Aggregated planning studies the balance between production and demand. The period of time considered is the medium run. In order to meet a fluctuating demand, at a minimum cost, we can change the company’s resources (employees, production, and inventory levels), we can influence the demand, or we can try to find a combination of both strategies.
As strategies to influence the demand, we have: advertising, sales, development of alternative products, etc. As strategies to influence production, we can highlight:
Most of the methods used to solve the aggregated planning problem consider the demand to be a deterministic factor, so, they only change the company’s productive resources. Thus, we can use the trial and error method trying to select the best option among a set of alternative production solutions, or a linear programming model to determine the problem’s optimal solution.
Linear programming (LP) models are being widely used to solve aggregated planning problems, in order to find the best combination of productive resources that minimizes the total labor, production, and storage costs. For T periods of time, the objective function can minimize the sum of costs related to: regular production, regular labor, hiring and firing employees, overtime, inventory, and/or outsourcing. The constraints are related to the total production and storage capacity, besides the use of labor. The problem can also be characterized as a nonlinear programming—NLP (nonlinear costs) model or as a binary programming—BP model (a choice among n alternative plans).
Buffa and Sarin (1987), Moreira (2006), and Silva Filho et al. (2009) present a general linear programming model for the aggregated planning problem. An adjusted formulation, for T periods of time (t = 1, …, T), is shown.
Model parameters:
Pt = total production in period t
Dt = demand in period t
rt = production cost per unit (regular hours) in period t
ot = production cost per unit (overtime) in period t
st = production cost per unit (with subcontracted/outsourced labor) in period t
ht = cost of an additional unit (regular hours) in period t by hiring employees from period t − 1 to period t
ft = cost of a cancelled unit in period t by firing employees from period t − 1 to period t
it = inventory cost per unit from period t to period t + 1
Itmax = maximum inventory capacity in period t (units)
Rtmax = maximum production capacity at regular hours in period t (units)
Otmax = maximum production capacity during overtime in period t (units)
Stmax = maximum subcontracted production capacity in period t (units)
Decision variables:
It = final inventory in period t (units)
Rt = regular production (regular hours) in period t (units)
Ot = overtime production in period t (units)
St = production with subcontracted labor in period t (units)
Ht = additional production in period t by hiring employees from period t − 1 to period t (units)
Ft = cancelled production in period t by firing employees from period t − 1 to period t (units)
General formulation:
For T periods of time, the model’s objective function tries to minimize the sum of costs related to regular production, overtime production, subcontracting or outsourcing, and hiring and firing employees, besides the costs with inventory maintenance.
Equation (1) of Expression (16.17) states that the final inventory in period t is the same as the final inventory in the previous period, added to the total produced in the same period, subtracting the demand for the current period.
The production capacity is specified in Equation (2) of Expression (16.17) as the sum of the total produced regularly in period t, the overtime production, and the total of subcontracted units for the same period.
Equation (3) of Expression (16.17) states that the total number of units produced with regular labor in period t is the same as the previous period (t − 1), adding the additional units produced with possible hiring, and subtracting the units cancelled due to possible dismissals from period t − 1 for period t.
Constraint 4 stipulates the maximum inventory capacity allowed for period t.
Constraint 5 guarantees that the regular production in period t will not be greater than the maximum limit allowed.
Constraint 6 stipulates the maximum production limit allowed using overtime in period t.
Constraint 7 sets a maximum production limit using outsourced labor for period t.
Finally, the non-negativity conditions of the model’s decision variables must also be met.
The formulation is based on a linear programming (LP) model to solve the respective aggregated planning problem. However, if we considered as a decision variable the number of employees to be hired and fired in each period, instead of the variation in production due to the hiring or firing of employees, we would find ourselves in a mixed-integer programming (MIP) problem, in which part of the decision variables is discrete. Similar to the production mix problem and to the production and inventory problem, when all the model’s decision variables are discrete (the quantities produced and stored can only assume integer values), we have an integer programming (IP) model.
Optimization models can help researchers and managers in their business decision-making process.
Among the existing optimization models, we can mention linear programming, network programming, integer programming, nonlinear programming, goal or multiobjective programming, and dynamic programming. Linear programming is one of the most widely used tools.
This chapter introduced and presented the main concepts of optimization models, especially, the modeling of linear programming problems (general formulation in the standard and canonical forms and business modeling problems).
The use of optimization models, mainly linear programming, is being more and more disseminated in academia and in the business world. It may be applied to several areas (strategy, marketing, finance, operations and logistics, human resources, among others) and to several sectors (transportation, automobile, aviation, naval, trade, services, banking, food, beverages, agribusiness, health, real estate, metallurgy, paper and cellulose, electrical energy, oil, gas and fuels, computers, telecommunications, mining, among others). The greatest motivation is the huge saving that may happen, around millions or even billions of dollars, for the industries that use them.
