1. Jym’s, a large sporting goods store is placing an order for bicycles with its
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Question
1. Jym’s, a large sporting goods store is placing an order for bicycles with its supplier. Four models can be ordered: Adult models M1 and M2; Children models M3 and M4. The profits from selling the bikes are 30, 25, 22, and 20, respectively. There are several conditions that the store needs to meet:
Space to hold the inventory. The store has 500 feet2 of storage area. The adult models need 2 feet2 of storage each, and the children models need 1 foot2 each.
Assembly of the bikes: Upon arrival, the bikes must be assembled. There are 1200 hours of assembly time available. The M1 model needs 5 hours of assembly each; the M2 model needs 6 hours of assembly; and each one of the children models needs 4 hours each.
Order quantity: At least 275 bikes of all types combined will be ordered.
Minimum order required per type: At least 30 bikes of each type need to be ordered.
The linear model that maximizes the store’s profit is provided in a Solver format below, as well as the sensitivity report. Based on the printouts provided, answer the following questions:
M1
M2
M3
M4
Max
30
25
22
20
Inventory
2
2
1
1
<=
500
Assembly
5
6
4
4
<=
1200
Combined Order
1
1
1
1
>=
275
Min M1
1
>=
30
Min M2
1
>=
30
Min M3
1
>=
30
Min M4
1
>=
30
Adjustable Cells
Final
Reduced
Objective
Allowable
Allowable
Cell
Name
Value
Cost
Coefficient
Increase
Decrease
$B$2
M1
40
0
30
1E+30
2.5
$C$2
M2
30
0
25
13
1E+30
$D$2
M3
175
0
22
2
2
$E$2
M4
30
0
20
2
1E+30
Constraints
Final
Shadow
Constraint
Allowable
Allowable
Cell
Name
Value
Price
R.H. Side
Increase
Decrease
$F$4
Inventory
345
0
500
1E+30
155
$F$5
Assembly
1200
8
1200
145
10
$F$6
Combined Order
275
-10
275
2.5
29
$F$7
Min M1
40
0
30
10
1E+30
$F$8
Min M2
30
-13
30
5
30
$F$9
Min M3
175
0
30
145
1E+30
$F$10
Min M4
30
-2
30
145
30
Questions:
How many units of each bike are purchased; calculate the total profit?
Can other items be stored at the same storage after the bikes were stored? If so, how much of the storage area was not used?
The supplier of M1 has notified the store that it needs to increase its selling price to cover increased transportation costs. Jym’s estimated its profit on an M1 bike would drop by 5% per bike, and therefore notified the supplier it was reducing the orders of M1 bikes from 40 to 35 units. Was it optimal for Jym’s to take this move? Base your answer on the sensitivity analysis results, and explain your answer.
The store offers overtime to assembly workers.
Should the store pay $4 more per overtime hour? Explain.
If the store pay $4 more, and use 100 hours of overtime, what is the total profit expected? Show your calculations. Don’t re-run.
Management is asking for your advice for making the following two simultaneous changes in the production plan: Allow the minimum number of M2 type bikes produced drop to 28, while increasing the minimum required production of M4 increase to 40. Would you recommend this move based on the sensitivity results you have? Don’t re-run.
Lawn Master produces 19-inch and 21-inch lawn mowers (LM), which it sells to membership warehouses and discount stores nationwide. Both lawn mowers are powered by Briggs and Stratton 3.5 horsepower engines. Upon receipt each model is assembled, tested, and packaged. The following information will help you formulate the linear programming model. Start by first defining your decision variables, and then relate to the information provided below.
The 19-inch model requires 40 minutes to assemble, test, and package combined. The 21-inch model requires one hour for the same operations. There are four production lines where the assembly + testing + packaging are performed. Each line operates 7.5 hours a day, five days a week. Write a constraint about the amount of time it takes to build 19-inch mowers and 21 inch mowers
Each week Lawn Master can receive up to 200 Briggs and Stratton engines. Write a constraint about the number of mowers built.
Lawn Master wants to produce at least twice as many LM19 as LM21. Write a constraint.
The profit on each 19-inch model is $50. The profit on each 21-inch model is $60. Formulate the objective function.
M1
M2
M3
M4
Max
30
25
22
20
Inventory
2
2
1
1
<=
500
Assembly
5
6
4
4
<=
1200
Combined Order
1
1
1
1
>=
275
Min M1
1
>=
30
Min M2
1
>=
30
Min M3
1
>=
30
Min M4
1
>=
30
Explanation / Answer
1). 40, 30, 175, 30 units of bike M1, M2, M3, M4 resp are purchased. Total profit = 30*40+25*30+22*175+20*30=6400
2). Refer $F$4, Final value is 345 therefore 345 SQ feet storage are is used. Storage not used = 500-345 =155 sq ft
3). 5% of 30 is 1.5, whereas allowable decrease in coefficient of M1 is 2.5 , there decrease of 5% is within the allowable range. Hence the current purchase is still the optimal. Therefore Jym's move to reduce the order of M1 bikes from 40 35 is not optimal.
4). Shadow price of assembly is 8 . 4 is still less than 8, therefore store should pay $4 per overtime hour for additional capacity (assembly hours). Each additional hour will increase the total profit by $ 4 (=8-4) net of overtime cost.
5). Allowable increase in assembly hours is 145, 100 is still less than that. Therefore shadow price is valid for these 100 hours. Hence total profit will increase by 8*100-4*100 = 400
New profit after adjusting the overtime cost of these 100 hours = 6400+400 = 6800
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