# The discipline of using management science

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Management science is the discipline that adapts the scientific approach for problem solving to executive decision making in order to accomplish the goal of best efficiency and best effectiveness. According to Lawrence and Pasternack, it involves:

Analyzing and building mathematical models of complex business situations.

Solving and refining the mathematical models typically using spreadsheets, or Window QSB to gain insight into the business situations.

Communicating or implementing the resulting insights and recommendations based on these models.

## Mathematical modeling

Management science employs mathematical modeling, the process that translates observed or desired phenomena into mathematical expressions. (Lawrence and Pasternack, 2002).

## The management science process

Figure A: The management science process

Source: Lawrence, John A. Jr., and Pasternack, Barry A., 2002, Applied Management Science: Modeling, Spreadsheet Analysis, and Communication for Decision Making, 2nd ed. USA: John Wiley & Sons, Inc., pp 9

## Step 1: Defining the problem

Problem definition

Observe operations

Ease into complexity

Political recognize realities

Determine what is really needed

Point out constraints

Find continuous feedback

Management science is generally applied in three situations:

Designing and implementing new procedures or operations

Evaluating set of procedures or operations

Deciding and proposing corrective action for procedures and operations that are creating unsatisfactory results

## Step 2: Building a mathematical model:

Mathematical modeling is a procedure that verbalizes and recognizes a problem and then sums it by turning the words into mathematical expressions.

Determining decision variables

Summing the objective and constraints

Building a shell of model

Collecting Data - cost issues/consider time

## Step 3: Solving a mathematical model:

Select a properly solution technique.

Build solutions of model.

Validate/Test/ results of model.

Carry out the analysis "what - if".

## Step 4: Communicating/monitoring the results

The final step is the last - solution phase in the processing management science. This step includes two functions:

Prepare a presentations or business reports.

Observe the progress of the performance.

## Definition

Linear programming model is a model in which looks for to minimize or maximize a linear objective function subject to a set of linear constraints.

## Why linear programming models are important

Many problems lend naturally themselves to a formulation of linear programming, and models with these structures can closely approximate many other problems.

This type of solving models has efficient solution techniques.

The output created from linear programming package supplies useful 'what - if" information linking the sensitivity of the range of optimal solution to alter in the coefficients model.

## Linear programming assumption

The parameter values are known with certainty

The above constrains and objective function shows the constant returns to scale.

Between the variables of decision don't have interactions.

## Application of LPM

In spite of these assumptions could occur to be restrictive, they provide frequently "close enough" estimations for some practical problems.

Figure B: Application of LPM

Source: Lawrence, John A. Jr., and Pasternack, Barry A., 2002, Applied Management Science: Modeling, Spreadsheet Analysis, and Communication for Decision Making, 2nd ed. USA: John Wiley & Sons, Inc., pp 50

## Limitation of linear programming model

With the computers are able to resolve linear programming problems with simple, the challenges are in the formulation of problems - translation the statement of problem linear equations to be resolved by computers. The problem is created from the statement of problem as follows:

Determine the problem objective that quantity is to be optimized.

Determine the constraints and decision variables on them.

Write the function of objectives and the constraints in the decision variables, applying information from the statement of problems to identify the properly coefficient of each term. Dispose of some unneeded information.

Extra some implicit constraints, like non-negative limitations.

Arrange the system of equations in the consistent form suitable for solving by computer.

## Decrease the errors risk in formulations of problem:

Initial conditions should be considered.

Make sure that each variable in the object function appears at least one in the constraints.

Consider constraints that might not be specified explicitly.

## Sensitivity analysis

Sensitivity analysis is the study of how changes in the coefficients of a linear programming problem affect the optimal solution. Sensitivity analysis is important to decision makers because real-world problems exist in a changing environment. (Sweeney, Dennis J., Williams, Thomas A., and Martin Kipp, 2008)

## Constraint

The Linear programming problems have restrictions or constraints that limit the degree to which the objective can be pursued.

## Range of optimality

Sensitivity analysis of an objective function coefficient focuses on answering the following question: "keeping all other factors the same, how much can an objective function coefficient change without changing the optimal solution?"

Assuming that there are no other changes to the input parameters:

The range of optimality is the range of values for an objective function coefficient in which the optimal solution remains unchanged.

The value of the objective function will change if this coefficient multiplies a variable whose value is possible.

## Reduced Cost

Assuming that there are no other changes to the input parameters:

The reduced cost for a variable that has a solution of 0 is the negative of the objective function coefficient increase necessary for the variable to be positive in the optimal solution.