Several real problems can be formulated through a linear programming model, including: a production mix problem, a mixture problem, a capital budget problem, an investment portfolio selection, production and inventory, an aggregated planning, among others.
The methods to solve a linear programming problem (graphical, analytical, by using the Simplex algorithm or by using computerized solutions) will be discussed in the next chapter.
Table 16.5
Sector | Time Necessary (Hours/Machine) to Manufacture 1 Unit | Time Available (Hours/Machine/Week) | ||||
---|---|---|---|---|---|---|
Refrigerator | Freezer | Stove | Dishwasher | Microwave oven | ||
Pressing | 0.2 | 0.2 | 0.4 | 0.4 | 0.3 | 400 |
Painting | 0.2 | 0.3 | 0.3 | 0.3 | 0.2 | 350 |
Molding | 0.4 | 0.3 | 0.3 | 0.3 | 0.2 | 250 |
Assembly | 0.2 | 0.4 | 0.4 | 0.4 | 0.4 | 200 |
Packaging | 0.1 | 0.2 | 0.2 | 0.2 | 0.3 | 200 |
Table 16.6
Sector | Total Number of Labor Hours to Manufacture 1 Unit | Employees Available | ||||
---|---|---|---|---|---|---|
Refrigerator | Freezer | Stove | Dishwasher | Microwave Oven | ||
Pressing | 0.5 | 0.4 | 0.5 | 0.4 | 0.2 | 12 |
Painting | 0.3 | 0.4 | 0.4 | 0.4 | 0.3 | 10 |
Molding | 0.5 | 0.5 | 0.3 | 0.4 | 0.3 | 8 |
Assembly | 0.6 | 0.5 | 0.4 | 0.5 | 0.6 | 10 |
Packaging | 0.4 | 0.4 | 0.4 | 0.3 | 0.2 | 8 |
Table 16.7
Product | Maximum Capacity (Units/Week) | Minimum Demand (Units/Week) | Profit Per Unit (US$/Unit) |
---|---|---|---|
Refrigerator | 1000 | 200 | 52 |
Freezer | 800 | 50 | 37 |
Stove | 500 | 50 | 35 |
Dishwasher | 500 | 50 | 40 |
Microwave oven | 200 | 40 | 29 |
Table 16.8
Food | Energy (cal/Portion) | Fibers (g/Portion) | % Vitamins and Minerals | Type of Food | Meals |
---|---|---|---|---|---|
Lettuce | 1 | 1 | 9 | V | 3, 5 |
Plums/prunes | 30 | 2.4 | 4 | F | 1, 2, 4 |
Rice | 130 | 1.2 | 0.5 | C | 3 |
Brown rice | 110 | 1.6 | 1 | C | 3 |
Olive oil | 90 | 0 | 0 | TF | 3, 5 |
Banana | 80 | 2.6 | 13 | F | 1, 2, 4 |
Cereal bar | 90 | 0.9 | 11 | C | 1, 2, 4 |
Crackers | 90 | 0.4 | 0.4 | C | 1, 2, 4 |
Broccoli | 10 | 2.7 | 15 | V | 3, 5 |
Meat | 132 | 0 | 1 | P | 3 |
Carrots | 31 | 2 | 19 | V | 3, 5 |
Cereal | 120 | 1.3 | 20 | C | 1 |
Chocolate | 150 | 0.2 | 0.5 | SS | 3, 5 |
Spinach | 18 | 2 | 28 | V | 3, 5 |
Beans | 95 | 7.9 | 6 | L | 3 |
Chicken | 112 | 0 | 1.5 | P | 3 |
Jello | 30 | 0.2 | 0 | SS | 3, 5 |
Chickpeas | 92 | 3.5 | 4 | L | 3 |
Yoghurt | 70 | 1.1 | 0.7 | MD | 1, 2, 4 |
Apples | 60 | 3 | 0.9 | F | 1, 2, 4 |
Papayas | 56 | 2.4 | 3.1 | F | 1, 2, 4 |
Eggs | 60 | 0.6 | 8.5 | P | 3 |
Butter | 100 | 0 | 0 | TF | 1, 5 |
Bread | 140 | 0.5 | 3.3 | C | 1, 5 |
Wholewheat bread | 142 | 0.8 | 12 | C | 1, 5 |
Turkey ham | 75 | 0.4 | 0.4 | P | 1, 5 |
Fish | 104 | 0.7 | 11 | P | 3 |
Pears | 88 | 4 | 1.2 | F | 1, 2, 4 |
Cottage cheese | 80 | 0.4 | 0.6 | MD | 1, 5 |
Arugula | 4 | 1 | 9.5 | V | 3, 5 |
Natural sandwiches | 240 | 1.4 | 19 | Mixed | 5 |
Soya | 85 | 3.9 | 8 | L | 3 |
Soup | 120 | 3.5 | 16 | Mixed | 5 |
Tomatoes | 26 | 1.5 | 5 | V | 3, 5 |
C, carbs; V, vegetables; F, fruit; P, protein; L, legumes; MD, milk and dairy products; TF, total fat; SS, sugar and sweets; 1, food that can be eaten at breakfast; 2 food that can be eaten as a morning snack; 3, food that can be eaten at lunch; 4, food that can be eaten as an afternoon snack; 5, food that can be eaten at dinner.
