the profit of standard desk to

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at least $ 200 is possible without changing the optimal solution. But, to produce positive
units of standard desks, the profit for one standard desk should be greater than $ 200.
Answer 2
1. The unbounded feasible region area is bounded below by the boundary ABC (Wulan,
Ramdhani, & Indriani, 2018).
2. The corner points of the feasible region are A, B, and C.
3Xa + Xb = 9
or, Xa
3 + Xb
9 = 1
Which intersects Xa axis at (3, 0) and Xb axis at (0, 9). So, we get A = (0, 9).
Point B is at the intersection of 3Xa + Xb = 9 ,Xb = 2
Solving the above two equations we get 3Xa = 7 => Xa=7/3=2 .33
Hence, the point B≡ ( 2 . 33 ,2 )
Again the point C is on the line Xb = 2 and located at infinity. Hence coordinate of
C≡ ( ∞ , 2 )
The value of the objective function at A, B, and C is,
Z ( 0,9 ) =0 .5∗0+ 0. 4∗9=3 . 6
Z ( 2 .33 , 2 ) =0. 5∗2. 33+0 . 4∗2=1. 965
Z ( ∞ , 2 ) =∞
Hence, the optimal (minimum) solution is at B (2.33, 2).
3. Minimized cost for the model is Z ( 2 .33 , 2 ) =0. 5∗2. 33+0 . 4∗2=1. 965 .