Single-period. This type of problem can be recognized when a probability distributio or empirical distribution for demand for a "perishable" item is given along with unit shortage and excess costs, or information that can be used to calculate them .Use these steps to solve the problem:

Question

1. Compute the optimal service level using Formula 13 - 21.

[latex]SL = \frac{C_s}{C_s + C_e}[/latex] ( 13 - 21)

2. If the given distribution is uniform or normal, use that to obtain the exact stocking level.

3. If the distribution is Poisson or empirical, SL will fall between two cumulative frequencies. Round up to the higher frequency to find the optimal stocking level. Not: If the distribution is empirical, first obtain the cumulative frequencies or probabilities.

A firm that installs cable TV systems uses acertain piece of equipment for which it carries two spare part. The part cost $500 each and have no salvage value. Part failures can be modeled by a Poisson distribultion with a mean of two failures during the useful life of the equipment. Holding and disposal costs are negligible. Estimate the apparent range of shortage cost.

Accepted Answer

[latex]C_s [/latex] is unknown [latex] C_e [/latex]= $500

The Poisson table (Appendix B, Table C)

Table C
Cumulative Poisson probabilities

µ\ x	0	1	2	3	4	5	6	7	8	9
0.05	0.951	0.999	1·000
0.1	0.905	0.995	1·000
0.15	0.861	0.99	0.999	1·000
0.2	0.819	0.982	0.999	1·000
0.25	0.779	0.974	0.998	1·000
0.3	0.741	0.963	0.996	1·000
0.35	0.705	0.951	0.994	1·000
0.4	0.67	0.938	0.992	0.999	1·000
0.45	0.638	0.925	0.989	0.999	1·000
0.5	0.607	0.91	0.986	0.998	1·000
0.55	0.577	0.894	0.982	0.998	1·000
0.6	0.549	0.878	0.977	0.997	1·000
0.65	0.522	0.861	0.972	0.996	0.999	1·000
0.7	0.497	0.844	0.966	0.994	0.999	1·000
0.75	0.472	0.827	0.96	0.993	0.999	1·000
0.8	0.449	0.809	0.953	0.991	0.999	1·000
0.85	0.427	0.791	0.945	0.989	0.998	1·000
0.9	0.407	0.772	0.937	0.987	0.998	1·000
0.95	0.387	0.754	0.929	0.984	0.997	1·000
1·0	0.368	0.736	0.92	0.981	0.996	0.999	1·000
1.1	0.333	0.699	0.9	0.974	0.995	0.999	1·000
1.2	0.301	0.663	0.88	0.966	0.992	0.998	1·000
1.3	0.273	0.627	0.857	0.957	0.989	0.998	1·000
1.4	0.247	0.592	0.833	0.946	0.986	0.997	0.999	1·000
1.5	0.223	0.558	0.809	0.934	0.981	0.996	0.999	1·000
1.6	0.202	0.525	0.783	0.921	0.976	0.994	0.999	1·000
1.7	0.183	0.493	0.757	0.907	0.97	0.992	0.998	1·000
1.8	0.165	0.463	0.731	0.891	0.964	0.99	0.997	0.999	1·000
1.9	0.15	0.434	0.704	0.875	0.956	0.987	0.997	0.999	1·000
2·0	0.135	0.406	0.677	0.857	0.947	0.983	0.995	0.999	1·000
2.2	0.111	0.355	0.623	0.819	0.928	0.975	0.993	0.998	1·000
2.4	0.091	0.308	0.57	0.779	0.904	0.964	0.988	0.997	0.999	1·000
2.6	0.074	0.267	0.518	0.736	0.877	0.951	0.983	0.995	0.999	1·000
2.8	0.061	0.231	0.47	0.692	0.848	0.935	0.976	0.992	0.998	0.999
µ\ x	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
3·0	0.05	0.199	0.423	0.647	0.815	0.916	0.966	0.988	0.996	0.999	1·000
3.2	0.041	0.171	0.33	0.603	0.781	0.895	0.955	0.983	0.994	0.998	1·000
3.4	0.033	0.147	0.34	0.558	0.744	0.871	0.942	0.977	0.992	0.997	0.999	1·000
3.6	0.027	0.126	0.313	0.515	0.706	0.844	0.927	0.969	0.988	0.996	0.999	1·000
3.8	0.022	0.107	0.259	0.474	0.668	0.816	0.909	0.96	0.984	0.994	0.998	0.999	1·000
4·0	0.018	0.092	0.238	0.433	0.629	0.785	0.889	0.949	0.979	0.992	0.997	0.999	1·000
4.2	0.015	0.078	0.21	0.395	0.59	0.753	0.868	0.936	0.972	0.989	996	0.999	1·000
4.4	0.012	0.066	0.135	0.359	0.551	0.72	0.844	0.921	0.964	0.985	0.994	0.998	0.999	1·000
4.6	0.01	0.056	0.163	0.326	0.513	0.686	0.818	0.905	0.955	0.98	0.992	0.997	0.999	1·000