Table 16.10
Stock 1 | Stock 2 | Stock 3 | Stock 4 | Stock 5 | Stock 6 | Stock 7 | Stock 8 | Stock 9 | Stock 10 | |
---|---|---|---|---|---|---|---|---|---|---|
Banking (%) | Banking (%) | Banking (%) | Banking (%) | Insurance (%) | Metallurgy (%) | Metallurgy (%) | Mining (%) | Paper-Cellulose (%) | Electrical Energy (%) | |
1 | 2.57 | 4.47 | 1.08 | 4.78 | 4.19 | 2.54 | 0.57 | 0.60 | 4.07 | 2.78 |
2 | 3.14 | 4.33 | 0.87 | 3.41 | 3.08 | 2.69 | 0.98 | 5.78 | 3.57 | 3.69 |
3 | 6.00 | 2.67 | 4.87 | 2.81 | 6.47 | 1.98 | 5.69 | 3.25 | 2.69 | − 2.14 |
4 | 2.14 | − 3.59 | 3.57 | 6.70 | 8.05 | − 3.14 | − 3.10 | − 0.88 | 2.02 | 4.01 |
5 | − 5.44 | 3.34 | − 2.78 | 2.08 | 5.04 | − 7.58 | − 3.28 | − 4.52 | − 1.57 | − 1.33 |
6 | 11.30 | 2.09 | − 5.69 | − 3.00 | − 3.47 | 6.85 | − 8.07 | − 2.88 | − 2.33 | 4.21 |
7 | 8.07 | − 7.80 | 6.44 | − 3.54 | − 2.09 | 4.70 | 2.67 | 0.58 | − 2.87 | 0.74 |
8 | 2.77 | − 6.14 | 6.87 | 2.97 | − 2.56 | 11.02 | 3.69 | − 3.69 | − 0.05 | 0.65 |
9 | 2.37 | 5.77 | 10.07 | 5.90 | 4.44 | − 5.99 | 6.47 | − 1.44 | 1.69 | 2.47 |
10 | 2.14 | − 3.23 | − 5.64 | − 7.01 | 6.07 | 0.14 | 0.22 | − 4.22 | 5.87 | − 3.54 |
11 | − 4.40 | − 1.04 | − 3.30 | − 2.04 | − 5.30 | − 2.36 | − 3.11 | 0.47 | 2.14 | − 2.58 |
12 | − 2.10 | − 3.02 | − 2.27 | 3.50 | − 2.07 | 2.14 | − 4.55 | 0.05 | 1.01 | 5.47 |
13 | 2.14 | 2.01 | − 5.47 | − 9.33 | 4.44 | 1.34 | 0.24 | − 6.95 | 3.99 | 3.54 |
14 | 4.69 | 3.67 | − 2.10 | − 8.07 | − 6.14 | 0.98 | − 3.50 | 8.41 | − 1.47 | 2.57 |
15 | 11.32 | − 5.69 | 2.07 | 2.77 | − 3.07 | 0.66 | − 2.78 | − 5.41 | 2.58 | − 4.78 |
16 | − 4.69 | − 2.00 | 3.47 | 5.48 | − 2.05 | 2.89 | − 8.40 | 0.22 | 3.57 | − 1.23 |
17 | 2.01 | 6.75 | 3.78 | − 3.50 | 2.67 | − 13.47 | − 7.55 | 9.54 | 0.88 | 0.27 |
18 | − 7.65 | 9.47 | 3.89 | 6.41 | 3.07 | − 4.23 | 0.07 | − 11.02 | − 2.34 | 3.55 |
19 | − 2.36 | − 5.33 | − 5.68 | 3.04 | 4.08 | − 0.28 | 9.56 | − 2.55 | − 1.09 | 2.67 |
20 | − 11.47 | − 6.01 | − 3.46 | 2.08 | 4.99 | 2.63 | 5.04 | − 12.23 | 7.03 | 0.74 |
21 | 3.39 | − 2.01 | − 3.09 | 3.64 | − 3.70 | − 3.63 | − 3.66 | − 2.00 | 4.33 | 3.69 |
22 | − 8.43 | 5.03 | 1.01 | − 6.80 | − 8.02 | 2.47 | − 4.40 | 4.47 | − 5.87 | − 0.25 |
23 | − 4.16 | 5.33 | − 5.61 | − 5.47 | − 7.35 | 0.50 | 2.57 | − 6.58 | 2.67 | − 0.98 |
24 | − 2.37 | − 3.36 | − 7.43 | − 6.17 | 2.44 | − 7.99 | − 3.01 | − 8.80 | 7.80 | 4.36 |
25 | 7.00 | 11.04 | 6.40 | 5.55 | 11.07 | 6.01 | 9.77 | 5.96 | 2.22 | 1.66 |
26 | 3.22 | 4.64 | 6.43 | 4.58 | − 2.47 | 14.15 | 6.41 | 3.22 | 1.49 | − 0.20 |
27 | 4.67 | 2.07 | 2.98 | − 2.07 | − 2.60 | 5.47 | − 2.60 | 4.74 | 1.42 | 1.59 |
28 | 3.20 | 3.68 | − 3.10 | − 2.65 | 3.18 | − 3.14 | − 3.01 | − 2.33 | − 0.77 | 5.67 |