The reduced cost is also the amount the objective function will change per unit increase in this variable.

Assuming that there are no other changes to the input parameters, the shadow price for a constraint is the change to the objective function value per unit increase to its right-hand side coefficient.

## Range of Feasibility

Assuming that there are no other changes to the input parameters:

The range of feasibility is the range of values for a right hand side value in which the shadow prices for the constraints remain unchanged.

In the range of feasibility, the value of the objective function will change by the amount of the shadow price times the change to the right hand side value.

## Minimum invest in trucks:

X1 = the number of pick up trucks.

X2 = the number of moving Van type trucks.

Table 1:

X1

24000

1

1

X2

60000

4

2.5

## Limitation

â‰¤ 40

â‰¤ 48

â‰¥ 36

Objective function: Minimum in investment in trucks:

24000X1 + 60000X2

Constraints:

X1 + 2.5X2 â‰¥ 36 (Total trucking capital)

X1 + 4X2 â‰¤ 48 (Total current workers)

X1 + X2 â‰¤ 40 (Total current facilities)

X1, X2 â‰¥ 0

Figure 1: Solution for Bay City Mover minimum in investment when purchase of pickup trucks and vans

In this case, we have 2 variables and 3 constraints. The minimum investment in trucks will be \$864000. With the amount of money, the company can buy 16 trucks and 8 vans.

## Sensitivity analysis:

Range of optimality:

The range of optimality for X1 is from 24000 to the infinity, means that the allowable decrease is 0, and the allowable increase is infinity. In contrast, the range of optimality for X2 is from negative infinity to 60000, means that the allowable decrease is infinity and the allowable increase is 0. Any changes out of range of optimality lead to change optimal solution.

Reduced cost:

According to the figure 1, the reduced cost is 0 means that is unnecessary to adjust the cost that invested in trucks and vans.

Figure 1 also indicates that the constraint C1 have 24000 in shadow price. That means if the company increases 1 unit of capacity (1 ton), it leads to increase \$24000 in investment. However, the company wish minimum in investment, so that the company will reduce capacity.

Moreover, The Shadow Price for C3 constraint (total number of Truck and Van) is 0 because it has a slack or unused capacity 16 units available. Additional number of Truck and Van will not improve the value of the objective function.

Range of Feasibility:

Figure 1 shows that Allowable Min RHS (right hand side) of capacity constraint (constraint C1) is 30, whereas Allowable Max RHS of capacity constraint is 44. Thus, the range of Feasibility is 14. The shadow price will be applied for every changes of capacity in the range of feasibility. For example, the company can add maximum 8 tons (the objective function will increase \$192000) and decrease maximum 16 tons (the objective function will decrease \$384000). Furthermore, allowable max RHS of the C2 constraint is 57.60, whereas allowable min RHS of this constraint is 36, so that the range of feasibility is 21 (the number of workers cannot odd). The shadow price will be applied for every changes of capacity in the range of feasibility, but shadow price is 0. The range of feasibility of constraint C3 is from 24 to infinity.

## Purchasing only pick up trucks:

Invest in trucks:

24000X1

Constraints:

X1 â‰¥ 36 (Total trucking capital)

X1 â‰¤ 48 (Total current workers)

X1 â‰¤ 40 (Total current facilities)

X1, X2 â‰¥ 0

Figure 2: Solution for Bay City Movers minimum in investment when purchasing only pickup trucks

The figure 2 shows solution optimal by using Linear Programming Model with 2 variables and 3 constraints in which variable X2 = 0. The lowest amount of money that Bay City Mover has to invest is \$864000, and this company can buy 36 trucks.

## Sensitivity analysis:

Reduced cost:

Similarly first case, the reduced cost is still 0.

The shadow price of capacity constraint is still 24000, means that if the company increases 1 unit of capacity (1 ton), it leads to increase \$24000 in investment. However other two constraints is 0.

The range of Feasibility:

In this case, the range of feasibility of constraint C1 is 40; with the allowable min RHS is 0 whereas the allowable max RHS is 40. The shadow price will be applied for every changes of capacity in the range of feasibility. For example, the company can add maximum 4 tons (the objective function will increase \$96000) and decrease maximum 36 tons (the objective function will decrease \$864000).

