Question

Wild West produces two types of cowboy hats. A Type 1 requires twice as much labor time as a Type 2. If all the available labor time is dedicated to Type 2 alone, the company can produce a total of 400 Type 2 hats a day. The respective market limits for the two types are 150 and 200 hats per year. The revenue is $8 per Type 1 hat and $5 per Type 2 hat. a) Determine the number of hats of each type that maximize revenue. b) Determine the dual price of the production capacity and the associated feasibility range. c) With a decrease of demand for Type 1 to 120, what would be the optimal revenue? d) What is the dual price of the market share of the Type 2 hat? You may use Excel, AMPL, or solve it graphically.

Answer 1

T1 and T2 be the number of Type1 and Type2 hats produced respectively.

Let L = required labor hours for producing one unit of a Type2 hat.
So, 2L = required labor hours for producing one unit of a Type1 hat.

So, total labor hours required = 2L.T1 + L.T2

Again, we know that if all the available labor hours were dedicated to Type 2 alone, the company can produce a total of 400 Type 2 hats a day.

So, available hours = 400.L

So, the labor hour constraint will be 2L.T1 + L.T2 ≤ 400L or, 2T1 + T2 ≤ 400 (canceling out L from both the side)

Objective function: Maximize.Z = Toal Revenue = 8T1 + 5T2

Subject to,

2T1 + T2 ≤ 400 (labor hours)
T1 ≤ 150 (market limit of Type1)
T1 ≤ 200 (market limit of Type2)

T1, T2 ≥ 0

-----------------------------------

(a)

	Type1	Type2
Qty.	100	200	Total
Revenue	$8	$5	1800

B C D Set Objective: SE54 To: O Max Min value of: Total SUMPRODUCT($C$3:$D$3,04:04) ---------- Type1 Type2 Qty. 100 200 Revenue 8 5 Subject to, Labor hours 2 1 Limit of T1 1 Limit of T2 By Changing Variable Cells: SCS3:5D53 =SUMPRODUCT($C$3:$D$3,C6:06) S 400 =SUMPRODUCT($C$3:$D$3,c7:07) S 150 =SUMPRODUCT($C$3:$D$3,C8:08) S 200 Subject to the Constraints: SE56:SES8 <= SGS6:SG58 Add Change Delete Reset All Load/Save Make Unconstrained variables Non-Negative Select a Solving Simplex LP Method: Options Solving Method Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP Simplex engine for linear Solver Problems, and select the Evolutionary engine for Solver problems that are non-smooth. Sheet1 Sheet1 (2) Sensitivity Report 1 Sheet3

(b)

		Final	Reduced	Objective	Allowable	Allowable
Cell	Name	Value	Cost	Coefficient	Increase	Decrease
$C$3	Qty. Type1	100	0	8	2	8
$D$3	Qty. Type2	200	0	5	1E+30	1
		Final	Dual	Constraint	Allowable	Allowable
Cell	Name	Value	Price	R.H. Side	Increase	Decrease
$E$6	Labor hours Total	400	4	400	100	200
$E$7	Limit of T1 Total	100	0	150	1E+30	50
$E$8	Limit of T2 Total	200	1	200	200	100

(c)

The maximum allowable decrease for Type1 demand is 50 i.e. up to 100. The given decease is 120 i.e. within the range of feasibility. So, the dual price of 0 will be the same.

So, revenue will be 1800 - 0 x (150 - 120) = 1800 (same as before)

(d)

Dual price = 1