29 | − 0.74 | − 0.58 | − 2.73 | 6.47 | 3.08 | − 3.25 | 7.78 | 4.01 | 0.59 | 4.90 |
30 | − 5.02 | − 7.04 | − 9.40 | 6.07 | 2.00 | − 1.08 | 8.36 | 4.32 | 3.07 | 3.92 |
31 | − 4.30 | 2.99 | 6.81 | 5.88 | − 6.47 | 5.47 | 2.04 | − 6.77 | − 2.55 | 2.14 |
32 | 2.64 | 7.66 | 6.90 | − 0.47 | 6.13 | 11.01 | 2.15 | − 2.64 | − 0.84 | − 0.71 |
33 | 6.77 | 7.16 | 5.87 | 8.09 | 2.47 | 5.71 | 3.19 | 5.74 | 5.98 | 2.04 |
34 | 6.70 | − 3.41 | 6.80 | 6.47 | 2.08 | − 14.33 | − 2.03 | 9.12 | 0.25 | 4.33 |
35 | 2.98 | − 2.01 | 5.32 | − 5.00 | 4.43 | − 5.44 | 6.07 | 8.40 | − 0.50 | − 2.36 |
36 | 5.70 | 11.52 | 6.00 | 0.27 | 2.29 | 2.47 | 5.73 | 6.47 | 1.00 | 1.60 |
Table 16.11
Fund | Annual Yield/Return (%) | Risk (%) | Initial Investment (US$) |
---|---|---|---|
Index-pegged fund A | 11.74 | 1.07 | 30,000.00 |
Index-pegged fund B | 12.19 | 1.07 | 100,000.00 |
Index-pegged fund C | 12.66 | 1.07 | 250,000.00 |
Fixed income fund A | 12.22 | 1.62 | 30,000.00 |
Fixed income fund B | 12.87 | 1.62 | 100,000.00 |
Fixed income fund C | 12.96 | 1.62 | 250,000.00 |
Commercial paper A | 16.04 | 5.89 | 20,000.00 |
Commercial paper B | 17.13 | 5.89 | 100,000.00 |
Multimarket fund | 18.10 | 5.92 | 10,000.00 |
Stock fund A | 19.53 | 6.54 | 1000.00 |
Stock fund B | 22.16 | 7.23 | 1000.00 |
Note: Since we have a binary decision variable (if product i is delivered by facility j to retailer k in period t, its value will be 1, otherwise, the value will be 0), we have a mixed programming problem.
Note: In this case, we must define a new decision variable that consists in determining the amount of product i to be transported from facility j to retailer k in period t.
The production data are:
Costs of regular production | US$ 1.50 per kg |
Cost to outsource labor | US$ 2.00 per kg |
Costs of regular labor | US$ 600.00/employee-month |
Cost to hire a worker | US$ 1000.00/worker |
Cost to fire a worker | US$ 900.00/worker |
Cost per overtime | US$ 7.00/overtime |
Inventory maintenance costs | US$ 1.00/kg-month |
Regular labor in the previous month | 10 workers |
Initial inventory | 600 kg |
Average productivity per employee | 16 kg/employee-hour |
Average productivity per overtime | 14 kg/overtime |
Maximum outsourced production capacity | 1 000 kg/month |
Maximum regular labor capacity | 20 workers |
Maximum inventory capacity | 2500 kg/month |
Each employee usually works 6 business hours a day, 20 business days a month, and he/she is only allowed to work 20 hours overtime a month, at the most. Formulate the aggregate planning model (mixed integer program) that will minimize the total production, labor and storage costs for the period analyzed, respecting the system’s capacity constraints.