just looking for the constraints and variables.
The Nestle’ Purina PetCare Company of St. Louis, Missouri, manufactures a gourmet cat food under the Fancy Feast brand. The brand includes a number of options for choices of meats, fish, and poultry in three ounce cans. The company is considering a new choice for fish to be labeled Tuna Classic. All Fancy Feast products are labeled with strict adherence to the minimum recommended daily allowance (RDA) per ounce established by the AAFCO trade group. The RDA per ounce requirements are: protein 2.6%; thiamine 13.7%; niacin 14.3%; calcium 5.7%; and iron 4.3%. The new product will be blended from two types of tuna: albacore and bonito, two types of supplements: C and D, and filler for stabilization. Albacore tuna contributes 20% of the RDA per ounce requirement for protein, 6% for calcium, and 5% for iron. Bonito tuna contributes 12% of the RDA per ounce requirement for protein, 5% for calcium, and 3% for iron. For supplement C, the contribution rates per ounce are: thiamine 42%; niacin 18%; calcium 22%; and iron 7%. For supplement D, the contribution rates per ounce are: thiamine 36%; niacin 40%; calcium 8%; and iron 9%. Filler does not contribute any of the RDA requirements. The costs per ounce for albacore tuna, bonito tuna, supplement C, supplement D, and filler are $0.15, $0.10, $0.20, $ 0.12, and $0.02 respectively. U.S. government regulations also specify that usage of the brand name tuna requires that at least 40% of the total product must consist of that ingredient.
a. Formulate a linear programming model that can be used to determine the optimal product ingredient mix that yields the minimum total contribution to cost while meeting the individual product RDA per ounce requirements and government branding criteria.
Problem #2 General Foods Corporation (20 points)
The cereal division of General Foods Corporation has had several new product launches that have proven successful: Oats and Rice, Toasted Oats, and Toasted Wheat. New products are only provided in standard 16 ounce boxes. If they prove to be in high demand, they are packaged in General Foods standard three box sizes: Single (16 ounce), Large (24 ounce), and Family (32 ounce). The company has experienced high demand for the new Toasted Wheat product and has forecast demand for next month of 11,500 family size boxes, 15,400 large size boxes, and 2,000 single size boxes. Company policy restricts actual production to vary by not more than 10% in either direction from demand forecasts. Toasted wheat is available in unlimited quantities and is milled and then packaged. Milling times (hours per box) are .009, .011, and .012 for the single, large and family box sizes respectively. The milling and packaging facility has 300 hours of milling time available for the month. Packaging can be done on any or all of three units. Unit 1 is available for 80 hours per month but can only package the large and family size boxes. Unit 2, which can package all three sizes, is available for 180 hours per month. Unit 3 can only package single and large sizes and is available for 160 hours per month. Profit contributions for General Foods are $0.40, $0.48, and $0.60 for the single, large and family box sizes, respectively.
a. Formulate a linear programming model that can be used to determine the optimal product milling and packaging mix that yields the maximum total contribution while meeting the individual product packaging size for the next month production requirements.
Problem #3 Edmonds Paper Products, Inc. (10 points)
Edmonds Paper Products manufactures a variety of products for commercial applications. In one of its processes, rolls of 100 inch wide paper are cut into smaller-width rolls of the same length. Orders for different width rolls are received at various times each week. Current economic conditions have forced Edmonds to abandon the prior practice of meeting demand on a short-time turnaround basis using machine operator estimates which have resulted in excessive waste. For Edmonds , waste is defined as both trim loss and surplus. Trim loss is defined as the leftover portion of a 100 inch roll after cutting to meet specific size requirements. Surplus waste is generated when more rolls of a specific width are cut than are demanded. Machine operators, intending to minimize the number of cuts per roll by creating an inventory of common sizes, have increased logistical costs as well as inventory costs for the company. Edmonds has implemented a policy to complete all orders on a weekly basis to meet demand and minimize waste. This week, the company has orders for 30, 50, 25, and 90 rolls of 60, 48, 36, and 24 inch widths respectively. Edmonds needs to determine all possible ways to cut these widths from 100 inch rolls and to minimize trim and surplus waste.
a. Formulate a linear programming model that can be used to determine the optimal product width type mix that yields the minimum total contribution to trim and surplus waste while meeting the individual product and weekly production requirements for Edmonds Paper Products, Inc .
These are very interesting problems. I would enjoy doing them. I hope you do too!
It would be best if you could give an attempt to solve each problem, and post your answers for verification. This way, you will get good marks for you assignment, but best of all, you will be better prepared for the exams.
this is what i got for problem 1. how does it look?
