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The Business Value of Process Flexibility –
An Optimization Model and its Application in the Service Sector
Patrick Afflerbach, Gregor Kastner, Felix Krause, Maximilian Röglinger
Business & Information System Engineering (2014) 6 (4)
Appendix (available online via
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- Derivation of the expected increases in cash inflows of making the inferior process more flexible:
/ (1)
- Derivation of the expected decreases in cash inflows of making the inferior process more flexible in case of excess demand for the inferior process:
/ (2)
- Derivation of the expected decreases in cash inflows of making the inferior process more flexible in case of a demand shortage for the inferior process given a level of realized flexibility :
/ (3)
As already explained, an additional case analysis is necessary to fully specify the cash inflow decreases. If the flexibility potential for the inferior process exceeds (Case 1.2.2), the realized process flexibility of the inferior process can obviously also exceed this threshold. Consequently, the reallocated capacity would then be larger than the maximal free capacity of the inferior process. In other words, the capacity of the inferior process would be reduced below the minimum demand. Clearly, the capacity reduction beyond the minimum demand lead to certain cash inflow reductions and have to be treated differently from capacity reductions up to the minimum demand which leads to uncertain cash inflow reductions. If the flexibility potential is smaller than (Case 1.2.1), the capacity of the inferior process is definitely not reduced below the minimum demand. As a consequence, the cash inflow reductions are always uncertain. A different treatment for realized process flexibilities is not mandatory.
First, we analyze those levels of the realized flexibility that are smaller than the threshold (Case 1.2.2). To obtain the expected cash inflow decreases, the function (the expected decreases in cash inflows given a realized level of flexibility of the inferior process) is integrated over the density function of the flexibility of the inferior process. This covers all excess demand realizations that can be covered by the chosen level of flexibility. Again, larger realizations are considered as well.
/ (4)Second, we analyze those levels of flexibility potentials of the inferior process that exceed the threshold (Case 1.2.2). As already stated, levels of realized flexibility of the inferior process below and above the separating threshold have to be treated differently. The expected cash inflow decreases are a combination of the formulas derived so far. For levels of realized flexibility of the inferior process smaller than the threshold, the decreases of the cash inflows are uncertain and function (4) can be applied. For flexibility realizations larger than the threshold, the additional capacity reductions beyond the minimum demand lead to certain decreases of the cash inflows from the sales of the inferior output. Therefore, formula (2) can be used because this equation considers certain reductions of cash inflows as well. The only difference is that formula (2) does not consider free capacity because it just does not occur in cases of excess demand for the superior process. As the free capacity does not decrease the cash inflows, we have to adjust formula (2) to fit it to the case of shortage demand. The expected free capacity for levels of realized flexibility of the inferior process exceeding the threshold is given by the uniform distribution of the free capacity and equals. In terms of expected values, the reductions of the cash inflows are certain after an adjustment of . Therefore, the expected reduction of the cash inflows for levels of flexibility of the inferior process exceeding equal:
/ (5)- Derivation of the expected increases of the cash inflows of the inferior process by making the superior process more flexible given a level of realizable flexibilization:
/ (6)
- Derivation of the periodic increases of cash inflows of the inferior process by making the superior process more flexible given a level flexibility potential:
/ (7)
- Derivations of the optimal levels of flexibility potentials
6.1 Optimal flexibility potential of the inferior process
For the derivation of the optimal level of flexibility potential of the inferior process, the objective function of the investment has to be determined first. Therefore, the corresponding periodic cash inflows have to be multiplied with the discount factor to obtain the risk adjusted present value from the cash inflows. Form the intermediate result, the cash outflows are subtracted to determine the risk adjusted net present value of a flexibility potential. The objective function can then be derived with respect to the flexibility potential. By setting the first derivative equal to zero and resolving the equation with respect to the flexibility potential of the inferior process. Because of the case distinction, this procedure has to be executed twice.
/ (8)Equation (8) can be resolved with respect to the flexibility potential of the inferior process by applying the solution formula for quadratic equations:
/ (9)Now the same approach is used for values of the flexibility potential exceeding the case distinction threshold:
/ (10)Again resolving equation (10) with respect to the optimal flexibility potential of the inferior process determines the optimum:
/ (11)6.2 Optimal flexibility potential of the superior process
For the derivation of the optimal level of flexibility potential of the superior process, the same approach is applied as for the optimal flexibility of the inferior process:
/ (12)Using again the solution formula for quadratic equations the optimal flexibility of the superior process can be determined:
/ (13)