The economic order quantity (EOQ) model is a classical model used for controlling inventory and satisfying demand. Costs included in the model are holding cost per unit, ordering cost, and the cost of goods ordered. The assumptions for that model are that only a single item is considered, that the entire quantity ordered arrives at one time, that the demand for the item is constant over time, and that no shortages are allowed. Suppose we relax the first assumption and allow for multiple items that are independent except for a restriction on the amount of space available to store the products. The following model describes this situation. Let D, annual demand for item j C, unit cost of item / = S, cost per order placed for item j w, space required for item j W = the maximum amount of space available for all goods /= inventory carrying charge as a percentage of the cost per unit. The decision variables are Q₁, the amount of item j to order. The model is [CD₁ + SPL + IC, Q Minimize s.t. j=1 Q₁ z 0. In the objective function, the first term is the annual cost of goods, the second is the annual ordering cost (D/Q, is the number of orders), and the last term is the annual inventory holding cost (Q/2 is the average amount of inventory). Construct and solve a nonlinear optimization model for the following data. w,Q, sw Annual Demand Item Cost ($) Order Cost ($) Space Required (sq. feet) W = 9,000 ¡ = 0.10 Complete the model. Item 1 2,000 2,000 100 150 Item 2 Item 3 1,000 50 50 135 25 80 125 40
The economic order quantity (EOQ) model is a classical model used for controlling inventory and satisfying demand. Costs included in the model are holding cost per unit, ordering cost, and the cost of goods ordered. The assumptions for that model are that only a single item is considered, that the entire quantity ordered arrives at one time, that the demand for the item is constant over time, and that no shortages are allowed. Suppose we relax the first assumption and allow for multiple items that are independent except for a restriction on the amount of space available to store the products. The following model describes this situation. Let D, annual demand for item j C, unit cost of item / = S, cost per order placed for item j w, space required for item j W = the maximum amount of space available for all goods /= inventory carrying charge as a percentage of the cost per unit. The decision variables are Q₁, the amount of item j to order. The model is [CD₁ + SPL + IC, Q Minimize s.t. j=1 Q₁ z 0. In the objective function, the first term is the annual cost of goods, the second is the annual ordering cost (D/Q, is the number of orders), and the last term is the annual inventory holding cost (Q/2 is the average amount of inventory). Construct and solve a nonlinear optimization model for the following data. w,Q, sw Annual Demand Item Cost ($) Order Cost ($) Space Required (sq. feet) W = 9,000 ¡ = 0.10 Complete the model. Item 1 2,000 2,000 100 150 Item 2 Item 3 1,000 50 50 135 25 80 125 40
Practical Management Science
6th Edition
ISBN:9781337406659
Author:WINSTON, Wayne L.
Publisher:WINSTON, Wayne L.
Chapter3: Introduction To Optimization Modeling
Section3.8: A Multiperiod Production Model
Problem 22P
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![The economic order quantity (EOQ) model is a classical model used for controlling inventory and satisfying demand. Costs included in the model are holding cost per unit, ordering cost, and the cost of goods ordered. The assumptions for that model are
that only a single item is considered, that the entire quantity ordered arrives at one time, that the demand for the item is constant over time, and that no shortages are allowed.
Suppose we relax the first assumption and allow for multiple items that are independent except for a restriction on the amount of space available to store the products. The following model describes this situation.
Let
= annual demand for item j
Cj = unit cost of item j
S; = cost per order placed for item j
= space required for item j
W;
W = the maximum amount of space available for all goods
i = inventory carrying charge as a percentage of the cost per unit.
The decision variables are Q₁, the amount of item j to order. The model is
Minimize
s.t.
N
j = 1
[c₂p₁ + SPI + IC; 2/1]
SjDj
CjDj
Q₁ ≥ 0.
In the objective function, the first term is the annual cost of goods, the second is the annual ordering cost (D;/Q; is the number of orders), and the last term is the annual inventory holding cost (Q;/2 is the average amount of inventory).
Construct and solve a nonlinear optimization model for the following data.
i = 0.10
ΣwjQ; ≤ W
j=1
Annual Demand
Complete the model.
Item Cost ($)
Order Cost ($)
Space Required (sq. feet)
W = 9,000
Item 1 Item 2
2,000
100
150
50
2,000
50
135
25
Item 3
1,000
80
125
40](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feeaa8045-e0ae-4f8f-bfb9-81e4f69114f4%2F9ce4c4b6-aeb4-45b7-9bac-f243f27dc699%2Ffsx5m8_processed.png&w=3840&q=75)
Transcribed Image Text:The economic order quantity (EOQ) model is a classical model used for controlling inventory and satisfying demand. Costs included in the model are holding cost per unit, ordering cost, and the cost of goods ordered. The assumptions for that model are
that only a single item is considered, that the entire quantity ordered arrives at one time, that the demand for the item is constant over time, and that no shortages are allowed.
Suppose we relax the first assumption and allow for multiple items that are independent except for a restriction on the amount of space available to store the products. The following model describes this situation.
Let
= annual demand for item j
Cj = unit cost of item j
S; = cost per order placed for item j
= space required for item j
W;
W = the maximum amount of space available for all goods
i = inventory carrying charge as a percentage of the cost per unit.
The decision variables are Q₁, the amount of item j to order. The model is
Minimize
s.t.
N
j = 1
[c₂p₁ + SPI + IC; 2/1]
SjDj
CjDj
Q₁ ≥ 0.
In the objective function, the first term is the annual cost of goods, the second is the annual ordering cost (D;/Q; is the number of orders), and the last term is the annual inventory holding cost (Q;/2 is the average amount of inventory).
Construct and solve a nonlinear optimization model for the following data.
i = 0.10
ΣwjQ; ≤ W
j=1
Annual Demand
Complete the model.
Item Cost ($)
Order Cost ($)
Space Required (sq. feet)
W = 9,000
Item 1 Item 2
2,000
100
150
50
2,000
50
135
25
Item 3
1,000
80
125
40
![Min
s.t.
+ 50.
+
Q1, Q2, Q3 ≥ 0.
=
=
2,000 +
=
+
1,000 +
-(₁
((
Q₁ +
Q₁
Q2
• 2,000
Q3
2,000)
]). 1,000)
Q₂ +
+ 0.10.
+0.10.
·(0
+0.10.
• Q39,000
N
Solve the model. (Round Q₁, Q₂, and Q₁ to three decimal places and total cost (in $) to two decimal places.)
Q1
Q2
Q3
Total cost = $
2
Q3
2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feeaa8045-e0ae-4f8f-bfb9-81e4f69114f4%2F9ce4c4b6-aeb4-45b7-9bac-f243f27dc699%2F9jxd2m5_processed.png&w=3840&q=75)
Transcribed Image Text:Min
s.t.
+ 50.
+
Q1, Q2, Q3 ≥ 0.
=
=
2,000 +
=
+
1,000 +
-(₁
((
Q₁ +
Q₁
Q2
• 2,000
Q3
2,000)
]). 1,000)
Q₂ +
+ 0.10.
+0.10.
·(0
+0.10.
• Q39,000
N
Solve the model. (Round Q₁, Q₂, and Q₁ to three decimal places and total cost (in $) to two decimal places.)
Q1
Q2
Q3
Total cost = $
2
Q3
2
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