4.8	0.008	0.048	0.143	0.294	0.476	0.651	0.791	0.887	0.944	0.975	0.99	0.996	0.999	1·000
5·0	0.007	0.04	0.125	0.265	0.441	0.616	0.762	0.867	0.932	0.968	0.986	0.995	0.998	0.999	1·000
5.2	0.006	0.034	0.109	0.238	0.406	0.581	0.732	0.845	0.918	0.96	0.982	0.993	0.997	0.999	1·000
5.4	0.005	0.029	0.095	0.213	0.373	0.546	0.702	0.822	0.903	0.951	0.978	0.99	0.996	0.999	1·000
5.6	0.004	0.024	0.032	0.191	0.342	0.512	0.67	0.797	0.886	0.941	0.972	0.988	0.995	0.998	0.999	1·000
5.8	0.003	0.021	0.072	0.17	0.313	0.478	0.638	0.771	0.867	0.929	0.965	0.984	0.993	0.997	0.999	1·000
6·0	0.003	0.017	0.062	0.151	0.285	0.446	0.606	0.744	0.847	0.916	0.957	0.98	0.991	0.996	0.999	0.999	1·000
6.2	0.002	0.015	0.054	0.134	0.259	0.414	0.574	0.716	0.826	0.902	0.949	0.975	0.989	0.995	0.998	0.999	1·000
6.4	0.002	0.012	0.046	0.119	0.235	0.384	0.542	0.687	0.803	0.886	0.939	0.969	0.986	0.994	0.997	0.999	1·000
6.6	0.001	0.01	0.04	0.105	0.213	0.355	0.511	0.658	0.78	0.869	0.927	0.963	0.982	0.992	0.997	0.999	0.999	1·000
6.8	0.001	0.009	0.034	0.093	0.192	0.327	0.48	0.628	0.755	0.85	0.915	0.955	0.978	0.99	0.996	0.998	0.999	1·000
7·0	0.001	0.007	0.03	0.082	0.173	0.301	0.45	0.599	0.729	0.83	0.901	0.947	0.973	0.987	0.994	0.998	0.999	1·000
7.2	0.001	0.006	0.025	0.072	0.156	0.276	0.42	0.569	0.703	0.81	0.887	0.937	0.967	0.984	0.993	0.997	0.999	0.999	1·000
7.4	0.001	0.005	0.022	0.063	0.14	0.253	0.392	0.539	0.676	0.788	0.871	0.926	0.961	0.98	0.991	0.996	0.998	0.999	1·000
7.6	0.001	0.004	0.019	0.055	0.125	0.231	0.365	0.51	0.648	0.765	0.854	0.915	0.954	0.976	0.989	0.995	0.998	0.999	1·000
7.8	·000	0.004	0.016	0.048	0.112	0.21	0.338	0.481	0.62	0.741	0.835	0.902	0.945	0.971	0.986	0.993	0.997	0.999	1·000
8·0	·000	0.003	0.014	0.042	0.1	0.191	0.313	0.453	0.593	0.717	0.816	0.888	0.936	0.966	0.983	0.992	0.996	0.998	0.999	1·000
8.2	·000	0.003	0.012	0.037	0.089	0.174	0.29	0.425	0.566	0.692	0.796	0.873	0.926	0.96	0.979	0.99	0.995	0.998	0.999	1·000
8.4	·000	0.002	0.01	0.032	0.079	0.157	0.267	0.4	0.537	0.666	0.774	0.857	0.915	0.952	0.975	0.987	0.994	0.997	0.999	1·000
8.6	·000	0.002	0.009	0.03	0.07	0.142	0.246	0.373	0.509	0.64	0.752	0.849	0.909	0.949	0.973	0.986	0.993	0.997	0.999	1·000
8.8	·000	0.002	0.007	0.024	0.062	0.128	0.226	0.348	0.482	0.614	0.729	0.822	0.889	0.935	0.964	0.981	0.99	0.995	0.998	0.999	1·000
9·0	·000	0.001	0.006	0.021	0.055	0.116	0.207	0.324	0.456	0.587	0.706	0.803	0.876	0.926	0.959	0.978	0.989	0.995	0.998	0.999	1·000
9.5	·000	0.001	0.004	0.015	0.015	0.04	0.089	0.165	0.269	0.392	0.522	0.645	0.752	0.836	0.94	0.967	0.982	0.991	0.996	0.998	0.999

provides these values for a mean of 2.0:

Number of Failures	Cmulative Probability
0	0.135
1	0.406
2	0.677
3	0.857
4	0.947
5	0.983
⁞	⁞

For the optimal stoking level , the service level must usully be rounded up to a feasible stocking level. Hence, you Know that the serice level must have been between. 406 and .677 in order to make two units the optinal level. By setting the service level lequal first to .406 and then to .677, you can establish bounds on the possible range of shortage costs.

[latex]\frac{C_s}{C_s + \$ 500 }=.406[/latex], so [latex]C_s = .406( \$ 500 +C_s)[/latex]

Solving, you find [latex]C_s= \$ 341.75[/latex].

Similarly,

[latex]\frac{C_s}{C_s + \$ 500 }=.677 [/latex] so [latex]C_s = .677( \$ 500 +C_s)[/latex]

Solving, you find [latex]C_s = \$ 1,047.99[/latex]. Hence, the apparent range of shortage cost is $341.75 to $ 1,047.99.

Question 13.P.9: Single-period. This type of problem can be recognized when a...

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