Min Z = 0.15X1 + 0.10X2 + 0.2X3 + 0.12X4 + 0.02X5
s.t.
0.2X1 + 0.12X2 + 0X3 + 0X4 > 0.026*3 (protein)
0X1 + 0X2 + 0.42X3 + 0.36X4 > 0.137*3 (thiamine)
0X1 + 0X2 + 0.18X3 + 0.40X4 >0.143*3 (niacin)
0.06X1 + 0.05X2 + 0.22X3 + 0.08X4 > 0.057*3 (calcium)
0.05X1 + 0.03X2 + 0.07X3 + 0.09X4 > 0.043*3 (iron)
X1 + X2 ≥ 0.4*3 (at least 40% of the total product must consist of tuna)
(X1 + X2 + X3 + X4 + X5 ) =3
All seems logical, although I have the following comments:
1. The blend does not depend on the packaging weight, so I would optimize the blend for a 1 oz. mix. This way, the formulation of the conditions will not be dependent on the packaging weight.
The actual cost will be the min.Z * packaging weight.
2. The minimum RDA protein, thiamine... requirements are ≥ instead of >.
To formulate the linear programming models for these three problems, we need to identify the decision variables, constraints, and objective function for each problem.
Problem #1 - Nestle' Purina PetCare Company
Decision Variables:
Let x1 be the ounces of albacore tuna used for Tuna Classic.
Let x2 be the ounces of bonito tuna used for Tuna Classic.
Let x3 be the ounces of supplement C used for Tuna Classic.
Let x4 be the ounces of supplement D used for Tuna Classic.
Let x5 be the ounces of filler used for Tuna Classic.
Constraints:
1. Protein requirement: 0.2x1 + 0.12x2 + 0.42x3 + 0.36x4 >= 2.6%*(x1 + x2 + x3 + x4 + x5)
2. Thiamine requirement: 0.36x4 + 0.42x3 >= 13.7%*(x1 + x2 + x3 + x4 + x5)
3. Niacin requirement: 0.4x4 + 0.18x3 >= 14.3%*(x1 + x2 + x3 + x4 + x5)
4. Calcium requirement: 0.06x1 + 0.05x2 + 0.22x3 + 0.08x4 >= 5.7%*(x1 + x2 + x3 + x4 + x5)
5. Iron requirement: 0.05x1 + 0.03x2 + 0.07x3 + 0.09x4 >= 4.3%*(x1 + x2 + x3 + x4 + x5)
6. Tuna ingredient requirement: x1 + x2 >= 0.4*(x1 + x2 + x3 + x4 + x5)
Objective Function:
Minimize the cost: 0.15x1 + 0.10x2 + 0.20x3 + 0.12x4 + 0.02x5
Problem #2 - General Foods Corporation
Decision Variables:
Let x1 be the number of single size boxes of Toasted Wheat produced.
Let x2 be the number of large size boxes of Toasted Wheat produced.
Let x3 be the number of family size boxes of Toasted Wheat produced.
Constraints:
1. Demand constraint: x3 = 11,500
2. Demand constraint: x2 = 15,400
3. Demand constraint: x1 = 2,000
4. Actual production constraint: x3 <= 1.1*11,500
5. Actual production constraint: x2 <= 1.1*15,400
6. Actual production constraint: x1 <= 1.1*2,000
7. Milling time constraint: 0.012x3 + 0.011x2 + 0.009x1 <= 300
8. Packaging time constraint: x2 + x3 <= 80
9. Packaging time constraint: x1 + x2 + x3 <= 180
Objective Function:
Maximize the profit: 0.60x3 + 0.48x2 + 0.40x1
Problem #3 - Edmonds Paper Products, Inc.
Decision Variables:
Let x1 be the number of rolls of 60-inch width.
Let x2 be the number of rolls of 48-inch width.
Let x3 be the number of rolls of 36-inch width.
Let x4 be the number of rolls of 24-inch width.
Constraints:
1. Production constraint: x1 + x2 + x3 + x4 = 30 + 50 + 25 + 90 = 195
2. Width constraint: x1 <= 30
3. Width constraint: x2 <= 50
4. Width constraint: x3 <= 25
5. Width constraint: x4 <= 90
Objective Function:
Minimize the waste: 100*(x1 + x2 + x3 + x4) - (60*x1 + 48*x2 + 36*x3 + 24*x4)
These are the generalized formulations for the given problems. To solve each problem, you can use any linear programming solver or software, and input the decision variables, constraints, and objective function according to the formulations above.