## Purchasing only Van Moving type trucks:

Invest in trucks:

60000X2

Constraints:

2.5X2 â‰¥ 36 (Total trucking capital)

4X2 â‰¤ 48 (Total current workers)

X2 â‰¤ 40 (Total current facilities)

X1, X2 â‰¥ 0

Figure 3: Solution for Bay City Mover minimum in investment when purchasing only Vans

The optimal solution to this 2-variable, 3-contraint Linear Program is shown in Figure 3. It is infeasible with this purchasing option.

## Purchasing the same number trucks as van

Minimum invest in trucks:

24000X1 + 60000X2

Constraints:

X1 + 2.5X2 â‰¥ 36 (Total trucking capital)

X1 + 4X2 â‰¤ 48 (Total current workers)

X1 + X2 â‰¤ 40 (Total current facilities)

X1 - X2 = 0 (number of trucks as van)

X1, X2 â‰¥ 0

Figure 4: Solution for Bay City Mover minimum in investment when purchasing the number of trucks as vans.

The optimal solution to this 2-variable, 4-contraint Linear Program is shown in Figure 4. It is infeasible with this purchasing option.

## Purchasing the minimum total number of trucks

Minimum total number of trucks:

X1 + X2

Constraints:

X1 + 2.5X2 â‰¥ 36 (Total trucking capital)

X1 + 4X2 â‰¤ 48 (Total current workers)

X1 + X2 â‰¤ 40 (Total current facilities)

24000X1 + 60000X2 = 8640000 (minimum in investment)

X1, X2 â‰¥ 0

Figure 5: Solution for Bay City Mover minimum in investment when purchasing minimum the number of trucks and vans

The figure 5 displays solution optimal by using Linear Programming Model with 2 variables and 4 constraints. The optimal solution depicts that 16 Pick Up Trucks and 8 Moving Vans should be bought. To satisfy all 4 constraints, the total 24 units is the smallest quantity of trucks and vans.

## Sensitivity analysis:

In this case, the shadow price of capacity constraint and current facilities are 0; and the allowable min RHS are 36 and 24 respectively. However, the current facilities constraint has 16 of slack.

The shadow price of total worker constraint (constraint C2) is -1 means that if Bay City Movers increase the constraint of workers from 48 to 49 persons, the total number of Pick Up Trucks and Moving Vans will down from 24 to 23 units. Moreover, The Range of Feasibility for C2 now is 21 (the allowable min is 36 and the allowable max is 57). In the range of feasibility for C2, the shadow price remains unchanged.

The range of feasibility of constraint C4 (minimum in investment) is \$144000.

## CONCLUSION AND RECOMMENDATION

It is clear that there are 3 solutions that the Bay City Mover should consider according to their options:

Figure 6: Solutions

## 1

Buy trucks and vans with minimum in investment

Number of trucks: 16

Number of vans: 8

Money: \$864000

Flexibility:Bay City Mover has 2 kinds of trucks and vans, so it is flexible to carry out their work.

Economy:the company can save fuel and workers as well as money when the load of goods is small by using pickup trucks.

Management: more kind of truck, more difficult to manage equipment and worker.

Equipment management: depreciation cost, maintain and repair costâ€¦

Worker management: human resource, salary, insurance, bonus, incentive money, reward system, and organizational behaviorâ€¦

## 2

Buy only trucks with minimum in investment

Number of trucks: 36

Number of vans: 0

Money: \$864000

Easier for Bay City Movers to manage 1 fleet of Pick Up Truck in calculating and checking depreciation cost, fuel cost, maintenance cost â€¦

More convenience for them to administrational works because the have only 1 team of workers.

Bay City Movers don't have the flexibility for different types of works.

In some cases, they have to use more pickup trucks for large quantity of goods. Consequently, they will waste the cost of fuel and loading capacity.

There are 12 workers that still available. That is a waste and the company still pays salary, insuranceâ€¦

## 3

Minimum in total trucks and van with minimum in investment

Number of trucks: 16

Number of vans: 8

Money: \$864000

This solution has the pros of the solution 1 including flexibility and economy.

Simple and convenient:with the objective is to minimize the total number of trucks, we can understand that the board of management of Bay City Movers emphasize on the simple and convenient administrational management.

Although this solution has a lot of advantages, but the pressure on this company is also heavy because they have to solve both types of restriction at the same time.

By analyzing pros and cons of these solutions, I would like to recommend solution 3 to Bay City Mover Company. Because this is the most suitable solution in the modern business. The company wants to increase productivity, cut cost, and approach flexibility, simple and convenient. On the other hands, the company also invests in advance technology to cope with restriction, and education and training workers as well.