A third class of analytical algorithms useful in designing and optimizing a process workflow is linear programming (LP) and related models. Table 5.13 lists several diverse applications of linear programming models. There are many others in the literature. LP models are characterized by minimization or maximization objec- tives and the model’s objective function limited by one or more system constraints. Typical system constraints relate to resource scarcity, minimum requirements rel- ative to demand or service, and other requirements. The basic characteristics of
table 5.12 Queuing analysis Study of Capacity utilization
Arrival
Rate of CapacityPercentage Service Rate
Probability of Zero Units Waiting Average Units Waiting Average Waiting
Time Probability of Waiting
10 20% 50 80% 0.05 0.005 20%
25 50% 50 50% 0.5 0.2 50%
40 80% 50 20% 3.2 0.1 80%
45 90% 50 10% 8.1 0.18 90%
49 98% 50 2% 48.02 1 98%
Note: As the system’s capacity utilization approaches 100%, waiting time signifi-
cantly increases. Assumption: Single-channel queue/Poisson arrival rate/ exponential service rate.
table 5.13 typical linear Programming applications
1. Maximizing service productivity 2. Minimizing network routing 3. Optimizing process control 4. Minimizing inventory investment 5. Optimally allocating investment 6. Optimizing product mix profitability 7. Minimizing scheduling cost
8. Minimizing transportation costs 9. Minimizing cost of materials mixtures
an LP model are shown in Table 5.14. As an example, supply-chain optimization problems require matching demand and supply when supply is limited and demand must be satisfied.
Table 5.14 shows that an LP optimization problem is comprised of four major components. These include decision variables whose levels are within an analyst’s control. Examples are decisions relative to when and how much to order, when to manufacture, and when and how much of a product to ship. Constraints are limitations placed upon the system. These include available capacity to manufacture raw materials or components, available production time, available hours, available overtime, and minimum sales. The objective of an LP analysis is to maximize or minimize an objective function. Examples include maximizing profits, minimizing cost, maximizing service levels, and maximizing production throughput. The LP
table 5.14 linear Programming (lP) Characteristics
What is linear programming?
1. An LP algorithm attempts to find a minimization, maximization, or solution when decisions are made with constrained resources as well as other system constraints. As an example, supply-chain optimization problems require matching demand and supply when supply is limited and demand must be satisfied. An LP problem is comprised of four major components:
a. Decision variables within analyst’s control: when and how much to order, when to manufacture, when and how much of the product to ship.
b. Constraints placed on the levels or amounts of decision variables that can be used in the final solution. Examples are capacity to produce raw materials or components, specified number of hours production can run, how much overtime a worker can work, a customer’s capacity to handle and process receipts.
c. Problem objective relative to minimization or maximization. Examples include maximizing profits, minimizing cost, maximizing service levels, and maximizing production throughput.
d. Mathematical relationships among the decision variables, constraints, and problem objectives.
When do we have a solution to a linear program?
1. Feasible solution: Satisfies all the constraints of the problem or objective function.
2. Optimum solution: The best feasible solution, relative to the decision variables and their levels, that achieves the objective of the optimization problem. Although there may be many feasible solutions, there is usually only one optimum solution.
Using Lean Methods to Design for Process Excellence n 155
model is used to evaluate mathematical relationships between decision variables and their constraints relative to the model’s objective. An optimum solution will be the best feasible solution, relative to the decision variables and their levels, that achieves the objective of the optimization problem, which is either minimization or maximization, without violating the constraints. Although there may be many feasible solutions, there is usually only one optimum (this may not always be true).
Figure 5.14 shows the basic elements of an LP model. The objective function represents the goal or objective of the optimization analysis. The problem is to determine the specific level or amount of each decision variable that should be used to optimize the objective function. Each decision variable Xi is weighted by an
objective coefficient Ci. In many LP problems the objective coefficient is a cost per
unit of the decision variable. The optimization of the objective function is limited by the constraints that specify the minimum or maximum amount of resources that can be used in the final solution. Each decision variable in a constraint is weighted by a coefficient that shows its relative contribution to usage of the con- straint toward optimization of the objective function. The right-hand side of each constraint can be of three types: less than or equal, equal, and greater than or equal. Finally, in standard LP models, all decision variables are constrained to be positive, i.e., Xi > 0.
An example of how the LP concept is applied in practice is shown in Figure 5.15: a simple transportation network consisting of three manufacturing facilities and four distribution centers. This is a special type of LP algorithm called the transportation model. The most common objective of a transportation model is to minimize transportation costs between manufacturing facilities and distribution
Maximize Z = C1X1 + C2X2 + …. + CnXn A11X1 + A12X2 + ……… + ≤ B1 A21X2 + ………. + ≤ B2 . . Am1X1 + ……… + AmnXn ≤ Bm Xi ≥ 0
Objective Function Decision Variables Xi
Objective Function Coefficients
Right-Hand-Side (RHS) Decision Variable Coefficient
centers. There are two types of constraints in a transportation model. These relate to the maximum available material that can be shipped from a given manufactur- ing facility and the required demand by a given distribution center for the material. If the available supply and required demand do not balance, then “dummy” manu- facturing facilities or distribution centers are incorporated into the model to bal- ance supply and demand constraints. The right-hand side (RHS) of each constraint also differs in that available supply is less than or equal to the RHS of the supply constraints, and demand is equal to required demand because it must be satisfied by the solution. The problem shown in Figure 5.15 is analyzed using Excel’s Solver
Solution Obtained UsingExcel’s ‘‘Solver”
From/To DC 1 DC 2 DC 3 DC 4 Supply Facility 1 Facility 2 Facility 3 Demand 10 20 10 5 100 5 15 20 30 200 3 5 20 5 300 150 200 50 200
Candidate Solution Shipped Facility 1 0 0 50 50 100 Facility 2 150 50 0 0 200 Facility 3 0 150 0 150 300 Supplied 150 200 50 200 Cost Facility 1 $ - $ - $ 500 $ 250 Facility 2 $ 750 $ 750 $ - $ - Facility 3 $ - $ 750 $ - $ 750 Total Cost $ 3,750 Min Z = X11 + X12 + X13 + X14 + X21 + X22 + X23 + X24 + X31 + X32 + X33 + X34 X11 + X12 + X13 + X14 ≤ 100 X21 + X22 + X23 + X24 ≤ 200 X31 + X32 + X33 + X34 ≤ 300 X11 + X21 + X31 = 150 X12 + X22 + X32 = 200 X13 + X23 + X33 = 50 X14 + X24 + X34 = 200 Xij≥0 for I = 1, 2, 3; j = 1, 2, 3, 4
Using Lean Methods to Design for Process Excellence n 157
algorithm, but many operations research books have software that can analyze LP models. The optimum solution is also shown in Figure 5.15. In the From/To matrix, the costs of every facility and distribution center combination are listed. As an example, the cost per unit from Facility 1 to DC 1 is $10. Also, the maxi- mum supply from Facility 1 is 100 units. The demand of each distribution center is shown as DC 1 = 150 units, DC 2 = 200 units, DC 3 = 50 units, and DC 4 = 200 units. The optimum solution is shown in the Candidate Solution matrix. As an example, Facility 1 ships 50 units to DC 3 and 50 units to DC 5. The total cost of the optimum solution is shown in the Cost matrix as $3750. An alternative and more general form of the model is shown at the bottom of Figure 5.15. LP models have proven useful in process workflow modeling and analysis in many diverse fields of business and science.